Document [original]

Universit¨at Paderborn

Fakult¨at f¨ur Elektrotechnik, Informatik und Mathematik

Dissertation

Realisability and Localisation

Realisierbarkeit und Lokalisierung

Birgit Huber

2007

REALISABILITY AND LOCALISATION

BIRGIT HUBER

Abstract. Let Abe a differential graded algebra with cohomology ring H∗A. A gra-

ded module over H∗Ais called realisable if it is (up to direct summands) of the form

H∗Mfor some differential graded A-module M. Benson, Krause and Schwede have

stated a local and a global obstruction for realisability. The global obstruction is given

by the Hochschild class determined by the secondary multiplication of the A∞-algebra

structure of H∗A.

In this thesis we mainly consider differential graded algebras Awith graded-com-

mutative cohomology ring. We show that a finitely presented graded H∗A-module X

is realisable if and only if its p-localisation Xpis realisable for all graded prime ideals

pof H∗A.

In order to obtain such a local-global principle also for the global obstruction, we

define the localisation of a differential graded algebra Aat a graded prime pof H∗A,

denoted by Ap, and show the existence of a morphism of differential graded algebras

inducing the canonical map H∗A→(H∗A)pin cohomology. The latter result actually

holds in a much more general setting: we prove that every smashing localisation on

the derived category of a differential graded algebra is induced by a morphism of

differential graded algebras.

Finally we discuss the relation between realisability of modules over the group

cohomology ring and the Tate cohomology ring.

Zusammenfassung. Sei Aeine differenziell graduierte Algebra mit Kohomologiering

H∗A. Ein graduierter H∗A-Modul heißt realisierbar, falls man ihn (bis auf direkte

Summanden) mit einem H∗A-Modul von der Form H∗Midentifizieren kann, wobei

Mein differenziell graduierter A-Modul ist. Benson, Krause und Schwede haben ein lo-

kales und ein globales Hindernis f¨

ur Realisierbarkeit angegeben. Das globale Hindernis

ist durch eine Hochschild Klasse gegeben, welche durch die sekund¨

are Mulitplikation

der A∞-Algebra-Struktur von H∗Abestimmt ist.

In dieser Doktorarbeit betrachten wir haupts¨

achlich differenziell graduierte Alge-

bren Amit graduiert-kommutativen Kohomologieringen. Wir zeigen, dass ein endlich

pr¨

asentierter graduierter H∗A-Modul Xgenau dann realisierbar ist, wenn dessen p-

Lokalisierung Xpf¨

ur alle graduierten Primideale pvon H∗Arealisierbar ist.

Um ein solches Lokal-Global Prinzip auch f¨

ur globale Realisierbarkeit zu formu-

lieren, definieren wir die Lokalisierung einer differenziell graduierten Algebra Aan

einem Primideal pvon H∗Aund bezeichnen sie mit Ap. Wir zeigen die Existenz ei-

nes Morphismus von differenziell graduierten Algebren, der in der Kohomologie die

kanonische Abbildung H∗A→(H∗A)pinduziert. Letzteres Resultat beweisen wir in

einem wesentlich allgemeineren Kontext: Wir zeigen, dass jede mit direkten Summen

kommutierende Lokalisierung der derivierten Kategorie einer differenziell graduierten

Algebra von einem Morphismus von differenziell graduierten Algebren induziert ist.

Abschließend diskutieren wir den Zusammenhang von Realisierbarkeit von Moduln

uber dem Gruppen-Kohomologiering und dem Tate-Kohomologiering.

2 BIRGIT HUBER

Contents

Danksagung 4

1. Introduction 6

Organisation 10

Notations and conventions 11

2. Triangulated categories 12

3. Graded rings and modules 15

3.1. Graded-commutative rings 17

4. Group and Tate cohomology rings 19

4.1. Group cohomology rings 20

4.2. Tate cohomology rings 21

5. The Hochschild Cohomology of a graded ring 22

5.1. Functoriality 23

5.2. The graded Bar resolution 24

5.3. Ring structure 24

5.4. The cup product pairing 26

6. Differential graded algebras and their derived categories 27

6.1. Differential graded algebras and modules 27

6.2. K(A) as stable category of a Frobenius category 29

6.3. Homotopically projective and homotopically injective dg modules 29

6.4. Derived functors 31

6.5. Cofibrant differential graded algebras 32

7. A∞-algebras 33

8. Localisation in triangulated categories 37

8.1. Categories of fractions 37

8.2. Localisation functors 38

8.3. Quotient categories 38

8.4. Localisation sequences 39

8.5. Recollements 40

8.6. Cohomological localisation 41

9. Realising smashing localisations by morphisms of dg algebras 44

9.1. Construction of a dg algebra morphism 45

9.2. The p-localisation of a dg algebra 51

10. Realisability 54

10.1. A local obstruction for realisability 54

10.2. Realisability and dg algebras 56

10.3. A global obstruction for realisability 57

11. Realisability and p-localisation 57

11.1. A motivation for p-localisation 58

11.2. Realisability is a local property 59

12. Localising the global obstruction 62

12.1. A map of Hochschild cohomology rings 63

12.2. Local-global principle for the global obstruction 65

13. Comparing realisability over group and Tate cohomology 68

13.1. Local realisability 68

13.2. Examples for the global obstruction 69

REALISABILITY AND LOCALISATION 3

13.3. Lifting H∗(G, k)→ˆ

H∗(G, k) to a morphism of dg algebras 74

13.4. Relating the global obstructions of H∗(G, k) and ˆ

H∗(G, k) 76

References 78

4 BIRGIT HUBER

Danksagung. An erster Stelle m¨ochte ich mich ganz besonders herzlich bei meinem

Dissertationsbetreuer Henning Krause f¨ur die Vergabe dieses Themas, die engagierte

Betreuung dieser Doktorarbeit und viele wichtige Anregungen f¨ur meine Arbeit be-

danken. Besonders danken m¨ochte ich ihm auch f¨ur die Unterst¨utzung außerhalb der

Mathematik, insbesondere f¨ur seinen großen Einsatz, die Finanzierung meiner Arbeit

stets zu sichern. Erw¨ahnen m¨ochte ich auch meine großartigen Dienstreisen zu Kon-

ferenzen oder Forschungsaufenthalten, die dank seines Einsatzes vom mathematischen

Institut oder der Forschungskommision der Universit¨at Paderborn finanziert wurden.

W¨ahrend eines dreimonatigen Forschungsaufenthalt an der NTNU Trondheim in Nor-

wegen wurde meine Arbeit von Øyvind Solberg betreut, und auch ihm m¨ochte ich f¨ur

sein großes Engagement und seine Gastfreundschaft sehr herzlich danken.

Besonders dankbar bin ich auch Dave Benson f¨ur eine Einladung zu einem einw¨ochigen

Forschungsaufenthalt nach Aberdeen. Ich m¨ochte ihm sehr herzlich f¨ur die vielen und

langen Diskussionen danken, in denen ich sehr viel gelernt habe und durch die ich die

Kapitel 4 and 13 ¨uber Gruppenkohomologie wesentlich verbessern konnte.

Mein sehr herzlicher Dank gilt auch Bernhard Keller, der sich bei jeder Konferenz, auf

der wir uns trafen, Zeit genommen hat, meine Fragen mit großer Geduld und Sorgfalt zu

beantworten. Dies hat entscheidend zur Entstehung des neunten Kapitels beigetragen.

Auch bei Ragnar-Olaf Buchweitz m¨ochte ich mich herzlich f¨ur Diskussionen w¨ahrend

seines Gastaufenthalts in Paderborn bedanken. Hierbei habe ich viel ¨uber Kommutative

Algebra gelernt und Anregungen erhalten, die mich bei dem Beweis von Ergebnissen in

Abschnitt 12.2 einen großen Schritt weiter gebracht haben.

F¨ur anregende mathematische Diskussionen m¨ochte ich mich auch herzlich bei Helmut

Lenzing und bei Dirk Kussin bedanken.

Herzlich danken m¨ochte ich Wolfgang Zimmermann f¨ur die Einf¨uhrung in das Ge-

biet der Algebra und Darstellungstheorie, die ich von ihm an der Universit¨at M¨unchen

erhalten habe.

Ich m¨ochte allen Mitgliedern und ehemaligen Mitgliedern der Paderborner Darstel-

lungstheorie Arbeitsgruppe und allen anderen wissenschaftlichen und nicht-wissenschaft-

lichen Mitarbeitern f¨ur die angenehme und freundliche Atmosph¨are am Mathematischen

Institut der Universit¨at Paderborn danken. Besonders danke ich in diesem Zusammen-

hang auch Kristian Br¨uning und Karsten Schmidt f¨ur die nette Atmosph¨are in unserem

B¨uro, viele mathematische Diskussionen und die ”L

EX-Unterst¨utzung“.

Herzlich danken m¨ochte ich auch Jes´us Soto und Stefan Wolf f¨ur das Korrekturlesen

meines Manuskripts.

Ich danke der Konrad-Adenauer-Stiftung f¨ur ein Graduiertenstipendium ¨uber 2,5

Jahre und besonders f¨ur die Finanzierung des Flugs zu einer Konferenz nach Mexiko.

W¨ahrend meines dreimonatigem Aufenthalts in Trondheim wurde ich durch ein Marie-

Curie-Stipendium finanziert, und ich m¨ochte daf¨ur dem Liegrits-Netzwerk und Idun

Reiten sehr herzlich danken.

Besonders danken m¨ochte ich auch der Universit¨at Paderborn f¨ur ein sechsmonatiges

Abschlussstipendium, welches mir erm¨oglicht hat, meine Doktorarbeit ohne finanzielle

Sorgen in Ruhe abzuschließen.

Sehr herzlich danke ich Claus Ringel f¨ur die Einladung zu einem zweimonatigen Gast-

aufenthalt am Sonderforschungsbereich ”Spektrale Strukturen und Topologische Metho-

den in der Mathematik“ (SFB 701) an der Universit¨at Bielefeld.

Last but not least m¨ochte ich mich bei meinen Eltern und bei Mart´ın Jim´enez f¨ur die

großartige und liebevolle Unterst¨utzung außerhalb der Mathematik bedanken.

REALISABILITY AND LOCALISATION 5

Ich danke den Gutachtern

Prof. Dave Benson

University of Aberdeen

Prof. Dr. Henning Krause

Universit¨at Paderborn

Prof. Dr. Alexander Zimmermann

Universit´e de Picardie, Amiens

6 BIRGIT HUBER

1. Introduction

In this thesis we connect two algebraic concepts which seem unrelated at first sight:

realisability and localisation. Using some advanced methods from Homological Algebra,

we establish a local-global principle for realisability. Before we discuss our main results

in detail, let us first explain the concepts we deal with.

The starting point is a cohomology theory which assigns to a mathematical object X

its cohomology group H∗X. Such cohomology theories arise for example in Algebraic

Topology, Algebraic Geometry, or in Representation Theory. Usually there exists some

commutative cohomology ring Esuch that H∗Xis naturally an E-module. Then an

E-module is realisable if it is up to isomorphism of the form H∗Xfor some object X.

The assignment described above can be expressed in the language of categories and

functors: If H∗:C → D is a functor between categories Cand D, then realisability is

concerned with deciding whether an object D∈ D is isomorphic to an object in the

image of the functor H∗.

In some specific representation theoretic context, Benson, Krause, and Schwede [5]

established a criterion for realisability. They investigated for a finite group Gand a

field kthe Tate cohomology functor

H∗(G, −): Mod kG →Modgr ˆ

H∗(G, k)

from the stable module category Mod kG into the category of Z-graded modules over

the Tate cohomology ring ˆ

H∗(G, k). In this setting, realisability deals with deciding

whether a graded ˆ

H∗(G, k)-module Xcan be written as ˆ

H∗(G, M) for some module M

over the group algebra kG.

The stable module category Mod kG has some additional structure: it is a triangulated

category. The functor ˆ

H∗(G, −) commutes both with arbitrary direct sums and with

arbitrary products.

More generally, Benson, Krause and Schwede [5] consider a compactly generated

triangulated category Tadmitting arbitrary direct sums, and a cohomological functor

H∗:T → Modgr E

into the category of Z-graded modules over a cohomology ring Ewhich preserves arbi-

trary direct sums and products.

For an arbitrary Z-graded E-module X, they have given a local obstruction

κ(X)∈Ext3,−1

E(X, X)

which is trivial if and only if Xis a direct summand of H∗Mfor some object M∈ T .

Moreover, Benson, Krause and Schwede [5] show that if there exists an infinite se-

quence of obstructions

κn(X)∈Extn,2−n

E(X, X), n ≥3,

where the class κn(X) is defined provided that the previous one κn−1(X) vanishes,

then it even holds X∼

=H∗M. In this sequence of obstructions all but the first one

depend on choices. Only κ3(X) is uniquely determined and actually, it equals the local

obstruction κ(X). It is remarkable that, despite of the necessity of an infinite sequence

of obstructions to decide if X∼

=H∗M, the first obstruction already tells whether Xis

a direct summand of H∗M.

REALISABILITY AND LOCALISATION 7

Since we mainly deal with the latter question, we call a Z-graded E-module Xreali-

sable if it is a direct summand of H∗Mfor some M∈ T . If X∼

=H∗M, then we refer

to a strictly realisable module.

The triangulated categories for which realisability is particularly interesting arise as

derived categories of differential graded algebras, or shortly, dg algebras. Such algebras

are complexes with an additional multiplicative structure. They have their origin in

Algebraic Topology [14] and encode topological invariants. Derived categories of dg

algebras were first studied systematically by Bernhard Keller [27].

If Ais a dg algebra, then realisability is concerned with deciding whether a graded

module over the cohomology ring H∗Ais (up to direct summands) isomorphic to a

cohomology module H∗M, where Mis a dg A-module. The functor in question is

H∗:D(A)→Modgr H∗A,

where D(A) denotes the derived category of the dg algebra A. Benson, Krause and

Schwede [5] show that this setting even admits a global obstruction for realisability. For

this purpose, they use a result of Kadeishvili [26] saying that H∗Aadmits an A∞-algebra

structure. The global obstruction arises as the Hochschild class µA∈HH3,−1(H∗A)

determined by the secondary multiplication mH∗A

3of H∗A: if the canonical class µAis

trivial, then all Z-graded H∗A-modules are realisable [5].

The other main concept we consider is localisation. This is an algebraic concept which

has its origin in Geometry. In Commutative Algebra, the localisation of a commutative

ring Rby a multiplicatively closed subset Sof Ris a uniquely determined ring of

fractions S−1Rwith the property that each s∈Sis made invertible in S−1R. Similarly,

one defines a module S−1Mwhich is a module over S−1R. One considers in particular

multiplicatively closed subsets S⊆Rwhich are the complement of a prime ideal pof

R. The ring of fractions is then denoted by Rpand the module S−1Mby Mp. The

ring of fractions Rpis a local ring, and many results on commutative rings or modules

over commutative rings can be proven more easily under the assumption that the ring

in question is local. The local-global principle says that an assertion holds if and only

it holds in localisation at every prime ideal. It is a classical principle of Commutative

Algebra.

Many rings which arise in Representation Theory of Groups or Algebraic Topology are

not commutative, but still, their elements commute up to a sign which depends on the

degree of the elements. Therefore these rings are called graded-commutative. Examples

are the group and Tate cohomology ring of a group or the singular cohomology ring

H∗Xof a topological space X. These examples arise as cohomology of a dg algebra.

Also many other dg algebras do not have a commutative, but still a graded-commutative

cohomology ring.

Although these rings are not strictly commutative, still many results from Commuta-

tive Algebra can also be proven in this more general setting. In particular, localisation

at prime ideals can be done similarly as in classical Commutative Algebra. This is

folklore knowledge for some experts, but there seems to be no published account. For

this reason, we provide some material on rings of fractions in the graded-commutative

setting in Section 3.1.

We will also consider localisation of triangulated categories. In the 1960s, Gabriel and

Zisman [20] introduced the Calculus of Fractions for arbitrary categories, generalising

the classical localisation of modules. This was used by Verdier in his th`ese [54] to

8 BIRGIT HUBER

study localisation of triangulated categories and in particular, to construct the Verdier

quotient.

In this thesis we show that there is a strong relation between realisability and locali-

sation. We prove relations in several settings.

In Chapter 11 we consider differential graded algebras Ahaving a graded-commutative

cohomology ring H∗A. We show that if a graded H∗A-module Xis realisable, then so

is Xpfor every graded prime ideal of H∗A. Our main result of Chapter 11 is

Theorem 1.1 (Local-global principle).Let Abe a dg algebra over a commutative ring

such that H∗Ais graded-commutative and coherent. The following conditions are equi-

valent for a finitely presented, graded H∗A-module X:

(1) Xis realisable.

(2) Xpis realisable for all graded prime ideals pof H∗A.

(3) Xmis realisable for all graded maximal ideals mof H∗A.

We will prove such a local-global principle also for global realisability in Chapter 12.

Before we give a precise formulation of this result, let us first explain the contents of

Chapter 9.

The results stated in Chapter 9 are joint work with K. Br¨uning [8] and also deal with

a realisability problem, however in a different setting. We consider smashing localisation

functors on derived categories of dg algebras, that is, localisation functors of triangulated

categories which commute with arbitrary direct sums. We show that every smashing

localisation on the derived category of a dg algebra can be realised by a morphism of

dg algebras. More precisely, we will prove

Theorem 1.2 (joint work with K. Br¨uning).Let Abe a dg algebra over a commutative

ring and L:D(A)→ D(A)a smashing localisation. Then there exists a dg algebra AL

with the property that D(AL)≃ D(A)/Ker L, and the map

D(A)(A, A)∗→ D(A)(LA, LA)∗, f 7→ L(f),

is induced by a zigzag of dg algebra maps

A∼

←− A0ϕ

−→ AL.

That is, there exists a dg algebra A0quasi-isomorphic to Aand a morphism of dg algebras

ϕ:A0→ALsuch that in cohomology, we have the commutative diagram

H∗A0

∼



H∗ϕ

))

D(A)(A, A)∗L//D(A)(LA, LA)∗.

Moreover, if Ais a cofibrant dg algebra, then there exists a morphism A→ALwhich

induces the algebra map D(A)(A, A)∗→ D(A)(LA, LA)∗in cohomology.

We focus in particular on the following special case: Let Abe a dg algebra with graded-

commutative cohomology ring, pa graded prime ideal of H∗Aand Lp:D(A)→ D(A)

the smashing localisation with the Lp-acyclic objects being those X∈ D(A) such that

(H∗X)p= 0. Then we denote

Ap=ALp

REALISABILITY AND LOCALISATION 9

and call the dg algebra Aplocalisation of Aat a prime pin cohomology. The cohomology

of Apsatisfies H∗(Ap)∼

=(H∗A)pas graded rings, and with this identification, the

canonical map H∗A→(H∗A)pis the cohomology of a zigzag of dg algebras

A∼

←− A0ϕ

−→ Ap.

In Chapter 12 we consider dg algebras Awith graded-commutative cohomology ring.

Our main result in this chapter is a local-global principle for global realisability. For this

purpose, we state a global obstruction for the p-local modules, i.e. those graded H∗A-

modules with the property that X∼

=Xp. Here we use the dg algebra Apconstructed

in Chapter 9: this obstruction arises from the A∞-structure of (H∗A)p∼

=H∗(Ap).

Actually, we show that the canonical Hochschild class µAp∈HH3,−1(H∗(Ap)), which

is a global obstruction for the graded H∗(Ap)-modules due to Benson, Krause and

Schwede [5], is also a global obstruction for the p-local H∗A-modules.

In order to relate the global obstruction µA∈HH3,−1(H∗A) for the H∗A-modules and

the global obstruction µAp∈HH3,−1(H∗Ap) for the p-local H∗A-modules, we construct

a map of Hochschild cohomology rings

Γ: HH∗,∗(H∗A)−→ HH∗,∗(H∗Ap)

which has the property Γ(µA) = µAp. This is the key to prove that also the global

obstruction behaves well under p-localisation:

Theorem 1.3 (Local-global principle).Let Abe a differential graded algebra over a

field ksuch that H∗Ais graded-commutative. Assume that the algebra H∗Aop ⊗k(H∗A)

is Noetherian. Then the following conditions are equivalent:

(1) µA∈HH3,−1(H∗A)is trivial.

(2) µAp∈HH3,−1(H∗Ap)is trivial for all graded prime ideals pof H∗A.

(3) µAm∈HH3,−1(H∗Am)is trivial for all graded maximal ideals mof H∗A.

In the last chapter of this thesis, we focus on realisability in the context of group

representation theory. We study the relation between realisability over the group coho-

mology ring H∗(G, k) and the Tate cohomology ring ˆ

H∗(G, k), where kis a field and G

a finite group. Note that H∗(G, k) can be viewed as a subalgebra of ˆ

H∗(G, k).

For the group cohomology ring, the appropriate realisability setting is given by the

functor

K(Inj kG)K(Inj kG)(ik,−)∗

//Modgr H∗(G, k),

where K(Inj kG) is the homotopy category of injective kG-modules.

The group cohomology ring H∗(G, k) has better properties than the Tate cohomology

ring ˆ

H∗(G, k) which, for instance, is not Noetherian in general. However, when it comes

to the source categories of realisability, the stable module category Mod kG is more

“handsome” than the homotopy category K(Inj kG). This is the reason why we are

interested in studying the relation of realisability over group and Tate cohomology.

The triangulated categories K(Inj kG) and Mod kG are related by a smashing locali-

sation

K(Inj kG)

Q//Mod kG

10 BIRGIT HUBER

and we are now concerned with finding a relation between realisability and this locali-

sation of triangulated categories.

We study realisability of fixed modules as well as global realisability. Note for the lat-

ter that both H∗(G, k) and ˆ

H∗(G, k) are the cohomology of a dg algebra and thus, they

admit an A∞-structure yielding global obstructions denoted by µG∈HH3,−1(H∗(G, k))

and ˆµG∈HH3,−1(ˆ

H∗(G, k)).

The canonical class ˆµGhas been computed for some groups Gby Benson, Krause and

Schwede [5], and by Langer [37]. We consider the same groups and compute the global

obstructions for the group cohomology rings. In many cases, the Hochschild classes

µG∈HH3,−1(H∗(G, k)) and ˆµG∈HH3,−1(ˆ

H∗(G, k)) turn out to behave surprisingly

similar. As a first explanation, we show that the algebra morphism H∗(G, k)→ˆ

H∗(G, k)

is the cohomology of a zigzag of dg algebra morphisms. Then we are ready to prove the

main result of this chapter, which is, in some parts, also an application of our results on

Hochschild cohomology from Chapter 12:

Theorem 1.4. Let Gbe finite group, ka field of characteristic p > 0and assume that

pdivides the order of G. If the Hochschild class ˆµG∈HH3,−1(ˆ

H∗(G, k)) is trivial, then

so is the Hochschild class µG∈HH3,−1(H∗(G, k)). If the p-rank of the group Gequals

one, then ˆµGis trivial if and only if µGis trivial.

In general, the last statement is not true for groups with p-rank at least two, as we

show by giving a counter-example.

Organisation. Our main results, as stated above, can be found in the Chapters 9, 11,

12 and 13. At the end of each of these chapters, we point out related open questions.

In the second chapter we recall facts about triangulated categories and introduce

briefly those triangulated categories that we deal with in this thesis.

A short review on graded rings and modules can be found in the third chapter. We

focus on localisation of graded-commutative rings in Section 3.1.

In Chapter 4 we introduce group and Tate cohomology rings and state their basic

properties.

Hochschild cohomology of graded rings is discussed in Chapter 5. In particular, we

study the multiplicative structure of the Hochschild cohomology ring HH∗,∗(Λ) of a

graded algebra Λ. We show that for elements ζ∈HHm,i(Λ) and η∈HHn,j(Λ), we have

the commutativity relation ζ·η= (−1)mn(−1)ijη·ζ, where the multiplication is given

by the Yoneda or the cup product. Hochschild cohomology rings of non-graded rings

are well-known to be graded-commutative, but we do not know of a published source of

this more general result. We will apply it to prove some of our results in Chapter 12.

In Chapter 6 we introduce differential graded algebras and discuss properties of their

derived categories.

A short introduction to A∞-algebas is given in Chapter 7. In particular, for a dg

algebra A, we present Kadeishvili’s construction [26] of the secondary multiplication

mH∗A

3:H∗A⊗3→H∗Aof the A∞-algebra H∗A.

Chapter 8 is about localisation in triangulated categories and contains important re-

quisites for our main results. After a short discussion of the Calculus of Fractions for

arbitrary categories due to Gabriel and Zisman [20] we focus on triangulated categories.

In particular, we introduce the Verdier quotient [54] and consider localisation sequences

REALISABILITY AND LOCALISATION 11

of triangulated categories. In Section 8.6 we state a theorem of Krause [36] on cohomo-

logical localisation and prove some results together with K. Br¨uning [8] which apply in

particular to cohomological p-localisation.

In Chapter 10 we give a short review of the results of Benson, Krause and Schwede [5]

and introduce their local and global obstruction for realisability, as discussed above. We

focus especially on realisability in the setting of dg algebras.

Notations and conventions. Unless otherwise stated, modules are always considered

to be right modules. In particular, we denote by Mod Rthe category of right R-modules

and by mod Rthe category of finitely generated right R-modules over a ring R. If R

is self-injective, then Mod Rresp. mod Rdenote the stable module categories of Mod R

resp. mod R.

When we talk about graded rings and modules, we always mean Z-graded rings and

modules. If Ris a graded ring and it is clear from the context that we mean graded R-

modules, then we sometimes speak of R-modules. We denote by Modgr Rthe category

of graded right R-modules, where the morphisms are the homogeneous graded R-linear

maps of degree zero. By Homi

R(M, N) we denote the homogeneous graded R-linear

maps M→Nrising the degree by i∈Z, and we write HomR(M, N) for Hom0

R(M, N).

Moreover, we set Hom∗

R(M, N) = `n∈ZHomn

R(M, N).

Cohomology of graded modules over a graded ring Ris bigraded; the first index gives

the cohomological degree and the second, internal degree arises from the grading of R.

For example, for i≥0 and j∈Zwe have

Exti,j

R(M, N) = Exti

R(M, N[j]),

where [j] denotes the j-fold shift on Modgr R.

In particular, Hochschild cohomology of a graded algebra Λ, denoted by HH∗,∗(Λ), is

bigraded. Note that only the first grading is changed by the differential.

For the homotopy category of complexes in an additive category Awe write K(A),

and the derived category of an abelian category Ais denoted by D(A). For the derived

category D(Mod R) of a ring Rwe write shortly D(R).

All dg algebras considered in this thesis are supposed to have a differential of de-

gree +1. So the homology of these dg algebras is, in fact, cohomology, and throughout

this thesis, we speak of cohomology. If Ais a dg algebra, then we denote its cohomology

ring by H∗A, and the homotopy resp. derived category of Awill be denoted by K(A)

resp. D(A).

The symbol ∼indicates a quasi-isomorphism and the symbol ∼

=is used for isomor-

phisms of objects in categories. Equivalences of categories are indicated by ≃.

The set of morphisms X→Yin a category Cis denoted by C(X, Y ) or HomC(X, Y ).

If Tis a triangulated category, then we denote its suspension functor by Σ or [1]. For

i∈Z, we write T(X, Y )ifor T(X, ΣiY) and we denote

T(X, Y )∗=a

i∈Z

T(X, Y )i.

The composition of maps f:A→Band g:B→Cis denoted by g◦for gf; similarly

for functors.

12 BIRGIT HUBER

2. Triangulated categories

Triangulated categories were introduced independently by Verdier in his th`ese [54],

and in Algebraic Topology by Puppe [44].

The purpose of this chapter is to state results on triangulated categories which we

will use later on. In particular, we recall examples of triangulated categories and fix

notation. For the definition of a triangulated category we refer to the book of Neeman

[43] or Krause’s notes [34].

Let Tbe a triangulated category. We denote the suspension functor by [1]: T → T

or Σ: T → T .A non-empty full subcategory Sis a triangulated subcategory if

(i) Sis closed under shifts, i.e. X∈ S if and only if X[1] ∈ S.

(ii) Sis closed under triangles, i.e. if in the exact triangle X→Y→Z→X[1] two

objects from {X, Y, Z}belong to S, then also the third.

A triangulated subcategory Sis called thick if it is closed under direct factors, that is,

a decomposition X=X0qX00 for X∈ S implies X0∈ S.

A triangulated subcategory Sadmitting arbitrary direct sums is called localising. If

Sis localising, then it is already a thick subcategory ([43, Rem. 3.2.7]).

Let Nbe a class of objects in T. The triangulated subcategory generated by Nis

the smallest full triangulated subcategory which contains N. We refer to [34, Ch. 2.8]

for an explicit construction.

If Tadmits arbitrary direct sums, then the triangulated subcategory generated by

Nis the smallest full triangulated subcategory which contains Nand is closed under

taking arbitrary direct sums. We denote this category Loc(N) since it is the smallest

localising subcategory that contains N.

An object X∈ T is called compact if the covariant Hom functor

T(X, −): T → Ab

into the category Ab of abelian groups commutes with arbitrary direct sums.

We provide a useful criterion to prove that a category is generated by compact objects.

Lemma 2.1. [50, Lemma 2.2.1] Let Tbe a triangulated category with arbitrary direct

sums and Ma set of compact objects. The following conditions are equivalent:

(1) Tis generated by M, i.e. T= Loc(M).

(2) An object X∈ T is trivial if and only if there are no graded maps from Mto

X, i.e. T(M, X[n]) = 0 for all M∈ M and n∈Z.

An exact functor T → S between triangulated categories is a functor preserving exact

triangles and shifts. More precisely, it is a pair (F, η), consisting of a functor F:T → S

and a natural isomorphism η:F◦[1]T

∼

−→ [1]S◦Fsuch that for every exact triangle

−→ Yg

−→ Zh

−→ X[1], the triangle

FX Ff

−−→ FY Fg

−−→ F Z ηX◦F h

−−−−→ (FX)[1]

is exact in S. The following proposition is useful to check whether an exact functor is

an equivalence. It is a version of ‘Beilinson’s Lemma’ [1].

Proposition 2.2. [48, Prop. 3.10] Let T,Sbe triangulated categories admitting arbitrary

direct sums and F:T → S an exact functor preserving arbitrary direct sums. Suppose

that Thas a compact generator Csuch that

REALISABILITY AND LOCALISATION 13

(1) FC is a compact generator of S, and

(2) the map

F:T(C, C[n]) → S(F C, FC[n])

is bijective for all n∈Z.

Then Fis an equivalence of triangulated categories.

Let Tbe a triangulated and Aan abelian category. A functor F:T → A is called co-

homological if it sends each exact triangle in Tto an exact sequence in A. In particular, if

X→Y→Z→X[1] is an exact triangle in T, then Fgives rise to an infinite exact

sequence

· · · → F(Y[−1]) →F(Z[−1]) →FX →FY →FZ →F(X[1]) →F(Y[1]) → · · ·

For a proof of the following well-known lemma, we refer to [34, Ch. 2.3].

Lemma 2.3. Let Tbe triangulated and Xin T. The representable functors

T(X, −): T → Ab and T(−, X): Top →Ab

are cohomological.

The following result is due to Neeman. It is a consequence of the Brown Repre-

sentability Theorem (see for example [34, Ch. 4.5]).

Proposition 2.4. [35, Prop. 3.3] Let F:S → T be a an exact functor between triangu-

lated categories, and suppose that Sis compactly generated.

(1) There is a right adjoint T → S if and only if Fpreserves arbitrary direct sums.

(2) There is a left adjoint T → S if and only if Fpreserves arbitrary products.

Example 2.5. We introduce briefly the triangulated categories which are considered in

this thesis, in particular to fix notation. For detailed definitions, we refer to [34].

(1) Let Abe an additive category and denote by C(A) the category of complexes in

A. The null-homotopic maps form an ideal in C(A) and the homotopy category K(A)

is the quotient of C(A) with respect to this ideal. Denote by Σ: K(A)→K(A) the

equivalence which takes a complex Xto its shifted complex ΣX, defined by

(ΣX)n=Xn+1 and dn

ΣX=−dn+1

Given a map α:X→Yof complexes, the mapping cone Cone(α) is the complex defined

in degree nby Xn+1 qYn, and endowed with the differential dn

Cone(α)=h−dn+1

αn+1 dn

Yi.It

fits into a mapping cone sequence

Xα

−→ Yβ

−→ Cone(α)γ

−→ ΣX,

given in degree nby

Xnαn

−−→ Ynh0

id i

−−−→ Xn+1 qYn[−id 0 ]

−−−−−→ Xn+1.

K(A) is triangulated, with the exact triangles being those isomorphic to a mapping cone

sequence as defined above.

(2) The derived category D(A) of an abelian category Ais obtained from K(A) by

formally inverting all quasi-isomorphisms. D(A) is a triangulated category, where the

triangulated structure is induced by the one of K(A). More precisely, D(A) carries a

unique triangulated structure such that the canonical functor K(A)→D(A) is exact.

14 BIRGIT HUBER

(3) In Chapter 6 we introduce differential graded algebras. If Ais a differential

graded algebra, then the homotopy category K(A) and the derived category D(A) are

triangulated categories. This generalises the homotopy resp. derived category of a non-

graded algebra, viewed as differential graded algebra concentrated in degree zero.

(4) Let Abe an exact category in the sense of Quillen [45]. Thus Ais an additive

category with a distinguished class of sequences

0→Xα

−→ Yβ

−→ Z→0

which are called exact. The exact sequences satisfy a number of axioms. In particular,

the maps αand βin each exact sequence as above form a kernel-cokernel pair. That

is, αis a kernel of βand βis a cokernel of α. A map in Aarising as the kernel in

some exact sequence is called admissible mono, and a map arising as a cokernel is called

admissible epi. A full subcategory Bof Ais extension-closed if every exact sequence in

Abelongs to B, provided that its end terms belong to B.

Let Abe an exact category. An object P∈ A is called projective if the induced

map HomA(P, Y )→HomA(P, Z) is surjective for every admissible epi Y→Z. Dually,

an object Iis injective if the induced map HomA(Y, I)→HomA(X, I) is surjective for

every admissible mono X→Y. The category Ahas enough projectives if every object

Zadmits an admissible epi Y→Zwith Yprojective, and it has enough injectives

if every object Xadmits an admissible mono X→Ywith Yinjective. Finally, Ais

called a Frobenius category if Ahas enough projectives and enough injectives and if both

coincide.

The stable category of a Frobenius category Ais denoted by S(A) and defined to be

the quotient of Awith respect to the ideal Iof morphisms factoring through an injective

object. Thus

HomS(A)(X, Y ) = HomA(X, Y )/I(X, Y )

for all X, Y in A.

We choose for each X∈ A an exact sequence

0→X→I(X)→ΣX→0

such that I(X) is injective. The morphism X→I(X) is called injective hull. One

easily checks that the assignment X7→ ΣXdefines an equivalence on S(A). Every

exact sequence 0 →X→Y→Z→0 fits into a commutative diagram

0//Xα//Yβ//



Z//



0//X//I(X)//ΣX//0

such that I(X) is injective. The category S(A) carries a triangulated structure, with

the exact triangles being those isomorphic to a sequence of maps

Xα

−→ Yβ

−→ Zγ

−→ ΣX

as in the diagram above.

(4)(a) If Ais a finite dimensional self-injective algebra, then Mod Ais a Frobenius

category and obviously, Σ equals the first cosyzygy Ω−1. We denote the stable category

S(Mod A) by Mod Aand conclude that it has a triangulated structure.

REALISABILITY AND LOCALISATION 15

(4)(b) The homotopy category K(A) of an additive category Aidentifies with the

stable category of C(A), where the exact structure is induced by the degree-wise split

short exact sequences of complexes, see [34, Ch. 7.2].

(4)(c) Similarly, if Ais a differential graded algebra, then the homotopy category K(A)

is the stable category of a Frobenius category. We provide more details in Section 6.2.

3. Graded rings and modules

In this section we introduce graded rings and modules, and state some properties of

the category of graded modules. In particular, we fix the sign convention we will use

throughout this paper. Unless otherwise stated, we mean graded right modules when

we speak of graded modules.

AZ-graded ring is a ring Rtogether with a decomposition of abelian groups

R=a

i∈Z

such that RiRj⊆Ri+j.

AZ-graded module over a Z-graded ring Ris an R-module Mtogether with a de-

composition of abelian groups

M=a

i∈Z

satisfying MiRj⊆Mi+j.

AZ-graded algebra over some commutative ring kis a graded ring Λ which also has

a graded k-module structure that makes Λ into a k-algebra. Note that the operation of

kon Λ has degree zero.

Throughout this thesis, we will talk about graded rings,graded algebras and graded

modules and always refer to Z-graded rings, algebras and modules.

Let Mbe a graded module over a graded ring R. The elements m∈Miare called

homogeneous elements of degree i, and we denote the degree of mby |m|. For n∈Z,

the n-fold shifted graded module M[n] is given by M[n]i=Mn+i. We use the notation

Σnmwhen we view m∈Mn+ias an element in M[n]i.

If M, N are graded R-modules and n∈Z, then an R-linear map f:M→Nis called

homogeneous graded map or shortly, graded map of degree nif f(Mj)⊆Nj+nfor all

j∈Z. Note that fcan also be considered as a graded map M→N[n] of degree zero.

We denote by Homn

R(M, N) the set of all graded maps M→Nof degree n, and we

define

Hom∗

R(M, N) = a

n∈Z

Homn

R(M, N).

The graded R-modules form a category denoted by Modgr R. The morphisms are the

graded maps of degree zero and we denote

HomR(M, N) = Hom0

R(M, N).

A graded R-module B⊆Mis a graded submodule of Mif the inclusion map is a

morphism in Modgr R. In this case, the quotient M/B also carries a natural grading. If

f:M→Nis a morphism in Modgr R, then Ker f, Im fand Coker fare graded modules.

Moreover, one can show that Modgr Ris a Grothendieck category [40, Ch. 2.2].

The graded submodules of Rare called graded right ideals. An arbitrary right ideal

Iof Ris graded if and only if it is generated by homogeneous elements.

16 BIRGIT HUBER

A graded R-module is graded free if it is a direct sum of shifted copies of R, or

equivalently, if it has an R-basis consisting of homogeneous elements. Note that it is not

enough to assume that the module is graded and free as non-graded module: Considering

R=Z×Zas graded ring concentrated in degree 0, the module F=Z×Zendowed

with the grading F0=Z×0, F1= 0 ×Z, and Fi= 0 otherwise, is not graded free since

it cannot have an R-basis consisting of homogeneous elements.

A graded R-module is called graded projective if it is a projective object in the category

Modgr R, or equivalently, if it is a direct summand of a graded free R-module. Since

Modgr Ris a Grothendieck category, it has enough injective objects. Those are the

graded injective modules.

Every graded R-module Madmits a graded free presentation F1→F0→M→0,

i.e. F0and F1are graded free modules. If both F0and F1can be chosen to be finitely

generated, then Mis called finitely presented.

A graded ring is right Noetherian if it is right Noetherian as a ring, i.e. every (not

necessarily graded) ideal is finitely generated. In this case, every finitely generated

graded R-module is already finitely presented.

If Nis a graded R-module and i≥0, j ∈Z, then one defines

Exti,j

R(M, N) = Exti

R(M, N[j]).

An element of Exti,j

R(M, N) can be represented by an exact sequence of graded R-

modules

0→N[j]→Xi→ · · · → X1→M→0.

Assume now that Ris a graded algebra over some commutative ring k. For graded

modules Mand Nthe tensor product M⊗kNis a graded module, where the degree i

component is given by

(M⊗kN)i=a

p+q=i

Mp⊗kNq.

Let f:M→M0and g:N→N0be graded maps. Note that due to the Koszul sign

rule, in the tensor product f⊗gthere appears a sign:

(3.1) (f⊗g)(m⊗n) = (−1)|g|·|m|f(m)⊗g(n),

where m∈Mis homogeneous and n∈N.

One also needs to involve signs to define the opposite algebra:Rop is again a graded

algebra, with multiplication

(3.2) r·r0= (−1)|r|·|r0|r0r.

A graded right R-module can be viewed as graded left Rop-module by setting

r·m= (−1)|r||m|mr.

If Sis a graded k-algebra, then R⊗kSis a graded k-algebra with multiplication

(3.3) (r⊗s)(r0⊗s0) = (−1)|r0||s|(rr0⊗ss0).

A graded (R, S)-bimodule Mis simultaneously a graded left R-module and a graded

right S-module such that (rm)s=r(ms). The graded (R, S)-bimodules correspond to

the graded right modules over Rop ⊗kS.

REALISABILITY AND LOCALISATION 17

If Mis a graded (R, S)-bimodule, then M[t] is a graded (R, S)-bimodule by setting

(3.4) r·(Σtm)·s= (−1)t|r|Σt(rms).

Note that the shift functor M7→ M[1] for graded right modules does not involve

any extra sign: we have (Σm)·r= Σ(mr). However, the sign in (3.4) appears when

translating graded right R-modules into graded left modules over Rop.

The graded rings we are particularly interested in arise as graded endomorphism rings

of objects of triangulated categories.

Example 3.1. Let Tbe a triangulated category with arbitrary direct sums and sus-

pension functor Σ. For objects M, N ∈ T we write T(N, M)i=T(N, ΣiM). Then

T(N, N)∗=a

i∈Z

T(N, N)i

is a graded ring, called the graded endomorphism ring of N, and T(N, M)∗is a graded

T(N, N)∗-module by composition of graded maps.

3.1. Graded-commutative rings. A graded ring Ris called graded-commutative if

rs = (−1)|r||s|sr for all homogeneous elements r, s ∈R. Although such a ring is not

strictly commutative, many results about graded and commutative rings (which are

studied for example in [10]) can still be carried over. However, “Commutative Algebra

over graded-commutative rings” is rarely treated in literature. We provide definitions

and results about localisation of graded-commutative rings that we will need later on.

We like to thank Dave Benson for pointing out Remark 3.3 and Lemma 3.5.

3.1.1. Prime and maximal ideals. A graded right ideal mof a graded ring Ris called

graded maximal right ideal if m6=Rand moreover, for any graded right ideal asuch

that m⊆a⊆R, it holds a=mor a=R. If b6=Ris a graded right ideal of R, then

there exists a maximal right ideal mcontaining b. If Ris graded-commutative, a graded

(maximal) right ideal is a graded (maximal) ideal.

For an arbitrary ring R, a prime ideal p Ris an ideal such that for a, b ∈R, it

holds a∈por b∈pwhenever aRb ⊆p. If Ris graded-commutative, then this definition

simplifies as in the case of strictly commutative rings:

Definition 3.2. Let Rbe a graded-commutative ring. An ideal p Ris a prime ideal

if ab ∈pimplies that a∈por b∈p. The set of graded prime ideals p⊆Ris called the

graded spectrum of Rand denoted by Specgr(R).

Actually, the prime spectrum of a graded-commutative ring can be identified with the

prime spectrum of a graded, strictly commutative ring:

Remark 3.3. Let Rbe a graded-commutative ring, and denote by nthe ideal generated

by the homogeneous nilpotent elements. If xis a homogeneous element of odd degree, it

holds 2x2= 0, and thus, 2xis nilpotent. Hence x≡ −xmod n, and the factor ring R/n

is a graded, strictly commutative ring. Since nis contained in all graded prime ideals,

it follows that

Specgr(R) = Specgr(R/n).

Lemma 3.4. Let Rbe a graded-commutative ring and a⊆Ra graded ideal.

(1) ais prime if and only if R/ais a domain.

18 BIRGIT HUBER

(2) ais a graded maximal ideal if and only if every non-zero homogeneous element

of R/ais invertible.

Proof. (1) is trivial. For (2) note that all non-zero homogeneous elements of a graded-

commutative ring Tare invertible if and only if the only graded ideals contained in T

are (0) and T.

If Ris graded commutative, then all non-zero homogeneous elements being invertible

implies that Ris a domain. Thus every graded maximal ideal is prime.

3.1.2. Rings and modules of fractions. Let Rbe a (not necessarily graded) ring and Sa

multiplicative subset with 1 ∈S. We define the right ring of fractions of Rwith respect

to Sas a ring R[S−1] together with a ring homomorphism ρ:R→R[S−1] satisfying

(F1) ρ(s) is invertible for each s∈S.

(F2) Every element in R[S−1] has the form ρ(r)ρ(s)−1with s∈S.

(F3) ρ(r) = 0 if and only if there exists an element s∈Ssuch that rs = 0.

It is not immediately clear from these axioms that R[S−1] is uniquely determined, but

it is, in fact, the case. We refer to [52, Ch. II].

Let Sbe a multiplicatively closed subset of R. The ring R[S−1] exists if and only if

the following conditions, called right Ore conditions, are satisfied:

(O1) If s∈Sand r∈R, then there exist s0∈Sand r0∈Rsuch that sr0=rs0

(O2) If r∈Rand s∈Swith sr = 0, then there exists s0∈Ssuch that rs0= 0.

If (O1) and (O2) are satisfied, then

R[S−1] = R×S/ ∼,

where the equivalence relation ∼is given by

s∼r0

if and only if there exist u, v ∈Rsuch that ru =r0vand su =s0v.

If the analogous left Ore conditions are satisfied, then there exists the left ring of

fractions [S−1]R. Furthermore, if both R[S−1] and [S−1]Rexist, then they are isomor-

phic [52, II, Cor. 1.3].

If we now assume that Ris graded and Sa multiplicative subset of homogeneous

elements with 1 ∈S, then it suffices to check the Ore conditions on homogeneous

elements. Moreover, if R[S−1] exists, then it is a graded ring, where

deg r

s= deg(r)−deg(s)

for any homogeneous element r∈Rand s∈S. [40, Ch. 8.1]

If Ris graded-commutative, then the Ore conditions are trivially satisfied and the

right ring of fractions R[S−1] exists. One might want to define the equivalence relation

as for strictly commutative rings, but the transitivity fails for elements s∈Sof odd

degree. However, one easily checks

Lemma 3.5. Let Rbe a graded-commutative ring and Sis a multiplicative closed subset

of homogeneous elements of R. Let Sev ⊆Sthe subset of even-degree elements of S.

Then R[S−1]∼

=R[S−1

ev ]as graded rings and the equivalence relation simplifies into r

s∼r0

with r, r0∈R, s, s0∈Sev if and only if there exists t∈Sev such that rs0t=r0st.

REALISABILITY AND LOCALISATION 19

Consequently, we may define the localisation of a graded-commutative ring Rwith

respect to a multiplicative subset Sof homogeneous elements to be the ring

R[S−1

ev ]

with the equivalence relation used in the strictly commutative case. Then addition and

multiplication are also defined as in that well-known case. We write S−1Rfor R[S−1

ev ].

If Mis a graded R-module, we define MS, the localisation of Mwith respect to Sas

S−1M=R×Sev/∼,

with m

s∼m0

s0if and only if ms0t=m0st for some t∈Sev. Obviously, S−1Mis a graded

S−1R-module with the canonical structure and grading

deg m

s= deg(m)−deg(s),

where m∈Mis homogeneous and s∈Sev.

Note that S−1Ris flat as both left and right R-module, and that M⊗RS−1R∼

=S−1M

as graded S−1R-modules.

Let abe a graded ideal of Rand let Sbe the subset of homogeneous elements of R\a.

Then we set Ma=S−1M. Similarly as in classical Commutative Algebra, we have

Proposition 3.6 (Local-global principle).Let Mbe a graded R-module. The following

conditions are equivalent:

(1) M= 0.

(2) Mp= 0 for all graded prime ideals p.

(3) Mm= 0 for all graded maximal ideals m.

Proof. We only need to show that (3) implies (1). Let x∈Mbe any element. We

consider Ann(x)∗, the largest graded ideal contained in Ann(x) = {r∈R|rx = 0}.

Assuming that Ann(x)∗is a proper graded ideal of R, we obtain a graded maximal

ideal mwhich contains Ann(x)∗. Since Mm= 0, there exists a homogeneous element

sin R\msuch that sx = 0. So sis contained in Ann(x)∗and thus in m, which is a

contradiction. 

4. Group and Tate cohomology rings

Homology and Cohomology of groups has been considered since the 1940s. Inspired

by a work of Hopf [23] from 1941 in which he considers what today is called the second

homology group H2(G, Z) of a group G, Eilenberg and Mac Lane [18] started to study

systematically homology and cohomology of groups.

The Tate cohomology ring was invented by Tate, but these results were never pub-

lished by himself; the first published account is contained in the book of Cartan and

Eilenberg [15].

In the first section of this chapter we study group cohomology rings, and in the second

Tate cohomology rings.

We thank Dave Benson for many useful comments and pointing out references to the

author.

20 BIRGIT HUBER

4.1. Group cohomology rings. Let kbe a commutative ring and Ga finite group.

The ring kbecomes a kG-module by trivial action of G. This module is called the trivial

module and also denoted by k.

If Mis a kG-module and n≥0, then the n-th cohomology of Gwith coefficients in

Mis defined to be

Hn(G, M) = Extn

kG(k, M).

The Yoneda splice multiplication yields a k-bilinear, associative map (see [13, Sect. 6])

Hn(G, M)×Hl(G, k)→Hn+l(G, M),

defined as follows: If ζ∈Hn(G, M) is represented by

Eζ: 0 →M→X0→ · · · → Xnπ

−→ k→0

and η∈Hl(G, k) by

Eη: 0 →kι

−→ Y0→ · · · → Yl→k→0,

then the Yoneda splice product of ζand ηis the class which is represented by the exact

sequence obtained by splicing together Eζand Eη:

0//M//X0//· · · //Xn



ι◦π//Y0//· · · //Yl//k//0



We denote by K(Inj kG) the homotopy category of Inj kG, which is the full subcate-

gory Mod kG formed of the injective kG-modules. The category Inj kG is additive, and

it is closed under arbitrary direct sums, provided that kG is noetherian.

Write iM ∈K(Inj kG) for an injective resolution of a kG-module M. With the

well-known identification

Extm

kG(k, X)∼

=K(Inj kG)(ik, Σm(iX))

for any X∈Mod kG and m≥0,one can also form a k-bilinear, associative product by

composition of chain maps of injective resolutions:

Hn(G, M)×Hl(G, k)→Hn+l(G, M),(f, g)7→ Σl(g)◦f.

This product coincides with the Yoneda splice product (see [13, Sect. 6]).

With any of the two multiplications,

H∗(G, k) = a

n≥0

Hn(G, k)

is a graded ring (concentrated in non-negative degrees) and

H∗(G, M) = a

n≥0

Hn(G, M)

is a graded module over H∗(G, k).

Note also that H∗(G, k) is a graded-commutative ring [13, Cor. 6.9]. Due to Evens

and Venkov, it is Noetherian whenever kis.

Theorem 4.1 (Evens, Venkov).If kis Noetherian, then H∗(G, k)is a finitely generated

k-algebra.

REALISABILITY AND LOCALISATION 21

4.2. Tate cohomology rings. Let kbe a field of characteristic p > 0 and Gbe a finite

group such that pdivides the order of G. We denote by Mod kG the stable module

category of kG. The objects are the same as in the module category Mod kG, but the

morphisms are given by

HomkG(M, N) = HomkG(M, N)/I(M, N),

where I(M, N) denotes the morphisms factoring through an injective object. The cat-

egory Mod kG is triangulated with shift functor Ω−1, the first cosyzygy, see Exam-

ple 2.5(4)(a). An elementary proof for the fact that Mod kG is triangulated can be

found in Carlson’s book [13, Thms. 5.6, 11.4].

ATate resolution or complete resolution of a kG-module Xis an exact sequence of

projectives

tX :· · · → P2→P1→P0δ

−→ P−1→P−2→ · · ·

with Im δ=X. It can be constructed by splicing together a projective and an injective

resolution of X.

If Mdenotes a kG-module and n∈Z, then the n-th Tate cohomology group of Gwith

coefficients in Mis defined to be the n-th cohomology of the complex HomkG(tk, M)

and denoted by

Hn(G, M) = d

Extn

kG(k, M).

The Tate cohomology groups identify with morphism groups in the stable module

category of kG: it holds

Hn(G, M)∼

=HomkG(k, Ω−nM),

see [13, Ch. 6]. Thus ˆ

H∗(G, k) = `n∈Zˆ

Hn(G, k) becomes a graded ring with multipli-

cation ˆ

Hn(G, k)×ˆ

Hm(G, k)→ˆ

Hn+m(G, k),(f, g)7→ Ω−n(f)◦g,

and ˆ

H∗(G, M) is a graded module over ˆ

H∗(G, k), also by composition of graded maps.

Since the non-negative part of the Tate resolution tk is a projective resolution of k,

we have ˆ

Hn(G, M) = Hn(G, M) for n > 0, and we obtain an exact sequence

0→ I(k, M)→H∗(G, M)→ˆ

H∗(G, M)→H−(G, M)→0,

where H−(G, M) denotes the negatively graded part of Tate cohomology. In particular,

since I(k, M) = 0 whenever Mhas no projective direct summands, we can view the

group cohomology ring as subring of the Tate cohomology ring. In fact, for the positively

graded part of ˆ

H∗(G, k), Yoneda splice multiplication and composition of graded maps

in the stable module category coincide [13, Sect. 6].

Proposition 4.2 (Tate duality).[15, XII, Cor. 6.5],[4, Sect. 2] Let D= Homk(−, k).

For any kG-module M, it holds

Hn−1(G, DM)∼

=Dˆ

H−n(G, M).

The Tate cohomology ring is graded-commutative [15, XII, Prop. 5.2]. In general, it

is not Noetherian. However, this is true in the so-called periodic case that we discuss

below. We omit the proof of the following well-known characterisation.

Lemma 4.3. The following are equivalent:

(1) The trivial module kadmits a periodic projective resolution.

22 BIRGIT HUBER

(2) There exists n > 0and an element x∈ˆ

Hn(G, k)such that the map

Hm(G, k)→ˆ

Hm+n(G, k), γ 7→ γ·x,

is an isomorphism for all m∈Z.

We call an element x∈ˆ

Hn(G, k) satisfying (2) a periodicity generator, and from

Lemma 4.3, we infer

Lemma 4.4. In the periodic case, the Tate cohomology ring is a localisation of the group

cohomology ring: ˆ

H∗(G, k) = S−1H∗(G, k),

where Sis the multiplicative subset generated by the periodicity generator of lowest

degree. In particular, ˆ

H∗(G, k)is Noetherian whenever it is periodic. 

Whether Tate cohomology is periodic or not depends on the p-rank of G, which is

the maximal rank of an elementary abelian p-subgroup of Gand denoted by rp(G).

Theorem 4.5. [15, XII, Prop. 11.1] Let ka be field of characteristic pand Ga group

with order divisible by p. Then the trivial module kadmits a periodic resolution if and

only if rp(G) = 1.

The p-groups of p-rank one are characterised as follows:

Theorem 4.6. [21, Chapter 5, Theorem 4.10] Let Gbe a p-group. Then rp(G) = 1 if

and only if Gis cyclic or p= 2 and Gis a generalised quaternion group.

The generalised quaternion group Q2nis given by the defining relations

Q2n=hh, g |h2n−1=g2=b, b2= 1, g−1hg =h−1i

and has order 2n+1, see [21, Ch. 2.6]. In the special case n= 2, the group is called

quaternion group.

Remark 4.7. The Tate cohomology ring comes in three different types: It is periodic

if and only if the p-rank of the group Gequals one. Whenever the depth of the group

cohomology ring H∗(G, k) is at least two, then ˆ

H∗(G, k) is a trivial extension of H∗(G, k)

by the negatively graded part ˆ

H−(G, k) of ˆ

H∗(G, k) [3, Thm. 3.3]. That is in particular

the case if the p-rank of the centre of a p-Sylow subgroup of Gis at least two ([17],

see also [3, Thm. 3.2]). But there are also cases where the Tate cohomology ring is

neither periodic nor a trivial extension: The smallest example is the semidihedral group

of order 16

SD16 =hg, h |g8= 1, h2= 1, h−1gh =g3i.

The 2-rank of SD16 is two and the depth of H∗(SD16, k) equals one (see [3]).

5. The Hochschild Cohomology of a graded ring

Hochschild cohomology HH∗(R) of an algebra Rover some commutative ring kwas

first defined by Hochschild in 1945 using the Bar resolution [22]. Cartan and Eilenberg

showed that HH∗(R) is isomorphic to Ext∗

Re(R, R) whenever Ris projective over k[15,

Ch. IX.6].

Hochschild cohomology of a graded algebra Λ is bigraded; it is denoted by HH∗,∗(Λ),

where the first index is the cohomological degree which is changed by the differential

and the second, internal one, arises from the grading of Λ. The difference between

REALISABILITY AND LOCALISATION 23

“usual” Hochschild cohomology and Hochschild cohomology of a graded algebras consists

basically in the occurrence of additional signs. However, sometimes it can be tedious to

figure them out.

In this chapter we introduce Hochschild cohomology of a graded algebra Λ with the

sign conventions as in [5]. In addition, we study in Section 5.3 the ring structure of

HH∗,∗(Λ). Hochschild cohomology HH∗(R) of a non-graded algebra Ris well-known to

be a graded-commutative ring (in the sense of Section 3.1) with multiplication given by

cup or Yoneda product. We show that for bigraded Hochschild cohomology HH∗,∗(Λ),

one additionally needs to take into account the internal degree: for ζ∈HHm,i(Λ) and

η∈HHn,j(Λ), we prove that ζη = (−1)mn(−1)ijηζ.

Throughout this chapter let Λ be a graded algebra over a field k. We write Λ⊗nfor

the n-fold tensor product over kand denote a tuple λ1⊗· · ·⊗λn∈Λ⊗nby (λ1,· · · , λn).

For any graded (Λ,Λ)-bimodule M, the Hochschild cohomology of Λ with coefficients

in Mis the cohomology of the bigraded complex C∗,∗(Λ, M) given by

(5.1) Cn,m(Λ, M) = Homm

k(Λ⊗n, M),

where n≥0 and m∈Z. Note that the differential δ:Cn,m(Λ, M)→Cn+1,m(Λ, M)

changes only the first grading. It is given by

(δϕ)(λ1, . . . , λn+1) = (−1)m|λ1|λ1ϕ(λ2, . . . , λn+1) +

i=1

(−1)iϕ(λ1, . . . , λiλi+1, . . . , λn+1)+(−1)n+1ϕ(λ1, . . . , λn)λn+1.

This construction obviously generalises the Hochschild complex in the non-graded case.

The Hochschild cohomology groups HH∗,∗(Λ, M) are the cohomology groups of the

complex C∗,∗(Λ, M),

HHs,t(Λ, M) = Hs(C∗,t(Λ, M)) .

HHs,t(Λ,Λ) is abbreviated by HHs,t(Λ).

Using the bimodule structure of M[t] given in (3.4), there is a natural isomorphism

of chain complexes

C∗,m(Λ, M)∼

=C∗,0(Λ, M[t]),

which induces a natural isomorphism of Hochschild cohomology groups

HHs,t(Λ, M)∼

=HHs,0(Λ, M[t]).

5.1. Functoriality. The complex C∗,∗(Λ, M) and its cohomology groups HH∗,∗(Λ, M)

are covariant functors in the (Λ,Λ)-bimodule M: If f:M→Nis a morphism of (Λ,Λ)-

bimodules, then we have a cochain homomorphism

Cs,t(Λ, M)→Cs,t(Λ, N), ϕ 7→ f◦ϕ,

which induces the map

HHs,t(Λ, M)→HHs,t(Λ, N),[ϕ]7→ [f◦ϕ].

Furthermore, the Hochschild groups are contravariant functors in the graded algebra.

Let α: Γ →Λ be a map of graded k-algebras. Then a (Λ,Λ)-bimodule Mcarries a

(Γ,Γ)-bimodule structure through the morphism α. We obtain a cochain homomorphism

Cs,t(Λ, M)→Cs,t(Γ, M), ϕ 7→ ϕ◦α⊗s,

24 BIRGIT HUBER

inducing the map

HHs,t(Λ, M)→HHs,t(Γ, M),[ϕ]7→ [ϕ◦α⊗s].

In general, we cannot expect α: Γ →Λ to give rise to a map HHs,t(Λ) →HHs,t(Γ).

However, in the case that αis a flat epimorphism of rings, we will construct an al-

gebra homomorphism HHs,t(Λ) →HHs,t(Γ) induced by αin Chapter 12.1. For this

construction, we need the graded Bar resolution.

5.2. The graded Bar resolution. Let Λe= Λop ⊗Λ. With the necessary precaution

on the signs (see (3.2) and (3.3)), this algebra is graded and we may identify graded

(Λ,Λ)-bimodules with graded right Λe-modules.

The graded bar resolution B=B(Λ) is a Λe-projective resolution of Λ defined as

Bn=Bn= Λ⊗(n+2), with Λe-module structure given by

(λ0, . . . , λn+1)(µ, µ0)=(−1)|µ||λ0···λn+1|(µλ0, λ1, . . . , λn, λn+1µ0),

and with differential

dn(λ0, . . . , λn+1) =

i=0

(−1)i(λ0, . . . , λiλi+1, . . . , λn+1).

For the fact that B(Λ) is indeed a graded projective Λe-resolution of Λ, we refer to [5,

Sect. 4] and [15, Ch. IX.6].

Lemma 5.1. The map

(5.2) Homt

k(Λ⊗s, M)→Homt

Λe(Λ⊗(s+2), M), f 7→ ˜

where ˜

f(λ0,· · · , λs+1) = (−1)t|λ0|λ0f(λ1,· · · , λs)λs+1,

is an isomorphism and extends to an isomorphism of chain complexes

(5.3) C∗,t(Λ, M)→Homt

Λe(B∗, M).

Consequently, we have

HHs,t(Λ, M) = Hs(C∗,t(Λ, M)) = Hs(Homt

Λe(B, M)),

and thus, it holds

HHs,t(Λ, M) = Exts,t

Λe(Λ, M).

Of course, the Hochschild cohomology groups can also be computed with an arbitrary

graded Λe-projective resolution of Λ.

5.3. Ring structure. G. Hochschild [22] proved that Hochschild cohomology HH∗(R)

of an algebra Radmits a ring structure, with multiplication given by the cup product.

It is well-known that the cup product coincides with the Yoneda product (see [12, Prop.

1.1]) and makes HH∗(R) into a graded-commutative ring (see for example [51]). In this

section we study the ring structure of bigraded Hochschild cohomology HH∗,∗(Λ) of a

graded algebra Λ.

There is a degree zero chain map ∆: B→B⊗ΛBlifting the identity map of Λ, given

∆(λ0,· · · , λn+1) =

i=0

(λ0,· · · λi,1) ⊗Λ(1, λi+1,· · · , λn+1).

REALISABILITY AND LOCALISATION 25

For the non-graded case, this can be found for example in [51]. It carries over to our

case without any additional sign adjustment.

Let ζ∈HHm,i(Λ) be represented by a cocycle Bm→Λ of degree i, and represent

η∈HHn,j(Λ) by a (n, j)-cocycle Bn→Λ. Then the cup product of ζand ηis given by

the composition of graded maps

B∆

−→ B⊗ΛBζ⊗η

−−→ Λ⊗ΛΛν

−→ Λ,

where ν: Λ⊗ΛΛ→Λ is the multiplication map. In order to write the cup product ζ∪η

explicitly as a (m+n, i+j)-cocycle Bm+n→Λ, we need to take into account the Koszul

sign rule (3.1) and obtain

(5.4) (ζ∪η)(λ0,· · · , λm+n+1) = (−1)(|λ0,··· ,λm|)·jζ(λ0,· · · λm,1)η(1, λm+1,· · · λm+n+1).

The cup product makes HH∗,∗(Λ) into a bigraded ring, that is, a Z×Z-graded ring.

The Yoneda product ζ∗ηis given through a graded lifting of η, i.e. a graded chain

map B→¯

ΣnBwhich lifts η. Here ¯

ΣnBdenotes the n-fold shift to the left of the complex

Bwithout changing the signs in the differential. Note that the ‘internal’ degree of this

chain map is the degree of η.

Now we adapt the proof for the non-graded case (see [12, Prop. 1.1]) to our setting

and show that cup product and Yoneda product coincide:

Proposition 5.2. Let ζ∈HHm,i(Λ) and η∈HHn,j(Λ) be represented by the (m, i)-

cocycle ζ:Bm→Λand the (n, j)-cocycle η:Bn→Λ, respectively. Then it holds

ζ∪η=ζ∗η.

Proof. In [12, Prop. 1.1], it is shown that ˜η:B→¯

ΣnB, defined by

B∆

−→ B⊗ΛBB⊗η

−−→ B⊗Λ¯

ΣnΛν

−→ ¯

ΣnB,

is a lifting of η. Actually, this is a graded lifting of internal degree j. Hence we may set

ζ∗η=ζ˜η=ζν(B⊗η)∆.

The Koszul sign rule (3.1) permits the equations

ζν =ν(ζ⊗¯

ΣnΛ) and (ζ⊗¯

ΣnΛ)(B⊗η) = ζ⊗η.

But then

ζ∗η=ζν(B⊗η)∆

=ν(ζ⊗¯

ΣnΛ)(B⊗η)∆

=ν(ζ⊗η)∆

=ζ∪η. 

Now we are ready to prove

Theorem 5.3. HH∗,∗(Λ) is a bigraded-commutative ring: For ζ∈HHm,i(Λ) and η∈

HHn,j(Λ), we have

ζη = (−1)mn(−1)ijηζ,

where the multiplication is given by the cup or Yoneda product. 1

1With another sign convention one obtains a different result: ζ˜

∪η= (−1)(m+i)(n+j)η˜

∪ζfor ζ∈

HHm,i(Λ) and η∈HHn,j (Λ). Here the multiplication ˜

∪arises by taking into account also the external

degree of the bigraded elements in the Koszul sign rule (3.1) and thus in the cup product (5.4). However,

starting with our sign convention, one can define a new multiplication by the rule η∗ζ= (−1)inηζ. This

new multiplication is then still associative, as is easily checked, and satisfies ζ∗η= (−1)(m+i)(n+j)η∗ζ.

26 BIRGIT HUBER

Proof. Let ζ∈HHm,i(Λ) and η∈HHn,j(Λ) be represented by the (m, i)-cocycle

ζ:Bm→Λ and the (n, j)-cocycle η:Bn→Λ, respectively. We carry over the proof of

[51, Thm. 2.1] to our bigraded case. There it is shown that the chain map ζ0:B→¯

ΣnB

given by

ζ0

p:Bm+p−→ Bp

(λ0,· · · , λp+m+1)7−→ (−1)mpζ(λ0,· · · , λm,1), λm+1,· · · , λp+m+1

is a lifting of ζ. Since all ζ0

pare homogeneous maps of degree i, we conclude that ζ0is

actually a graded lifting. Moreover, we have that

η∗ζ(λ0,· · · , λm+n+1) = ηζn(λ0,· · · , λm+n+1)

= (−1)mnηζ(λ0,· · · , λm,1), λm+1,· · · , λn+m+1

= (−1)mn(−1)|ζ(λ0,··· ,λm,1)|·jζ(λ0,· · · , λm,1)η(1, λn+1,· · · , λn+m+1)

= (−1)mn(−1)i+|(λ0,··· ,λm)|·jζ(λ0,· · · , λm,1)η(1, λm+1,· · · , λn+m+1)

On the other hand, it holds

(ζ∪η)(λ0,· · · , λm+n+1) = (−1)|(λ0,··· ,λm)|·jζ(λ0,· · · λm,1)η(1, λm+1,· · · λm+n+1).

We infer

η∗ζ= (−1)mn(−1)ijζ∪η,

and the claim follows since cup and Yoneda product coincide by Proposition 5.2. 

Remark 5.4. We may identify the graded ring HH0,∗(Λ) with the graded centre of Λ,

Zgr(Λ) = {x∈Λ|xλ = (−1)|x||λ|λx for all λ∈Λ}, by the evaluation map

HH0,∗(Λ) →Zgr(Λ), f 7→ f(1),

which is easily checked to be well-defined and bijective.

5.4. The cup product pairing. Let ϕ∈HomΛe(Bs,Λ) be a (s, t)-Hochschild cocycle

B:· · · //Bs//



Bs−1//· · · //B0//Λ//0

Tensoring ϕover Λ with a homomorphism of graded right Λ-modules f:X→Yyields

the map

f⊗Λϕ:X⊗ΛBs→Y

which is homogeneous of degree t. The complex B⊗ΛXis a projective resolution of

Xin Modgr Λ, and since this construction commutes with the differential, f⊗Λϕis a

(s, t)-cocycle:

B⊗ΛX:· · · //Bs⊗ΛX//

f⊗Λϕ



Bs−1⊗ΛX//· · · //B0⊗ΛX//X//0

The cohomology class f∪ϕof f⊗Λϕonly depends on the cohomology class of the

cocycle ϕand hence, we obtain a well-defined map

(5.5) ∪: HomΛ(X, Y )⊗ΛHHs,t(Λ) −→ Exts,t

Λ(X, Y ),

called the cup product pairing.

REALISABILITY AND LOCALISATION 27

6. Differential graded algebras and their derived categories

Differential graded algebras, or shortly, dg algebras were introduced by Cartan [14]

in 1956. They arise as complexes with an additional multiplicative structure. For

example, the endomorphism complex End(C) of a complex Ccarries a natural dg algebra

structure. Derived categories of dg algebras or more generally, of dg categories, were

first studied systematically by Bernhard Keller in ‘Deriving DG Categories’ [27].

6.1. Differential graded algebras and modules. A graded algebra over a commu-

tative ring

A=a

n∈Z

is called a differential graded algebra or dg algebra if it carries a differential d:A→A,

i.e. a graded k-linear map of degree +1 with the property d2= 0, which is required to

satisfy the Leibniz rule

d(xy) = d(x)y+ (−1)nxd(y) for all x∈Anand y∈A.

The cohomology of Ais a graded associative algebra over kand denoted by H∗A.

Adg A-module is a graded (right) A-module Xendowed with a differential d:X→X

satisfying the Leibniz rule

d(xy) = d(x)y+ (−1)nxd(y) for all x∈Xnand y∈A.

Amorphism of dg A-modules is an A-linear map which is homogeneous of degree

zero and commutes with the differential. We denote the category of dg A-modules by

Moddg A.

A map f:X→Yof dg A-modules is null-homotopic if there is a graded A-linear

map ρ:X→Yof degree −1 such that f=dY◦ρ+ρ◦dX. The null-homotopic maps

form an ideal and the homotopy category K(A) is the quotient of Moddg Awith respect

to this ideal. The homotopy category carries a triangulated structure which is defined

in the same way as for the homotopy category K(C) of an additive category C.

A map X→Yof dg A-modules is a quasi-isomorphism if it induces an isomorphism

HnX→HnYin each degree n∈Z. The derived category of the dg algebra Ais the

localisation of K(A) with respect to the class Sof all quasi-isomorphisms,

D(A) = K(A)[S−1].

Note that Sis a multiplicative system and compatible with the triangulation. Therefore

D(A) is triangulated and the localisation functor K(A)→ D(A) is exact.

Remark 6.1. (1) Any graded algebra is a dg algebra Awith differential zero. A non-

graded algebra Rcan be viewed as a dg algebra Awith A0=Rand An= 0, otherwise.

In this case, Moddg Acan be identified with the category of complexes of R-modules

C(Mod R). Furthermore, K(A) identifies with K(Mod R), the homotopy category of

C(Mod R), and D(A) with D(Mod R), the derived category of complexes of R-modules.

In particular, the results we state in the following sections carry over to the derived

category of a module category over a non-graded algebra.

(2) Let X, Y be complexes in some additive category C. Then the complex HomC(X, Y )

is given by a

n∈ZY

p∈Z

HomC(Xp, Y p+n),

28 BIRGIT HUBER

with differential

dn(φ) = dY◦φ−(−1)nφ◦dX

for φ= (φp)p∈Z∈HomC(Xp, Y p+n). Note that

HnHomC(X, Y )∼

=HomK(C)(X, ΣnY)

because Ker dnidentifies with HomC(C)(X, ΣnY) and Im dn−1with the ideal of null-

homotopic maps X→ΣnY. The composition of graded maps yields a dg algebra

structure for

EndC(X) = HomC(X, X),

and HomC(X, Y ) is a dg module over EndC(X).

(3) If Ais a dg algebra and X, Y dg A-modules, then the homomorphism complex

HomA(X, Y ) is defined in an analogous way:

HomA(X, Y ) = a

n∈Z

Homn

A(X, Y ),

where Homn

A(X, Y ) denotes the homogeneous graded A-linear maps X→Yrising the

degree by n∈Z. The differential dn: Homn

A(X, Y )→Homn+1

A(X, Y ) is defined to be

dn(f) = dY◦f−(−1)nf◦dX.

Also in this case, we have an isomorphism

HnHomA(X, Y )∼

=HomK(A)(X, ΣnY).

The endomorphism ring EndA(X) = HomA(X, X) is a dg algebra and HomA(X, Y ) a

dg module over EndA(X) by composition of graded maps.

The following well-known lemma shows that the functor

D(A)(A, −)∗=a

i∈Z

D(A)(A, −)i

is naturally isomorphic to the cohomology functor H∗.

Lemma 6.2. Let Abe a dg algebra and H∗Aits cohomology algebra. For any X∈ D(A),

the evaluation map

D(A)(A, X)∗→H∗X, f 7→ f(1),

is a natural isomorphism of graded H∗A-modules, where D(A)(A, X)∗becomes a graded

H∗A-module via the isomorphism D(A)(A, A)∗∼

=H∗A.

It follows in particular that D(A) is compactly generated by A, the free dg A-module

of rank one.

J. Rickard [46] proved that the compact objects of D(Mod R), where Ris a ring, are

the perfect complexes. A complex of R-modules is called perfect if it is quasi-isomorphic

to a bounded complex of finitely generated projective R-modules. The full subcategory

of perfect complexes of D(Mod R) is denoted by Dper(Mod R).

This characterisation of the compact objects was extended to the derived category

D(A) of a dg algebra Aby Neeman [41]. Here Dper(A) denotes the smallest thick

subcategory of D(A) containing A.

Proposition 6.3. [41],[34, Ch. 5.5]. A dg module is compact in D(A)if and only if it

is contained in Dper(A).

REALISABILITY AND LOCALISATION 29

6.2. K(A)as stable category of a Frobenius category. Let Abe a dg algebra over

a commutative ring k. Then Moddg Ais an exact category with respect to the exact

sequences of dg A-modules

0→X→Y→Z→0

which are split considered as sequences of graded A-module maps. Furthermore, Moddg A

is a Frobenius category (see Example 2.5(4) for a definition). The projective-injective

objects are the dg A-modules M⊕M[1] with differential 0 id

0 0 , where M∈Moddg A.

Since the maps factoring through an injective object are precisely the nullhomotopic

maps, the associated stable category coincides with the homotopy category K(A). We

refer to [28, Sect. 8.2.3] and [27, Sect. 2.2] for more details.

Lemma 6.4. Let ϕ:X→Ybe any morphism in K(A). Then ϕcan be represented

by a morphism X→ˆ

Yin Moddg Awhich is a split monomorphism in the category of

graded A-modules.

Proof. Since K(A) = S(Moddg A), we can choose a map of dg A-modules f:X→Y

that induces ϕin the stable category. Let i:X→I(X) be the injective hull and

s:I(X)→Xthe graded A-linear map satisfying s◦i= idX. We set ˆ

Y=Y⊕I(X).

Then the map f

i:X→Y⊕I(X)

is clearly a monomorphism of dg A-modules that induces ϕin the homotopy category,

and the graded A-module map

[0s] : Y⊕I(X)→X

satisfies [ 0s]◦f

i= idX.

6.3. Homotopically projective and homotopically injective dg modules. We

use the terminology of Bernhard Keller, as presented in [28]. Throughout this section

let Abe a dg algebra over some commutative ring k. We say that a dg A-module Xis

homotopically projective if

K(X, Y ) = 0

for all acyclic dg A-modules Y.

Dually, Xis called homotopically injective if

K(Y, X) = 0

for all acyclic dg A-modules Y.

We denote by Kp(A) (resp. Ki(A)) the full subcategory of homotopically projective

(resp. homotopically injective) dg A-modules of K(A).

Theorem 6.5. [28, Sect. 8.1.6]

(1) For any dg Amodule X, there is a triangle

pX→X→aX→ΣpX

in K(A), where pXis homotopically projective and aXis acyclic. Any triangle

Z→X→Y→ΣZwith homotopically projective Zand acyclic Yis isomorphic

to (pX, X, aX), and there is a unique such isomorphism extending the identity

of X.

30 BIRGIT HUBER

(2) For any dg Amodule X, there is a triangle

a0X→X→iX→Σa0X

in K(A), where iXis homotopically injective and a0Xis acyclic. Any triangle

Z→X→Y→ΣZwith acyclic Zand homotopically injective Yis isomorphic

to (a0X, X, iX), and there is a unique such isomorphism extending the identity

of X.

In particular, each dg Amodule Xis quasi-isomorphic to a homotopically projective

(resp. homotopically injective) dg A-module and we call

pX→Xresp. X→iX(6.1)

an homotopically projective resp. homotopically injective resolution. In particular, the

assignments pand ican be shown to be functorial. Since they vanish on acyclic com-

plexes, they extend to functors D(A)→ K(A). In fact, we have

Theorem 6.6. [28, Sect. 8.2.6]

(1) The composition

Kp(A),→ K(A)can

−−→ D(A)

is an equivalence of triangulated categories with quasi-inverse given by

p:D(A)→ Kp(A).

More precisely, pinduces a fully faithful left adjoint to the quotient functor

can: K(A)→ D(A).

(2) The composition

Ki(A),→ K(A)can

−−→ D(A)

is an equivalence of triangulated categories with quasi-inverse given by

i:D(A)→ Ki(A).

More precisely, iinduces a fully faithful right adjoint to the quotient functor

can: K(A)→ D(A).

Corollary 6.7. For all dg A-modules Xand Y, we have

K(A)(X, iY)∼

=D(A)(X, iY)∼

=D(A)(X, Y )∼

=D(A)(pX, Y )∼

=K(A)(pX, Y ).

Remark 6.8. A dg A-module is homotopically projective if and only if it is chain

homotopy equivalent to a cofibrant dg module (see [48, Rem. 3.17]). A dg A-module X

is cofibrant if there exists an exhaustive increasing filtration by dg A-submodules

0 = X0⊆X1⊆ · · · ⊆ Xn⊆ · · ·

such that each subquotient Xn+1/Xnis a direct summand of shifted copies of A. The

derived category D(A) can also be defined in the following way: The objects are the

cofibrant modules, and the morphisms are chain homotopy classes of dg A-module mor-

phisms.

REALISABILITY AND LOCALISATION 31

6.4. Derived functors. Let Aand Bbe two dg algebras over a commutative ring k.

Adg (A, B)-bimodule is a graded (A, B)-bimodule which carries in addition a k-linear

differential dof degree +1 satisfying

d(axb) = (da)xb + (−1)pa(dx)b+ (−1)p+qax(db)

for all a∈Ap, x ∈Xq, b ∈B.

Let Mbe any dg A-module. To define the tensor product M⊗AXof dg modules,

we first observe that the tensor product M⊗kXis a dg B-module. As for graded rings,

the degree ncomponent is defined to be

(M⊗kX)n=a

p+q=n

Mp⊗Xq.

Additionally, we now have the differential

d(m⊗x) = (dm)⊗x+ (−1)|m|m⊗dx.

Since the k-submodule generated by all differences ma ⊗x−m⊗ax is stable under

both dand multiplication with elements of B, the quotient modulo this submodule is a

well-defined dg B-module which we denote by M⊗AX. Moreover, this construction is

functorial in Mand X.

Let Nbe a dg B-module. Then HomB(X, N), as defined in Remark 6.1, is a right dg

A-module by setting

(fa)(x) = f(ax).

Observe that − ⊗AXand HomB(X, −) induce functors between K(A) and K(B)

which form an adjoint pair

K(A)

−⊗AX//K(B)

HomB(X,−)

We define the total left derived functor − ⊗L

AXas the composition

D(A)p

−→ K(A)−⊗AX

−−−−→ K(B)can

−−→ D(B),

and the total right derived functor RHomB(X, −) as

D(B)i

−→ K(B)HomB(X,−)

−−−−−−−−→ K(A)can

−−→ D(A).

Then the total derived functors also form an adjoint pair

D(A)

−⊗L

//D(B)

RHomB(X,−)

oo.

In particular, we deduce from Proposition 2.4 that − ⊗L

AXpreserves arbitrary direct

sums and RHomB(X, −) preserves arbitrary direct products.

32 BIRGIT HUBER

6.5. Cofibrant differential graded algebras. The category of dg algebras dga/k over

a commutative ring kadmits a model category structure [49]. A model category is a

category with three distinguished classes of morphisms, the fibrations,cofibrations and

weak equivalences. These are required to satisfy certain axioms. An object Cin a model

category is called cofibrant if the morphism 0 →Cis a cofibration. We refer to [24, Ch.

1.1] for details.

In the category dga/k, the fibrations are the degree-wise surjective dg algebra mor-

phisms and the weak equivalences equal the quasi-isomorphisms. A dg algebra is called

cofibrant if it is a cofibrant object in the model category dga/k, that is:

Definition 6.9. A dg algebra Ais cofibrant if for any morphism of dg algebras f:A→C

and every surjective quasi-isomorphism of dg algebras g:B→C, there exists a lift

h:A→B. That is, we have a commutative diagram



∼



Af//

h??

A direct consequence of the model category axioms for dga/k is

Lemma 6.10. [24, Ch. 1.1] If Ais any dg algebra over a commutative ring k, then there

exists a cofibrant dg algebra Acof and a quasi-isomorphism

Acof ∼//A.

We remark that from the model theory axioms, it follows moreover that the quasi-

isomorphism Acof ∼

−→ Aabove is a surjective map [24, Ch. 1.1].

Examples for cofibrant dg algebras are the Sullivan algebras [19, Ch. 12] which we

define in the following. From now on we assume that kis a field of characteristic zero.

We recall the definition of the free graded-commutative algebra:

Let Vbe a graded vector space over k. The elements v⊗w−(−1)|v||w|w⊗vgenerate

an ideal Iin the tensor algebra TV . The free graded-commutative algebra ΛVis quotient

of the Tensor algebra TV by the ideal I,

ΛV=TV/I.

If v1,· · · , vnis a k-basis of V, one also writes Λ(v1,· · · , vn) for ΛV.

Definition 6.11. ASullivan algebra is a dg algebra of the form (ΛV, d), where

(1) V=`p≥1Vpis a positively graded vector space

(2) V=Sl≥0V(l), where V(0) ⊆V(1) ⊆ · · · is an increasing sequence of graded

subspaces such that

d= 0 on V(0) and d(V(l)) ⊆ΛV(l−1) for all l≥1.

Lemma 6.12. [19, Lemma 12.4] Every Sullivan algebra (ΛV, d)is a cofibrant dg algebra.

For an arbitrary dg algebra A, the quasi-isomorphism Acof ∼

−→ Awhich exists by

Lemma 6.10 is not easy to compute. However, for a certain class of dg algebras, one can

construct explicitly quasi-isomorphic Sullivan algebras:

REALISABILITY AND LOCALISATION 33

Proposition 6.13. [19, Prop. 12.1] Assume that Ais a graded-commutative dg algebra

concentrated in non-negative degrees which satisfies H0(A) = k. Then there exists a

Sullivan algebra (ΛV, d)and a quasi-isomorphism

(ΛV, d)∼//A.

Example 6.14. [19, Ch. 12, Exm. 4] Not every dg algebra of the form (ΛV, d) is a

Sullivan algebra: Consider (Λ(v1, v2, v3), d), where |vi|= 1, and the differential is given

by dv1=v2v3,dv2=v3v1, and dv3=v1v2. This dg algebra is not a Sullivan algebra.

However, we can state a Sullivan algebra which is quasi-isomorphic to (Λ(v1, v2, v3), d):

there is a quasi-isomorphism

σ: (Λ(w),0) ∼//(Λ(v1, v2, v3), d),

where wis of degree 3; the map σis given by σ(w) = v1v2v3.

7. A∞-algebras

A∞-algebras are generalisations of dg algebras. They were invented by J. Stasheff at

the beginning of the 1960s as a tool in the study of ‘group-like’ topological spaces. In

the 1990s, the relevance of A∞-algebras in algebra became more and more apparent.

We focus on a result of Kadeishvili stating that the cohomology of a dg algebra is an

A∞-algebra. Instead of Kadeishvili’s Russian original paper [26] we refer the reader to

the articles [29] and [30] by Bernhard Keller.

Throughout this chapter let kbe a field and write shortly ⊗for ⊗k.

Definition 7.1. An A∞-algebra is a Z-graded vector space

A=a

p∈Z

together with a family of homogeneous k-linear maps

mn:A⊗n→A, n ≥1,

of degree 2 −nsatisfying the relations

(i) m1m1= 0.

(ii) m1m2=m2(m1⊗1+1⊗m1).

(iii) More generally, for all n≥1,

X(−1)r+stmu(id⊗r⊗ms⊗id⊗t) = 0,

where the sum runs over all decompositions n=r+s+t, and we set u=r+1+t.

Note that (A, m1) is a differential complex due to (i). Condition (ii) means that m1is

a graded derivation with respect to the multiplication m2, and equation (iii) with n= 3

shows that the multiplication m2is associative only up to homotopy. The map m3is

called the secondary multiplication.

Remark 7.2. (1) In general, an A∞-algebra is not associative. However, its cohomology

H∗Awith respect to the differential m1is an associative Z-graded algebra with the

multiplication induced by m2.

34 BIRGIT HUBER

(2) If Ais concentrated in degree zero, then A=A0is just an associative algebra.

That is because mnis of degree 2 −nand consequently, all mnother than m2have to

vanish.

(3) If mnis trivial for all n≥3, then Ais a dg algebra. Conversely, each dg algebra

carries an A∞-structure with m1the differential, m2the multiplication, and all other

mntrivial.

A morphism between two A∞-algebras Aand Bis in general not just a map A→B,

but something quite more complicated:

Definition 7.3. A morphism of A∞-algebras f:A→Bis a family of graded maps

fn:A⊗n→B

of degree 1 −nsuch that

(i) f1m1=m1f1, i.e. f1:A→Bis a chain map.

(ii) f1m2=m2(f1⊗f1) + m1f2+f2(m1⊗id + id ⊗m1).

(iii) More generally, for n≥1, we have

X(−1)r+stfu(id⊗r⊗ms⊗id⊗t=X(−1)smr(fi1⊗fi2⊗ · · · fir),

where the first sum runs over all decompositions n=r+s+t, and we set

u=r+ 1 + t. The second sum runs over 1 ≤r≤nand all decompositions

n=i1+· · · +ir. Furthermore, the sign on the right hand side is given by

s= (r−1)(i1−1) + (r−2)(i2−1) + · · · + 2(ir−2−1) + (ir−1−1).

Note that equation (ii) means that f1commutes with the multiplication m2up to a

homotopy given by f2.

An A∞-morphism f:A→Bis

•aquasi-isomorphism if the chain map f1is a quasi-isomorphism,

•strict if fi= 0 for all i6= 1,

•the identity morphism if f:A→Ais strict with f1= idA.

The composition of two A∞-morphisms g:A→Band h:B→Cis defined as

(h◦g)n=X(−1)shr◦(gi1⊗ · · · ⊗ gir),

where the sum and the sign are as in Definition 7.3 (iii).

Theorem 7.4 (Kadeishvili [26], see also [30]).Let Abe an A∞-algebra. Then the

cohomology H∗Ahas an A∞-algebra structure such that

1) mH∗A

1= 0 and mH∗A

2is induced by mA

2, and

2) there is a quasi-isomorphism of A∞-algebras f:H∗A→Alifting the identity in

cohomology, i.e. H∗f1= idH∗A.

Moreover, this structure is unique up to (non unique) isomorphism of A∞-algebras.

In particular, the cohomology H∗Aof a dg algebra Ais an A∞-algebra. We now show

how to construct the secondary multiplication mH∗A

3of H∗Aand at the same time, the

first three terms of the quasi-isomorphism f:H∗A→Alifting the identity of H∗A:

Construction 7.1. Let Abe a dg algebra with differential mA

1and multiplication mA

We view H∗Aas a complex with zero differential. Since we are working over a field,

we can choose a quasi-isomorphism f1:H∗A→Ainducing the identity in cohomology.

REALISABILITY AND LOCALISATION 35

This amounts to choosing a representative cocycle for each cohomology class, in a linear

way. Note that f1cannot be chosen to be multiplicative, but it does commute with

multiplication up to coboundaries. So we can choose a k-linear map of degree −1,

f2:H∗A⊗H∗A→A,

satisfying

(7.2) mA

1f2(x, y) = f1(xy)−f1(x)f1(y).

So it holds indeed

(7.3) f1mH∗A

2=mA

2(f1⊗f1) + mA

1f2.

Now we look for f3and m3such that

f1◦mH∗A

3+f2◦(mH∗A

2⊗id −id ⊗mH∗A

+f3◦(mH∗A

1⊗id⊗2+ id ⊗mH∗A

1⊗id + id⊗2⊗mH∗A

=mA

3◦(f1⊗f1⊗f1) + mA

2◦(f1⊗f2−f2⊗f1) + mA

1f3.

Since mH∗A

1= 0, this simplifies into

(7.4) f1◦mH∗A

3=mA

2◦(f1⊗f2−f2⊗f1)−f2◦(mH∗A

2⊗id −id ⊗mH∗A

2) + mA

1◦f3.

Now one checks that the map

(7.5) Φ3=mA

2◦(f1⊗f2−f2⊗f1)−f2◦(mH∗A

2⊗id −id ⊗mH∗A

has its image in the cycles Z∗Aof A. So we define

(7.6) mH∗A

3=π◦Φ3,

where πdenotes the quotient map Z∗A→H∗A. Then

f1◦mH∗A

3−Φ3= (f1◦π−id)Φ3

has its image in the coboundaries and thus we can indeed choose a k-linear map

f3:H∗A⊗3→A

of degree −2such that

f1◦mH∗A

3−Φ3=mA

1◦f3

as desired.

This construction depends on some choices and the secondary multiplication is not

uniquely determined. However, it determines a Hochschild class which is independent

of all choices:

Proposition 7.5. [5, Prop. 5.4] Let Abe a dg algebra over a field k. Then the secondary

multiplication mH∗A

3of the A∞-algebra H∗Ais a (3,−1)-Hochschild cocycle. Moreover,

its Hochschild class is independent of all choices in defining the maps f1and f2.

The Hochschild class of any choice of mH∗A

3is denoted by µA∈HH3,−1(H∗A). We are

particularly interested in this Hochschild class since it determines a global obstruction

for realisability ([5], see Section 10.3). Because of this special property it is also referred

to as canonical class. In Chapter 13.2 we will compute the secondary multiplication and

its Hochschild class in some examples.

36 BIRGIT HUBER

The following proposition has applications in the Chapters 12 and 13.

Proposition 7.6. [5, Cor. 5.7] Let α:A→Bbe a morphism of dg algebras and

H∗α:H∗A→H∗Bthe induced morphism in cohomology.

(1) In the Hochschild group HH3,−1(H∗A, H∗B), it holds

H∗α◦µA=µB◦(H∗α)⊗3,

where H∗Bis a (H∗A, H∗A)-bimodule through H∗α.

(2) If αis a quasi-isomorphism, then the class µAis mapped to µBunder the induced

isomorphism between the Hochschild cohomology of H∗Aand H∗B.

Hence for any choice of secondary multiplications mH∗A

3and mH∗B

3, the difference

H∗α◦mH∗A

3−mH∗B

3◦(H∗α)⊗3

is a (3,−1)-Hochschild coboundary. If we assume in addition that the algebra map

H∗α:H∗A→H∗Bis a monomorphism, we can obtain equality of H∗α◦mH∗A

3and

mH∗B

3◦(H∗α)⊗3not only on the level of Hochschild classes, but even on the level of

k-linear maps:

Proposition 7.7. Let α:A→Bbe a morphism of dg algebras and assume that

H∗α:H∗A→H∗Bis a monomorphism. Given the choices in defining mH∗A

3, we

can define mH∗B

3such that

H∗α◦mH∗A

3=mH∗B

3◦(H∗α)⊗3in Hom−1

k(H∗A⊗3, H∗B).

Proof. Let fA

1:H∗A→Aand f2:H∗A⊗H∗A→Abe any choices of the first two

components of a quasi-isomorphism f:H∗A→Alifting the identity. We define a

graded degree zero map fB

1:H∗B→Bby

1=α◦fA

1◦(H∗α)−1

on the image of the monomorphism H∗αand extend this map k-linearly to a graded map

inducing the identity in cohomology. This is indeed possible because we are working

over a field and since

H∗fB

1(H∗α(x)) = H∗α◦fA

1(x) = H∗α(x).

Now we define fB

2. On the image of H∗α⊗H∗αwe set

2=α◦f2◦(H∗α⊗H∗α)−1.

We then extend fB

2k-linearly to a degree −1 map satisfying

dfB

2(x, y) = fB

2(x)fB

2(y)−fB

2(xy)

for all x, y ∈H∗B. Our choices for fB

1and fB

2then automatically yield

H∗α◦mH∗A

3=mH∗B

3◦(H∗α)⊗3.

Remark 7.8. [5, Exm. 7.7] Let Aand Bdg algebras over a field kand mH∗A

3resp. mH∗B

secondary multiplications of their cohomology. Then under the K¨unneth isomorphism

H∗(A⊗kB)∼

=H∗A⊗kH∗B, the canonical class µA⊗Bis represented by the cocycle

m(x1⊗y1, x2⊗y2, x3⊗y3) =

(−1)|x3||y1|+|x3||y2|+|x2||y1|mH∗A

3(x1, x2, x3)⊗y1y2y3+x1x2x3⊗mH∗B

3(y1, y2, y3).

REALISABILITY AND LOCALISATION 37

8. Localisation in triangulated categories

The classical localisation S−1Rof a commutative ring Rwith respect to a multiplica-

tively closed subset S⊆Rgives rise to the functor − ⊗RS−1R: Mod R→Mod S−1R.

It assigns to an R-module Mthe S−1R-module S−1M, whose elements are fractions

s, m ∈M, s ∈S. The tensor functor −⊗RS−1Ris right adjoint to HomS−1R(S−1R, −),

and it is well-known that the latter functor is fully faithful.

This calculus of fractions has been generalised by Gabriel and Zisman [20] to arbitrary

categories. In his th`ese [54], Verdier applied this to introduce localisation of triangulated

categories. In particular, he invented the Verdier quotient which is a quotient category

T/Bof a triangulated category Tby a triangulated subcategory B.

In the first two sections we recall categories of fractions and localisation functors for

arbitrary categories. Localisation of triangulated categories, in particular the Verdier

quotient, will be introduced in Section 8.3. Localisation functors of triangulated cate-

gories give rise to localisation sequences, which we define in Section 8.4, and those

localisation sequences which are at the same time co-localisation sequences, the re-

collements, are considered in Section 8.5. In the last section of this chapter we study

cohomological localisations. These localisations are a key tool to prove our results stated

in the Chapters 9 and 11.

8.1. Categories of fractions. A functor F:C → D is said to invert a morphism σof

Cif Fσ is invertible. For a category Cand any class of morphisms Σ of Cthere exists

(after taking the necessary set-theoretic precautions) the category of fractions C[Σ−1],

together with a canonical functor

QΣ:C → C[Σ−1]

having the following properties:

(Q1) QΣmakes the morphisms in Σ invertible.

(Q2) If a functor F:C → D makes all morphisms in Σ invertible, then there is a

unique functor G:C[Σ−1]→ D such that F=G◦QΣ.

An explicit construction of the category C[Σ−1] can be found in the book of Gabriel

and Zisman [20].

Let C,Dbe categories and

F//D

an adjoint pair of functors, that is, F:C → D and G:D → C are a pair of functors

such that Gis right adjoint to F. By η: idC→G◦Fwe denote the unit, and by

ε:F◦G→idDthe counit of the adjunction.

Lemma 8.1. [20, Ch. I, Prop. 1.3] Let Σbe the set of morphisms σof Csuch that F σ

is invertible. The following are equivalent:

(1) The functor Gis fully faithful.

(2) The counit of the adjunction ε:F◦G→idDis invertible.

(3) The functor ¯

F:C[Σ−1]→ D satisfying F=¯

F◦QΣis an equivalence.

38 BIRGIT HUBER

8.2. Localisation functors. Let F:C → D and G:D → C be an adjoint pair of

functors satisfying the equivalent conditions of Lemma 8.1, and set L=G◦F. The

following well-known lemma shows that the pair (F, G) can be reconstructed from Land

the adjunction unit Ψ: idC→L.

Lemma 8.2. Let L:C → C be a functor and Ψ: idC→La natural transformation.

The following are equivalent:

(1) The map LΨ: L→L2is invertible and LΨ = ΨL.

(2) There exists a pair of functors F:C → D and G:D → C such that Fis left

adjoint to G,Gis fully faithful, L=G◦F, and Ψ: idC→G◦Fis the unit of

the adjunction.

For a proof we refer to [36]. However, we sketch how the pair (F, G) can be constructed

from the functor L: Given L:C → C, we define Dto be the full subcategory of Cformed

by those objects Xsuch that ΨX:X→LX is invertible. The functor Fis given by

F:C → D, FX =LX, and G:D → C is defined to be the inclusion. Note also that Ψ

equals the adjunction unit η: idC→G◦F.

Definition 8.3. Let Cbe an additive category.

(1) We call a pair (L:C → C,Ψ: idC→L) a localisation functor if it satisfies the

conditions of Lemma 8.2.

(2) An object X∈ C is called L-acyclic if L(X) = 0, and the full subcategory of

L-acyclic objects is denoted by Ker L.

(3) An object X∈ C is called L-local if X∼

=LX0for some X0∈ C. The full

subcategory of L-local objects is denoted by CL.

Remark that an object X∈ C is L-local if and only if ΨXis invertible, see [36, Lemma

1.5]. Thus the category Dconstructed in Lemma 8.2 equals CL.

Justified by Lemma 8.2, a functor F:C → D admitting a fully faithful right adjoint

Gis also called localisation functor.

8.3. Quotient categories. If the category Cadmits an abelian (resp. triangulated)

structure and Bis a Serre (resp. triangulated) subcategory, then we can form a quotient

category by inverting a special class of morphisms. We present this construction for

triangulated categories.

Let Cbe triangulated and Ba triangulated subcategory. We define ΣBto be the class

of all morphisms σ∈ C such that there exists an exact triangle

X→Yσ

−→ Z→X[1],

with X∈ B.

The category of fractions C[Σ−1

B] is called the Verdier Quotient and denoted by C/B.

The following properties of the Verdier Quotient are well-known.

Lemma 8.4. Let QBdenote the canonical functor QΣB:C → C/B.

(1) C/Bcarries a unique triangulated structure such that the functor QB:C → C/B

is exact.

(2) The kernel of QBconsists of the thick closure of B, i.e. those objects X∈ C

such that there exists Y∈ C with XqY∈ B.

(3) For any exact functor F:C → D satisfying F(B) = 0, there exists a unique

functor ¯

F:C/B → D such that F=¯

F◦QB.

REALISABILITY AND LOCALISATION 39

A pair (L:T → T ,Ψ: idC→L) is a localisation functor of triangulated categories if

Lis an exact localisation functor and Ψ commutes with the suspension functor in the

sense that Ψ ◦[1]C∼

=[1]C◦Ψ.

Then Ker Land TLare triangulated subcategories of T. Moreover, the functor

T → TL, X 7→ LX, is exact and induces an equivalence

T/Ker L≃

−→ TL.

It follows from Lemma 8.2 that we have a bijection between localisation functors

(L:T → T ,Ψ: idT→L) and quotient functors Q:T → T /Bhaving a fully faithful

right adjoint R(which are also called localisation functors). Observe that the adjunction

unit η: idT→RQ satisfies η◦[1]T∼

=[1]T◦ηbecause Ris fully faithful.

Definition 8.5. A localisation functor (L:T → T ,Ψ: idC→L) is called smashing if

Lcommutes with arbitrary direct sums.

A localising subcategory Bof Tis called smashing if Q:T → T /Badmits a fully

faithful right adjoint Rwhich commutes with arbitrary direct sums. Then the quotient

functor Q:T → T /Bis also called smashing localisation.

Note that the composition R◦Qcommutes with arbitrary direct sums if and only if

Rdoes. Hence we have a bijection between the smashing localisation functors in the

two different senses.

8.4. Localisation sequences. A sequence of exact functors

−→ B G

−→ C

between triangulated categories is called localisation sequence if the following conditions

hold:

(L1) The functor Fhas a right adjoint Fρ:B → A satisfying Fρ◦F∼

=idA.

(L2) The functor Ghas a right adjoint Gρ:C → B satisfying G◦Gρ∼

=idC, i.e. Gis

a localisation functor.

(L3) Let Xbe an object in B. Then GX = 0 if and only if X∼

=FX0for some

X0∈ A0.

The sequence (F, G) is called colocalisation sequence if the sequence (Fop, Gop) of op-

posite functors is a localisation sequence.

We recall the basic properties of a localisation sequence

Lemma 8.6 (Verdier [54], see also [35]).Let AF

−→ B G

−→ C be a localisation sequence.

Identify A= Im Fand C= Im Gρ.

(1) The functors Fand Gρare fully faithful.

(2) For given objects X, Y ∈ B, we have

X∈ A ⇐⇒ HomB(X, C) = 0,

Y∈ C ⇐⇒ HomB(A, Y ) = 0.

(3) The functor Ginduces an equivalence B/A≃C. Hence every triangulated func-

tor G0:B → C0satisfying G0◦F= 0 factors over G.

(4) For each X∈ T there is an exact triangle

(F◦Fρ)(X)→X→(Gρ◦G)X→Σ(F◦Fρ)X

which is functorial in X.

40 BIRGIT HUBER

(5) The sequence

CGρ

−−→ B Fρ

−→ A

is a colocalisation sequence.

8.5. Recollements. We say that a sequence

−→ B G

−→ C

of exact functors between triangulated categories induces a recollement

A//B

Fλ

Fρ

oo//C

Gλ

Gρ

if it is at the same time a localisation and a colocalisation sequence.

This means that the functors Fand Gadmit left adjoints Fλand Gλas well as right

adjoints Fρand Gρsuch that the adjunction morphisms

Fλ◦F∼

−→ idA

∼

−→ Fρ◦Fand G◦Gρ

∼

−→ idB

∼

−→ G◦Gλ

are isomorphisms.

Let Λ be a Noetherian ring. We denote by

I:Kac(Inj Λ) →K(Inj Λ)

the inclusion functor, and by Qthe canonical functor given by the composition

K(Inj Λ) inc

−−→ K(Mod Λ) can

−−→ D(Mod Λ).

Theorem 8.7. [35, Cor. 4.3] The sequence

Kac(Inj Λ) I

−→ K(Inj Λ) Q

−→ D(Mod Λ)

induces a recollement

(8.1) Kac(Inj Λ) //K(Inj Λ)

Iρ

Iλ

oo//D(Mod Λ).

Qλ

Qρ

Remark 8.8. [6, Sect. 5] The recollement (8.1) provides two embeddings of D(Mod Λ)

into K(Inj kG): The fully faithful functor Qρassigns to X∈D(Mod Λ) its homotopically

injective resolution iX introduced in Chapter 6.3. The other embedding is given by the

functor Qλ, which identifies D(Mod Λ) with the localising subcategory of K(Inj kG)

generated by iΛ.

Assume now that Λ is self-injective. Then Qλmaps a complex Xof Λ-modules to

its homotopically projective resolution pX. Furthermore, for every Λ-module M, the

canonical triangle

(8.2) pM →iM →tM →Σ(pM)

is isomorphic to the triangle

(8.3) (Qλ◦Q)( ¯

M)→¯

M→(I◦Iλ)¯

M→Σ(Qλ◦Q)¯

where ¯

M=QρM.

REALISABILITY AND LOCALISATION 41

Krause proved Theorem 8.7 more generally for a locally Noetherian Grothendieck

category Asuch that D(A) is compactly generated. In order to prove that (I, Q)

induces a localisation sequence, the essential point is the following proposition which is

also stated more generally in [35].

Proposition 8.9. [35, Prop. 2.3] The triangulated category K(Inj Λ) is compactly gen-

erated by the injective resolutions iM of the Noetherian modules M. Moreover, the

canonical functor K(Mod Λ) →D(Mod Λ) induces an equivalence

Kc(Inj Λ) ≃

−→ Db(mod Λ),

where Kc(Inj Λ) denotes the full subcategory of compact objects in K(Inj Λ).

The following lemma is well-known and easy to check.

Lemma 8.10. Let Λbe a Noetherian self-injective ring. The functor

Z0:Kac(Inj Λ) →Mod Λ

is an equivalence with quasi-inverse M7→ tM, where tM denotes a Tate resolution of

any representative of M∈Mod Λ.

Over a finite dimensional cocommutative Hopf algebra, the adjoints in the recollement

can be written down explicitly:

Proposition 8.11. [6] Let Hbe a finite dimensional cocommutative Hopf algebra. Then

the adjoints in the recollement (8.1) take the form

Mod H≃Kac(Inj H)//K(Inj H)

−⊗ktk

Homk(tk,−)

oo//D(Mod H).

Homk(pk,−)

−⊗kpk

In this case, the triangles (8.2) and (8.3) are isomorphic to

(8.4) M⊗kpk →M⊗kik →M⊗ktk →Σ(M⊗kpk).

8.6. Cohomological localisation. Cohomological localisations were first studied by

Bousfield [7]. Hovey, Palmieri and Strickland [25, Thm. 3.3.7] applied them in the

context of axiomatic stable homotopy theory. We refer to a paper of Krause [36] for a

more algebraic and detailed approach.

Throughout this section let Tbe a triangulated category which admits arbitrary

direct sums and is generated by a set of compact elements. We fix a compact object

A∈ T and write H∗for the functor T(A, −)∗.

Theorem 8.12. [36] Every exact localisation functor (L, Ψ) on Modgr Γextends to an

exact localisation functor (ˆ

L, ˆ

Ψ) on Tsuch that the diagram



H∗

//Modgr Γ



TH∗

//Modgr Γ

commutes up to a natural isomorphism. More precisely, it holds

42 BIRGIT HUBER

(1) The morphisms LH∗ˆ

Ψ,ΨH∗ˆ

Land

LH∗LH∗ˆ

Ψ//LH∗ˆ

(ΨH∗ˆ

L)−1

//H∗ˆ

are invertible.

(2) An object Xin Tis ˆ

L-acyclic if and only if H∗Xis L-acyclic.

(3) If X∈ T is ˆ

L-local, then H∗Xis L-local.

If Ais a generator of Tand Lpreserves arbitrary direct sums, then ˆ

Lpreserves arbitrary

direct sums.

Remark 8.13. [36, Rem. 2.4] Let L: Modgr Γ→Modgr Γ be an exact localisation

functor and denote by ˆ

L:T → T the exact localisation functor which exists by Theo-

rem 8.12. Write Cfor the ˆ

L-acyclic objects. By Lemma 8.2, ˆ

Land Lgive rise to adjoint

pairs of functors

Q//T/C

ooand Modgr Γ

F//(Modgr Γ)L

satisfying ˆ

L=R◦Qand L=G◦F. The diagram below commutes up to natural

isomorphism.



T(A,−)∗

//Modgr Γ



T/C



T/C(QA,−)∗

//(Modgr Γ)L



TT(A,−)∗

//Modgr Γ

Now assume that Γ is graded-commutative and consider the localisation functor

L: Modgr Γ→Modgr Γ given by localisation with respect to a multiplicatively closed

subset Sof Γ. See Chapter 3.1.2 for localisation of graded-commutative rings. The

following results in this section are joint work with K. Br¨uning [8].

Proposition 8.14. Suppose that the ring T(A, A)∗is graded-commutative and let

L: Modgr T(A, A)∗→Modgr T(A, A)∗be localisation with respect to a multiplicatively

closed subset of homogeneous elements S⊆ T (A, A)∗. Denoting C= Ker ˆ

L, the diagram



T(A,−)∗

//Modgr T(A, A)∗

can



T/CT/C(QA,−)∗

//Modgr S−1T(A, A)∗

commutes up to natural isomorphism. Furthermore, T/C(QA, QA)∗and S−1T(A, A)∗

are isomorphic not only as graded T(A, A)∗-modules, but also as graded rings.

REALISABILITY AND LOCALISATION 43

Proof. The diagram commutes by Remark 8.13. Writing again H∗for T(A, −)∗, the

naturality of H∗Ψ yields a commutative square

H∗AΨH∗A//

H∗ˆ

ΨA



LH∗A

LH∗ˆ

ΨA

∼



H∗ˆ

LA ΨH∗ˆ

∼

=//LH∗ˆ

in which the lower and the right hand side morphism are bijective by Theorem 8.12.

Now note that H∗ˆ

ΨAis up to isomorphism given by the canonical map

Q:T(A, A)∗→ T /C(QA, QA)∗, f 7→ Qf,

and that ΨH∗Aequals up to isomorphism the canonical ring homomorphism

can: T(A, A)∗→S−1T(A, A)∗.

Since Q:T/(A, A)∗→ T /C(QA, QA)∗is a multiplicative map inverting all elements in

S, we obtain a ring homomorphism r:S−1T(A, A)∗→ T /C(QA, QA)∗which makes the

upper triangle in the modified diagram

T(A, A)∗can //



S−1T(A, A)∗

S−1Q

∼



T/C(QA, QA)∗ν

∼

=//S−1T/C(QA, QA)∗

commute. We now show that ris bijective by proving the commutativity of the lower

triangle.

Since both the maps ν◦rand S−1Qmake the following diagram of T(A, A)∗-modules

T(A, A)∗µ//

ν◦Q



S−1T(A, A)∗

ν◦r

uullllllllllllll

S−1Q

uullllllllllllll

S−1T/C(QA, QA)∗

commute, the universal property of localisation of modules yields ν◦r=S−1Qand

hence the claim. 

Proposition 8.15. Suppose that the ring T(A, A)∗is graded-commutative and let

L: Modgr T(A, A)∗→Modgr T∗(A, A)∗be localisation with respect to a multiplicatively

closed subset of homogeneous elements S⊆ T (A, A)∗. If the compact object A∈ T is a

generator, then the category C= Ker ˆ

Lis generated by compact objects of T.

Proof. We show that Cis generated by {Cone(σ)|σ:A→A[n]∈S}. Let Mbe any

object of C. Using Lemma 2.1, it is enough to show that T(Cone(σ), M)∗= 0 for all

σ∈Simplies M= 0. By Lemma 2.3, every triangle

Aσ

−→ A[n]→Cone(σ)→A[1]

gives rise to an exact sequence

T(Cone(σ), M)∗→ T (A[n], M)∗T(σ,M)∗

−−−−−→ T (A, M)∗→ T (Cone(σ)[−1], M)∗.

44 BIRGIT HUBER

By assumption, we have T(Cone(σ)[−1], M)∗= 0 = T(Cone(σ), M)∗. Hence the map

T(A[n], M)∗T(σ,M)∗

−−−−−→ T (A, M)∗

is an isomorphism for all σ∈Sand thus, T(A, M)∗is S-local. On the other hand,

T(A, M)∗is S-acylic and so we conclude that T(A, M)∗= 0. It follows that M= 0

because Ais a compact generator. 

8.6.1. Cohomological p-Localisation. Let Abe a dg algebra such that H∗Ais graded-

commutative and let pbe a graded prime ideal of H∗A. Denote by Cpthe full subcategory

of objects Xin D(A) such that (H∗X)p= 0. In other words, Cpis the kernel of the

cohomological functor

(− ⊗H∗A(H∗A)p)◦ D(A)(A, −)∗.

From the previous discussion we obtain

Corollary 8.16. The localisation

D(A)

Q//D(A)/Cp

is smashing, and there is an isomorphism r:D(A)(A, A)∗

∼

−→ D(A)/Cp(QA, QA)∗of

graded rings making the diagram

D(A)(A, A)∗can //



D(A)(A, A)∗

∼

vvlllllllllllll

D(A)/Cp(QA, QA)∗

commutative. Furthermore, the squares

D(A)



D(A)(A,−)∗

//Modgr H∗A

−⊗H∗A(H∗A)p



D(A)D(A)(A,−)∗

//Modgr H∗A

D(A)/Cp

D(A)/Cp(QA,−)∗

//Modgr H∗ApD(A)/Cp

D(A)/Cp(QA,−)∗

//Modgr H∗Ap

inc

commute up to natural isomorphism. 

9. Realising smashing localisations by morphisms of dg algebras

The results in this chapter are joint work with K. Br¨uning [8], with substantial con-

tributions of Bernhard Keller. We show that every smashing localisation on a derived

category of a dg algebra can be realised by a morphism of dg algebras. More precisely,

if Ais a dg algebra and L:D(A)→ D(A) a smashing localisation, we prove the exis-

tence of a dg algebra ALwith the property D(A)/Ker L≃ D(AL), a dg algebra A0

quasi-isomorphic to Aand a zigzag of dg algebra morphisms

A∼

←− A0→AL

which identifies in cohomology with the algebra map L:D(A)(A, A)∗→ D(A)(LA, LA)∗.

If the dg algebra Ais cofibrant, then the algebra map Lidentifies with the cohomology

of a morphism A→AL, and the quotient functor is naturally isomorphic to the left

derived functor − ⊗L

AAL.

REALISABILITY AND LOCALISATION 45

As an application, we consider in Section 9.2 dg algebras with graded-commutative

cohomology ring. For such a dg algebra A, we introduce the localisation of Aat a prime p

in cohomology and denote this dg algebra by Ap. It has the property H∗(Ap)∼

=(H∗A)p.

Moreover, we show that with this identification of graded algebras, the canonical mor-

phism H∗A→(H∗A)pis induced by a zigzag of dg algebra morphisms.

9.1. Construction of a dg algebra morphism. Let Abe a differential graded algebra

over some commutative ring kand let

L:D(A)→ D(A)

be a smashing localisation. Denoting by Cthe category of L-acyclic objects, we have an

adjoint pair of functors

D(A)

Q//D(A)/C

satisfying R◦Q=L. The right adjoint Ris fully faithful and commutes with arbitrary

direct sums.

Our first aim is to write the quotient category D(A)/Cas derived category of a

differential graded algebra AL. Then we construct a zigzag of dg algebra morphisms

A∼

←− A0→ALwhich induces the algebra morphism

D(A)(A, A)∗→ D(A)(LA, LA)∗, f 7→ Lf,

in cohomology. For this purpose, we identify throughout this chapter the functors

H∗:D(A)→Modgr H∗Aand D(A)(A, −)∗:D(A)→Modgr H∗A(see Lemma 6.2).

The following lemma which we learned from Dave Benson is the key to our construc-

tion.

Lemma 9.1. Let A,Bdg algebras and Ma dg (B, A)-bimodule. Let α:A→Mand

β:B→Mbe maps of dg modules which satisfy α(1) = β(1). Then

X={(a, b)∈A×B|α(a) = β(b)}

is a dg algebra with differential dX= (dA, dB)and the projections p1, p2in the pullback

diagram

Xp2//



Aα//M

are dg algebra morphisms. If βis a surjective quasi-isomorphism, then the diagram

induces a pullback diagram in cohomology.

Proof. The first assertions are immediately checked. For the last one we show that

H∗X={(a, b)∈H∗A×H∗B|H∗α(a) = H∗β(b)}.

A pair (a, b)∈H∗Xtrivially satisfies the property H∗α(a) = H∗β(b) and conse-

quently, the inclusion ⊆is always fulfilled.

For the other inclusion we need to assume that βis a surjective quasi-isomorphism.

Let (a, b)∈H∗A×H∗Bsuch that H∗α(a) = H∗β(b). We choose representing cocycles

aof aand bof b. Then α(a)−β(b) = mfor some coboundary m∈M. Since βis a

surjective quasi-isomorphism, there is a coboundary b0∈Bsuch that β(b0) = m. Hence

the tuple (a, b +b0) satisfies α(a) = β(b+b0) and thus (a, b) = (a, b +b0)∈H∗X. 

46 BIRGIT HUBER

The following lemma ensures that the cohomology of the dg algebra ALwhich we

construct below is independent of all choices that we will make.

Lemma 9.2. Let X, Y be dg A-modules and let ν:X→Ybe an isomorphism in K(A).

(1) Denote by X→I(X)the injective hull of Xin the Frobenius category Moddg A.

There exists a dg algebra Sand a zigzag of quasi-isomorphisms of dg algebras

EndA(X)∼

←− S∼

−→ EndA(Y⊕I(X)).

(2) Let Ibe any injective module in the Frobenius category Moddg A. There is a dg

algebra Tand a zigzag of quasi-isomorphisms of dg algebras

EndA(Y)∼

←− T∼

−→ EndA(Y⊕I).

(3) There exists a zigzag of quasi-isomorphisms of dg algebras from EndA(X)to

EndA(Y).

Proof. (1) By Lemma 6.4, we can choose a representing dg A-module map

¯ν:X→Y⊕I(X)

of ν∈ K(A)(X, Y ) which is split as map of graded A-modules. Hence the map

¯ν∗:EndA(Y⊕I(X)) → HomA(X, Y ⊕I(X)), f 7→ f◦¯ν,

is surjective. Applying Lemma 9.1, the pullback

Sp2//



EndA(Y⊕I(X))



∼

¯ν∗



EndA(X)¯ν∗

∼//HomA(X, Y ⊕I(X))

gives rise to a pullback diagram in cohomology, and the object Sis actually a dg algebra.

In particular, we obtain quasi-isomorphisms of dg algebras

EndA(X)S

∼

oo∼

//EndA(Y⊕I(X)).

(2) The dg A-module map ι:Y[id 0 ]

−−−→ Y⊕Iis obviously a split monomorphism

inducing idYin the homotopy category. Hence we obtain a pullback diagram

Tp2

∼//

p1∼



EndA(Y⊕I)



∼

ι∗



EndA(Y)ι∗

∼//HomA(Y, Y ⊕I)

yielding the claim.

(3) is a trivial consequence of (1) and (2). 

The proof of the following lemma is immediate.

Lemma 9.3. The object QA is a compact generator of D(A)/C.

Fix a homotopically projective replacement of RQA ∈ D(A). By abuse of notation

we denote the replacement also by RQA.

REALISABILITY AND LOCALISATION 47

Proposition 9.4. The functor RHom(RQA, R−): D(A)/C → D(EndA(RQA)) is an

equivalence of triangulated categories.

Proof. We use Proposition 2.2 and first note that RHom(RQA, R−) preserves arbitrary

direct sums because for any family (Xi)i∈Iin D(A)/C, the map

i∈I

RHom(RQA, RXi)→RHom(RQA, R a

i∈I

Xi)

identifies in cohomology with the isomorphism

i∈I

D(A)(RQA, RXi)∼

i∈I

D(A)/C(QA, Xi)∼

=D(A)/C(QA, a

i∈I

Xi)∼

=D(A)(RQA, R a

i∈I

Xi).

Moreover, the functor RHom(RQA, R−) maps the compact generator QA of D(A)/C

to EndA(RQA) which compactly generates D(EndA(RQA)). Finally, the map

D(A)/C(QA, QA[n]) RHom(RQA,R−)

−−−−−−−−−−−→ D(EndA(RQA))(EndA(RQA),EndA(RQA)[n])

is an isomorphism for all n∈Zsince RQA being homotopically projective implies that

the diagram

D(A)/C(QA, QA[n])

R∼



RHom(RQA,R−)//D(EndA(RQA))(EndA(RQA),EndA(RQA)[n])

D(A)(RQA, RQA[n]) ∼

=//Hn(EndA(RQA))

∼

is commutative. 

Hence we have shown that the quotient category D(A)/Cis equivalent to the de-

rived category of the dg algebra EndA(LA), where LA was chosen to be homotopically

projective. Note that Lemma 9.2 provides a zigzag of quasi-isomorphisms between the

endomorphism dg algebras of two different homotopically projective replacements of an

object in D(A).

In order to construct a zigzag A∼

←− A0→ EndA(LA) of dg algebra morphisms inducing

D(A)(A, A)∗→ D(A)(LA, LA)∗

in cohomology, we need to make another choice on the dg A-module representing LA.

Let η: id →RQ be the unit and ε:QR →id the counit of the adjunction

D(A)

Q//D(A)/C.

Since Ais homotopically projective, we have

ηA∈ D(A)(A, RQA)∼

=K(A)(A, RQA).

Lemma 9.5. For any map ¯ηAin Moddg Athat represents ηA∈ K(A)(A, RQA)and any

dg A-module M, the map

¯η∗

A:HomA(RQA, RQM)→ HomA(A, RQM), f 7→ f◦¯ηA,

is a quasi-isomorphism.

48 BIRGIT HUBER

Proof. Since Ris fully faithful, the usual adjunction isomorphism (see [38, Ch. IV.1])

gives rise to the mutually inverse maps

Hn(¯η∗

A): D(A)(RQA, RQM[n]) → D(A)(A, RQM[n]), f 7→ f◦ηA,

and

D(A)(A, RQM[n]) → D(A)(RQA, RQM[n]), g 7→ R(εQA)◦RQ(g).

Remark 9.6. By Lemma 6.4, we may represent ηA:A→LA by a monomorphism of

dg A-modules

¯ηA:A→c

LA,

which is a split as map of graded A-modules. Remember that c

LA =LA ⊕I(A), where

A→I(A) is the injective hull of Ain the Frobenius category Moddg A, and LA was

already chosen to be homotopically projective. By Lemma 9.2, we have a zigzag of

quasi-isomorphisms

EndA(LA)∼

←− T∼

−→ EndA(c

LA).

We define the dg algebra ALto be EndA(c

LA). By abuse of notation we write AL=

EndA(LA). Note that from Lemma 9.2 and Proposition 9.4, it follows that

D(AL)≃ D(A)/C.

Theorem 9.7. The algebra map

D(A)(A, A)∗→ D(A)(LA, LA)∗, f 7→ L(f),

is induced by a zigzag of dg algebra maps

A∼

←− A0ϕ

−→ AL.

That is, there exists a dg algebra A0quasi-isomorphic to A, a morphism of dg algebras

ϕ:A0→ALand in cohomology, we have the commutative diagram

H∗A0

∼



H∗ϕ

((

D(A)(A, A)∗L//D(A)(LA, LA)∗

Proof. We identify the dg algebras EndA(A) and Athrough the isomorphism given by

evaluation at 1. Let

A0p2//

p1∼





∼¯η∗



EndA(A)¯ηA∗//HomA(A, LA)

be a pullback diagram.

The map ¯ηA∗is a quasi-isomorphism (Lemma 9.5), and surjective since ¯ηAis a split

monomorphism of graded A-modules (Remark 9.6). We infer from Lemma 9.1 that A0

is a dg algebra quasi-isomorphic to A, and we set ϕ=p2.

REALISABILITY AND LOCALISATION 49

In cohomology, we obtain the commutative diagram

H∗A0H∗(p2)//

H∗(p1)∼



H∗EndA(LA)

∼

=H∗(¯η∗



H∗EndA(A)H∗(¯ηA∗)//H∗(HomA(A, LA))

and thus it remains to show that the composition

H∗(¯η∗

A)−1◦H∗(¯ηA∗)

identifies with the map

D(A)(A, A)∗→ D(A)(LA, LA)∗, f 7→ L(f).

In fact, for f∈ D(A)(A, A)∗we have

H∗(¯η∗

A)−1◦H∗(¯ηA∗)(f) = R(εQA)◦RQ(ηA)◦RQ(f)∈ D(A)(RQA, RQA).

But it is well-known that εQA ◦Q(ηA)∼

=idQA (see [38, Ch. IV.1]) and hence the claim

follows. 

Observe that the map ϕ:A0→ALis a monomorphism: Since ¯ηAis split as map of

graded A-modules, ¯ηA∗is injective and so is ϕ=p2.

If we assume in addition that Ais a cofibrant dg algebra (see Section 6.5), then the

algebra map L:D(A)(A, A)∗→ D(A)(LA, LA)∗is not only induced by a zigzag of dg

algebra maps, but by a morphism A→AL.

Corollary 9.8. Let Abe a cofibrant dg algebra. The algebra morphism

D(A)(A, A)∗→ D(A)(LA, LA)∗, f 7→ L(f),

lifts to a dg algebra morphism ψ:A→AL.

Proof. Since Ais cofibrant, the map p1:A0→Ain the pullback diagram

A0p2//



p1∼





∼¯η∗



A¯ηA∗//HomA(A, LA)

splits: There is a morphism of dg algebras s:A→A0such that p1◦s= idA. We define

ψto be the composition

p2◦s:A→AL.

Then H∗ψidentifies with the canonical map L:D(A)(A, A)∗→ D(A)(LA, LA)∗.

Now our aim is to show that if Ais cofibrant, then we can identify the functors

Q:D(A)→ D(A)/C ≃ D(AL) and −⊗L

AAL:D(A)→ D(AL), where ALis a dg (A, AL)-

bimodule through the morphism ψ:A→AL.

Lemma 9.9. There exists a natural transformation

λ:RHomA(A, −)→RHomA(LA, L−)

in D(A)which commutes with the suspension functor. For every M∈ D(A),λMinduces

the map

D(A)(A, M)→ D(A)(LA, LM), f 7→ Lf,

in cohomology.

50 BIRGIT HUBER

Proof. By Lemma 9.5, the adjunction unit ηA:A→LA induces a natural isomorphism

RHomA(ηA, LM). Therefore we can define the morphism λMto be the composition

RHomA(A, M)RHom(A,ηM)//RHomA(A, LM)RHom(ηA,LM)−1

∼

=//RHomA(LA, LM),

which obviously induces L:D(A)(A, M)∗→ D(A)(LA, LM) in cohomology. The natu-

rality of λMfollows from the naturality of RHom(A, ηM) and RHom(ηA, LM).

The unit ηof the adjoint pair (Q, R) commutes with the suspension functor [1], hence

so does RHom(A, ηM). Since RHom(ηA, LM) commutes with [1], we conclude that

λ◦[1] ∼

=[1] ◦λ.

Note that if Ais cofibrant, then λAequals the dg algebra morphism ψ:A→AL

constructed in Corollary 9.8. In addition, RHomA(LA, LM) becomes an object in

D(A) through the dg algebra morphism ψ.

Proposition 9.10. Suppose that Ais a cofibrant dg algebra. Then the diagram

D(A)



−⊗L

AAL//D(AL)

D(A)/C

RHomA(RQA,R−)

≃66

commutes up to natural isomorphism.

Proof. We show that the functors RHomA(LA, L−) and −⊗L

AALare naturally isomor-

phic. A natural transformation

τ:− ⊗L

AAL−→ RHomA(LA, L−)

is given as composition of the three natural maps in the diagram

M⊗L

AAL

τM//

canM⊗L

AAL∼



RHomA(LA, LM)

RHomA(A, M)⊗L

AAL

λM⊗L

AAL//RHomA(LA, LM)⊗L

AAL

νM

where canMis the canonical identification and νMis defined by

νM:RHomA(LA, LM)⊗L

AEndA(LA)→RHomA(LA, LM).

f⊗g7→ f◦g

Note that τcommutes with the suspension functor since this holds for λby Lemma 9.9,

and obviously for can and ν. In order to prove that τis an isomorphism, it suffices to

show that the full subcategory A={M∈ D(A)|τMis an isomorphism}of D(A) is a

localising subcategory containing A.

First we point out that Ais closed under triangles by the Five-Lemma for triangulated

categories, and that it is easy to check that A∈ A.

Furthermore, Ais closed under taking shifts because τcommutes with the suspension

functor.

REALISABILITY AND LOCALISATION 51

Finally, we show that Ais closed under taking arbitrary direct sums. To that end,

recall that the functor RHomA(LA, R−) commutes with arbitrary direct sums (Propo-

sition 9.4) and hence, so does RHomA(LA, L−). Since − ⊗L

AALobviously commutes

with arbitrary direct sums, the claim follows. 

Remark 9.11. Let Tbe an algebraic triangulated category in the sense of Keller. We

refer to [34, Sect. 6.5] for a definition. If Tis generated by a single compact object, then

there is a dg algebra Asuch that T ≃ D(A), see [34, Sect. 6.5]. Since we can choose the

dg algebra Ato be cofibrant (see Lemma 6.10), every smashing localisation L:T → T

is induced by a morphism of dg algebras A→AL.

9.2. The p-localisation of a dg algebra. Let Abe a dg algebra over a commutative

ring kand assume throughout this section that the cohomology algebra H∗Ais graded-

commutative. We fix a graded prime ideal pof H∗A. By Cpwe denote the full subcate-

gory of objects Min D(A) such that (H∗M)p= 0. The localisation Lp:D(A)→ D(A),

given by the adjoint pair

D(A)

Q//D(A)/Cp,

is smashing by Corollary 8.16. Now we apply the results of Section 9.1 to this special

case. We define

Ap=ALp,

and we call Aplocalisation of Aat a prime pin cohomology.

From Lemma 9.2 and Proposition 9.4, we infer that D(A)/Cp≃ D(Ap). For this

special smashing localisation, we have

Theorem 9.12. Let Abe a dg algebra over a commutative ring ksuch that H∗Ais

graded-commutative and let pbe a graded prime ideal of H∗A. The dg algebra Aphas

the property H∗(Ap)∼

=(H∗A)p. Moreover, with this identification of graded algebras,

the canonical map

can: H∗A→(H∗A)p

is induced by a zigzag of dg algebra maps

A∼

←− A0ϕ

−→ Ap.

That is, we have a commutative diagram

H∗A0

∼



H∗ϕ

H∗Acan //(H∗A)p∼

=//H∗(Ap)

Proof. Since D(A)(A, A)p∼

=D(A)(LpA, LpA) by Corollary 8.16, the dg algebra Apsatis-

fies H∗(Ap)∼

=(H∗A)p.Theorem 9.7 shows that the zigzag A∼

←− A0ϕ

−→ Apinduces the

map

Lp:D(A)(A, A)∗→ D(A)(LpA, LpA)∗, f 7→ Lp(f)

in cohomology. But we may identify the algebra maps can and Lpby Corollary 8.16. 

The following result is an immediate consequence of Corollary 9.8 and Theorem 9.12.

52 BIRGIT HUBER

Corollary 9.13. Let Abe a cofibrant dg algebra such that H∗Ais graded-commutative

and let pbe a graded prime ideal of H∗A. Then the canonical algebra morphism

can: H∗A→(H∗A)plifts to a dg algebra morphism

ψ:A//Ap.

A class of cofibrant dg algebras with graded-commutative cohomology are the Sullivan

algebras (ΛV, d) introduced Section 6.5.

Remark 9.14. The smashing localisation Lp:D(A)→ D(A) can be interpreted as

p-localisation on the derived category. It satisfies a local-global principle:

For M∈ D(A) and a graded prime pof H∗A, we define

Mp=RHomA(LpA, LpM)∈ D(Ap)

and call Mplocalisation of Mat a graded prime ideal pof H∗A. If Ais a cofibrant dg

algebra, then we have Mp=M⊗L

AApby Proposition 9.10. Since (H∗M)p∼

=H∗(Mp),

the following conditions are equivalent for M∈ D(A):

(1) M= 0.

(2) Mp= 0 for all graded prime ideals p.

(3) Mm= 0 for all graded maximal ideals m.

9.2.1. A universal property of Ap.Let Abe a dg algebra over a commutative ring k

such that H∗Ais graded-commutative and p∈Specgr(H∗A).

The cohomology of the dg algebra Apsatisfies a universal property since H∗(Ap)

is isomorphic to the ring of fractions S−1(H∗A)=(H∗A)p, where Sis the subset of

homogeneous elements in H∗A\p. If β:A→Bis a morphism of dg algebras such

that H∗βmakes Sinvertible, then H∗βfactors uniquely over the canonical morphism

can: H∗A→(H∗A)p.

Without loss of generality, we assume from now on that Ais cofibrant. Then can is

induced by a morphism of dg algebras ψ:A→Apand the universal property yields a

unique algebra morphism g:H∗(Ap)→H∗Bmaking the following diagram commute:

H∗AH∗β//

H∗ψ



H∗B

H∗(Ap)

The dg algebra morphisms β:A→Band ψ:A→Apgive rise to functors

Fβ:D(A)−⊗L

−−−−→ D(B) and Fψ:D(A)−⊗L

AAp

−−−−−→ D(Ap).

Now we prove a universal property on the level of derived categories.

Proposition 9.15. There is a unique functor G:D(Ap)→ D(B)making the following

diagram commute:

D(A)Fβ//

Fψ



D(B)

D(Ap)

REALISABILITY AND LOCALISATION 53

Proof. We first note that by Proposition 9.10, the functor Fψis nothing but the quotient

functor Q:D(A)→ D(A)/Cpcomposed with the equivalence D(A)/Cp≃ D(Ap). Thus

we can use the universal property of Qand only need to show that Fβ(Cp) = 0.

In Proposition 8.15 we have shown that

M={Cone(σ)|σ:A→A[n]∈S}

is a set of compact generators of Cpand thus it suffices to check that Fβvanishes on M.

Any element of Mfits into an exact triangle

Ax·

−→ A[n]→Cone(x·)→A[1]

in D(A), where x·denotes multiplication with an element x∈Awhose cohomology H∗x

belongs to S. Applying the functor Fβto this triangle, we obtain a triangle in D(B)

naturally isomorphic to

Bβ(x)·

−−−→ B[n]→Fβ(Cone(x·)) →B[1].

Since H∗β(x) is invertible, we infer that Fβ(Cone(x·)) is contractible and consequently,

the object Fβ(Cone(x·)) is zero in D(B). 

Since Cpis generated by compact elements, the quotient functor Q:D(A)→ D(A)/Cp

gives rise to a quotient functor Dper(A)→ Dper(A)/Cper

p, where Cper

p=Cp∩ Dper(A).

Furthermore, this quotient functor identifies with the functor

Dper(A)−⊗L

AAp

−−−−−→ Dper(Ap).

This proves

Corollary 9.16. There is a unique functor G:Dper(Ap)→ Dper(B)which makes the

following diagram commute:

Dper(A)Fβ//

Fψ



Dper(B)

Dper(Ap)



Remark. The discussion above raises the question whether the functor G:D(Ap)→

D(B) and with it the algebra map g:H∗(Ap)→H∗Bcan be lifted to a zigzag of dg

algebra morphisms. Our construction in Section 9.1 does not apply since in general,

we cannot expect that Gis a smashing localisation. It remains to enlighten the rela-

tion of our construction with DG quotients, which have a universal property and were

introduced by Drinfeld [16].

There is also a construction by To¨en [53] (see also in [31]) which seems to be related:

Let dgcatkbe the category of small dg categories over a commutative ring k. The

localisation of dgcatkwith respect to the quasi-equivalences is denoted by Hqe. If Ais

a small dg category and Sa set of morphisms in H0(A), then a morphism F:A → B

in Hqe is said to make Sinvertible if the induced functor H0(A)→H0(B) takes each

s∈Sto an isomorphism. To¨en constructs a morphism Q:A→A[S−1] in Hqe which

makes Sinvertible. This morphism has a universal property: Each morphism in Hqe

making Sinvertible factors uniquely through Q.

54 BIRGIT HUBER

However, if Ais a dg algebra, viewed as dg category with a single object, then the

object A[S−1] is in general not a dg algebra, but a dg category with more than one

object.

10. Realisability

In this chapter we introduce the concept of realisability as considered in a paper of

Benson, Krause and Schwede [5]. They are concerned with deciding whether a graded

module over the Tate cohomology ring ˆ

H∗(G, k), where Gis a finite group and ka field,

is isomorphic to ˆ

H∗(G, M) for some kG-module M.

More generally, they consider a triangulated category Tadmitting arbitrary direct

sums and a cohomological functor H∗:T → Modgr Einto the category of graded mod-

ules over a graded ring E. The functor H∗is required to preserve arbitrary direct sums

and products. Then realisability deals with deciding whether a graded E-module is

isomorphic to a module in the image of this cohomological functor.

Benson, Krause and Schwede [5] have stated a local obstruction for realisability up to

direct summands, and a criterion for realisability which is given by an infinite sequence

of obstructions.

If Ais a dg algebra over a field k, then the functor in question is the cohomology

functor H∗:D(A)→Modgr H∗A. In this setting, Benson, Krause and Schwede also

prove the existence of a global obstruction for realisability up to direct summands.

In the first section we introduce the general setup of Benson, Krause and Schwede [5]

and recall the construction of the local obstruction. After focusing on realisability in

the setting of dg algebras in the second section, we study the global obstruction and its

basic properties in Section 10.3.

10.1. A local obstruction for realisability. Let Tbe a triangulated category ad-

mitting arbitrary direct sums. We denote the suspension functor by Σ. Let Nbe a

compact object in Tand E=T(N, N)∗=`i∈ZT(N, ΣiN) the graded endomorphism

ring of N(see Example 3.1). If Xis a graded E-module, then we denote by X[n] the

n-fold shifted module.

If Mis an object in T, then we obtain a graded E-module T(N, M)∗by composition of

graded maps. On the other hand, given any graded E-module X, when is Xisomorphic

to T(N, M)∗for some M∈ T ?

We will mainly consider this question only up to direct summands. Therefore, from

now on, we use the following terminology for realisability2:

Definition 10.1. Let Xbe a graded E-module. We call Xrealisable if there exists

an object M∈ T such that Xis isomorphic to a direct summand of T(N, M)∗. If

X∼

=T(N, M)∗for some M∈ T , then we call Xstrictly realisable.

Remark 10.2. Note that the functor T(N, −)∗occurs in a very natural way: For every

graded ring Λ and every cohomological functor H∗:T → Modgr Λ which preserves

arbitrary direct sums and products, there exists a compact object C∈ T such that H∗

is naturally isomorphic to T(C, −)∗[36, Lemma 3.2].

Benson, Krause and Schwede [5] have constructed a local obstruction for realisability:

2Our terminology is different from the one in [5]. Benson, Krause and Schwede call an E-module

realisable if X∼

=T(N, M)∗for some M∈ T .

REALISABILITY AND LOCALISATION 55

Theorem 10.3. [5, Thm. 3.7] Let Tbe a triangulated category with arbitrary direct

sums, N∈ T a compact object and E=T(N, N)∗the graded endomorphism algebra of

N. For each graded E-module X, there exists an element

κ(X)∈Ext3,−1

E(X, X)

determining the realisability: Xis realisable if and only if κ(X)is trivial.

For the proof of Theorem 10.3 and more details we refer to [5]. However, for our

purposes we sketch the construction of the local obstruction κ(X):

Construction 10.1. AT-presentation of a graded E-module Xconsists of a distin-

guished triangle

(10.2) Σ−1Bδ

−→ R1α

−→ R0π

−→ B

in Ttogether with an epimorphism of graded E-modules ε:T(N, R0)∗→Xsuch that

the sequence

T(N, R1)∗α∗

−→ T (N, R0)∗ε

−→ X→0

is exact. If the objects R0and R1are assumed to be N-free, that is, isomorphic to a

direct sum of shifted copies of N, then we refer to an N-special T-presentation.

Given an N-special T-presentation

(Σ−1Bδ

−→ R1α

−→ R0π

−→ B, ε:T(N, R0)∗→X),

we obtain an exact sequence of graded E-modules

(10.3) 0 →X[−1] η[−1]

−−−→ T (N, B)∗[−1] δ∗

−→ T (N, R1)∗α∗

−→ T (N, R0)∗ε

−→ X→0

by applying the functor T(N, −)∗to the triangle. The monomorphism η:X→ T (N, B)∗

is determined by η◦ε=π∗.

This sequence is called associated extension of the N-special T-presentation.

In [5, Prop. 3.4] it is shown that there exists an N-special T-presentation for each

graded E-module X, and that the Yoneda-class of the associated extension, denoted by

κ(X)∈Ext3,−1

E(X, X), is independent of the choice of the N-special T-presentation.

Since T(N, R0)∗and T(N, R1)∗are free, the extension κ(X) is trivial if and only if

the monomorphism

X[−1] η[−1]

−−−→ T (N, B)∗[−1]

is split. Thus κ(X) determines, in fact, the realisability of X.

Remark 10.4. Let Tbe a triangulated category admitting direct sums and N∈ T

a compact object. Denote by E=T(N, N)∗the graded endomorphism algebra of N.

Benson, Krause and Schwede extend their theory by an infinite sequence of obstructions

which decides whether a graded E-module Xis strictly realisable.

They show that if there exists an infinite sequence of obstructions

κn(X)∈Extn,2−n

E(X, X), n ≥3,

where the class κn(X) is defined provided that the previous one κn−1(X) vanishes, then

it even holds X∼

=T(N, M)∗. In this sequence of obstructions all but the first one

depend on choices. Only κ3(X) is uniquely determined and equals the local obstruction

κ(X).

56 BIRGIT HUBER

In view of the need for an infinite sequence of obstructions to decide whether

X∼

=T(N, M)∗for some M∈ T , it is remarkable that the first obstruction of this

sequence already tells whether Xis a direct summand of T(N, M)∗.

In our results we will only consider this first obstruction.

10.2. Realisability and dg algebras. Let Abe a dg algebra over some commuta-

tive ring k. Remember that the functor D(A)(A, −)∗is naturally isomorphic to the

cohomology functor H∗(Lemma 6.2). Hence a graded H∗A-module is realisable if and

only if it is a direct summand of H∗M, where M∈ D(A) is a dg A-module. Moreover,

Ais a compact object in D(A). Hence Theorem 10.3 applies, and we can decide whether

Xis a direct summand of H∗Mfor some dg A-module M∈ D(A).

Example 10.5. (1) Let kbe a Noetherian ring and Ga finite group. Remember that

the group cohomology ring H∗(G, k) is actually a graded endomorphism ring of an object

in a triangulated category: We have H∗(G, k)∼

=K(Inj kG)(ik, ik)∗, where ik denotes

an injective resolution of k. Moreover, K(Inj kG) admits arbitrary direct sums since

Inj kG does. Since ik is compact in K(Inj kG) by Proposition 8.9, the assumptions of

Theorem 10.3 are satisfied.

On the other hand, we know from Remark 6.1(2) that K(Inj kG)(ik, ik)∗is the coho-

mology of the endomorphism dg algebra End(ik) of the complex ik,

H∗End(ik)∼

=H∗(G, k),

and we can also consider realisability in the setting of dg algebras. Now note that we

have a commutative square

K(Inj kG)

Hom(ik,−)



K(Inj kG)(ik,−)∗

//Modgr H∗(G, k)

∼



D(End(ik)) H∗

//Modgr H∗End(ik)

Since the exact functor Hom(ik, −): K(Inj kG)→ D(End(ik)) commutes with arbitrary

direct sums and is dense (see for example [6, Prop. 3.1]), a graded H∗(G, k)-module is

realisable by a complex C∈K(Inj kG) if and only if it is realisable by a dg End(ik)-

module M.

Observe that H∗(G, k) is also isomorphic to the graded endomorphism ring

D(kG)(k, k)∗. However, except in trivial cases, the stalk complex kis not a compact

object in D(kG) and so the assumptions for Theorem 10.3 are not fulfilled.

(2) Let kbe a field and Gbe a finite group. The Tate cohomology ring ˆ

H∗(G, k) is

the graded endomorphism ring HomkG(k, k)∗and kis compact in Mod kG. With the

equivalence Z0:Kac(Inj kG)≃

−→ Mod kG (Lemma 8.10), we can write ˆ

H∗(G, k) also as

graded endomorphism ring Kac(Inj kG)(tk, tk)∗and conclude that

H∗End(tk)∼

=ˆ

H∗(G, k).

REALISABILITY AND LOCALISATION 57

Similarly as in (1), we have a commutative square

Kac(Inj kG)

Hom(tk,−)



Kac(Inj kG)(tk,−)∗

//Modgr ˆ

H∗(G, k)

∼



D(End(tk)) H∗

//Modgr H∗End(tk)

and Hom(tk, −) is exact, dense and preserves arbitrary direct sums. Hence a graded

H∗(G, k)-module is realisable by an acyclic complex (or equivalently, by an object of

Mod kG) if and only if it is realisable by an object of D(End(tk)).

Note that if kis a field of characteristic p > 0 and Gap-group, then the functors

Hom(ik, −) and Hom(tk, −) are even equivalences [6].

Considering realisability in the setting of dg algebras has a striking advantage, as the

following section indicates.

10.3. A global obstruction for realisability. Let Abe a dg algebra over a field k.

We have seen in Chapter 7 that H∗Ais an A∞-algebra whose secondary multiplication

determines a Hochschild class µA∈HH3,−1(H∗A), called the canonical class. Now the

importance of this result for the realisability theory becomes evident:

Theorem 10.6. [5, Thm. 6.2] Let Xbe a graded H∗A-module. The realisability ob-

struction

κ(X)∈Ext3,−1

H∗A(X, X)

is given by the cup product pairing

idX∪µA.

In particular, if the class µA∈HH3,−1(H∗A)is trivial, then all graded H∗A-modules

are realisable.

Because of the last property the class µAis referred to as global obstruction.

Remark. The converse of the last statement in Theorem 10.6 is not true in general:

Benson, Krause and Schwede provide an example of a dg algebra Awith the property

that all H∗A-modules are realisable, but with non-trivial canonical class µA[5, Exm.

5.15].

11. Realisability and p-localisation

In this chapter we study the relation between realisability of modules over graded-

commutative cohomology rings and p-localisation.

A first motivation for our use of p-localisation is the problem when a module over the

graded-commutative ring H∗(G, k) is isomorphic to H∗(G, M) for some kG-module M.

This is discussed in Section 11.1.

More generally, we consider dg algebras with graded-commutative cohomology rings

in Section 11.2. In our main result of this chapter we prove that the classical local-global

principle of Commutative Algebra applies for realisability.

58 BIRGIT HUBER

11.1. A motivation for p-localisation. Let Gbe a finite group and ka field such that

Char(k) divides the order of G. Let Xbe a graded module over the group cohomology

ring H∗(G, k). With Theorem 10.3 we can determine whether Xis realisable by a

complex in the homotopy category K(Inj kG). However, we rather want to know when

Xis realisable by a module, i.e. Xis a direct summand of H∗(G, M) with M∈Mod kG,

or even X∼

=H∗(G, M).

The category Mod kG is embedded in K(Inj kG), but with Theorem 10.3 we cannot

decide whether an arbitrary realisable H∗(G, k)-module can actually be realised by a

module. Now we show that one can say more about realisability of p-local modules,

where pis a non-maximal prime.

Throughout this section we denote by H∗(G) the group cohomology ring of G. Re-

member that H∗(G) is graded-commutative (Section 4.1).

Let pbe a graded prime ideal of H∗(G) and denote by Cpthe kernel of the cohomo-

logical functor

HomK(ik, −)∗

p=K(Inj kG)(ik, −)∗⊗H∗(G)H∗(G)p:K(Inj kG)→Modgr H∗(G)p.

From Theorem 8.12 we obtain a smashing localisation

K(Inj kG)

//K(Inj kG)/Cp.

On the other hand, we have a smashing localisation

K(Inj kG)

//Mod kG,

where ˆ

Q(C) = Z0(C⊗ktk) and ˆ

R(M) = tM, see Chapter 8.5.

Lemma 11.1. If pis non-maximal, then Qpfactors over ˆ

Q, i.e. there is a functor

Q: Mod kG →K(Inj kG)/Cp

such that Qp=¯

Q◦ˆ

Q. Moreover, ¯

Qhas a right adjoint ¯

Rsatisfying ¯

R¯

Q∼

=id and it

holds Rp=ˆ

R¯

Proof. By Proposition 8.11, we have a localisation sequence

D(Mod kG)−⊗kpk

−−−−→ K(Inj kG)ˆ

Q=Z0(−⊗ktk)

−−−−−−−−−→ Mod kG.

Consequently, in order to obtain the functor ¯

Q, it suffices to show that the composition

Qp◦(− ⊗kpk) is zero (see Lemma 8.6(3)).

Now observe that HomK(ik, kG)∗∼

=HomkG(H0(ik), kG)∼

=k. Since pis non-

maximal, it holds kp= 0 and hence kG is contained in Cp. We infer that the composition

Qp◦(− ⊗kpk) vanishes on kG, and by d´evissage on all objects of D(Mod kG).

The functor ¯

Q: Mod kG →K(Inj kG)pwe have obtained that way commutes with

arbitrary direct sums since this holds for ˆ

Qand Qp, and because ˆ

Qis dense. Further-

more, the category Mod kG is compactly generated by the finite dimensional modules

(see [47]), thus ¯

Qhas a right adjoint ¯

Rby Proposition 2.4. Obviously it holds Rp=ˆ

R¯

and we conclude that ¯

R¯

Q= id. 

Remember that a H∗(G)-module Xis p-local if X∼

=Xpas H∗(G)-modules.

REALISABILITY AND LOCALISATION 59

Proposition 11.2. Let pbe a non-maximal prime and assume that X∈Modgr H∗(G, k)

is a p-local module. If Xis realisable, then it can be realised by a Tate resolution tM of

some kG-module M. Furthermore, Xpis isomorphic to a direct summand of H∗(G, M)p.

Proof. From Theorem 8.12 and Remark 8.13 we obtain diagrams

K(Inj kG)



Hom(ik,−)∗

//Modgr H∗(G)

−⊗H∗(G)p



K(Inj kG)Hom(ik,−)∗

//Modgr H∗(G)

K(Inj kG)/Cp

Hom(Qp(ik),−)∗

//Modgr H∗(G)pK(Inj kG)/Cp

Hom(Qp(ik),−)∗

//Modgr H∗(G)p

inc

which commute up to isomorphism. Let Xbe a graded p-local module over H∗(G).

If Xis realisable by some complex C∈K(Inj kG), then Xpis realisable by an object

Qp(C)∈K(Inj kG)/Cp. Hence X∼

=inc(Xp) is realisable by RpQp(C). We infer from

Lemma 11.1 that Xcan be realised by ˆ

R(M), where Mdenotes ¯

RQp(C)∈Mod kG.

But the functor ˆ

Rassigns to Mits Tate resolution tM.

For the second claim, we show that HomK(ik, tM)∗

p∼

=H∗(G, M)p. Applying the

cohomological functor HomK(ik, −)pto the canonical triangle

pM →iM →tM →pM[1]

yields an exact sequence

HomK(ik, pM[−1])∗

p→HomK(ik, iM)∗

p→HomK(ik, tM)∗

p→HomK(ik, pM)∗

of graded H∗(G)-modules. In the proof of Lemma 11.1 we have shown that pM and

pM[−1] lie in the kernel of HomK(ik, −)p. Thus H∗(G, M)pand HomK(ik, tM)∗

pare,

in fact, isomorphic. 

Observe that the proof also shows that for a strictly realisable, graded H∗(G, k)-

module Xwhich is moreover p-local, there exists a module M∈Mod kG such that

X∼

=HomK(ik, tM) and Xp∼

=H∗(G, M)p.

Remark. If the module H∗(G, M) in Proposition 11.2 is p-local, then Xis realisable by

the kG-module M. However, in general we cannot expect H∗(G, M) to be p-local.

11.2. Realisability is a local property. In this section we show that the classical

local-global principle applies for realisability. Let Abe a differential graded algebra

over a commutative ring k, and assume that the cohomology ring H∗Ais graded-

commutative. Under some finiteness assumptions we prove that a graded H∗A-module

is realisable if and only if Xpis realisable for all graded prime ideals pof H∗A.

Lemma 11.3. Let Abe a dg algebra with graded-commutative cohomology ring H∗A

and fix a graded prime ideal pof H∗A. If a graded H∗A-module Xis (strictly) realisable,

then Xpis (strictly) realisable.

Proof. We use the commutative diagrams

D(A)



D(A)(A,−)∗

//Modgr H∗A

−⊗H∗A(H∗A)p



D(A)D(A)(A,−)∗

//Modgr H∗A

D(A)/Cp

D(A)/Cp(QA,−)∗

//Modgr H∗ApD(A)/Cp

D(A)/Cp(QA,−)∗

//Modgr H∗Ap

inc

60 BIRGIT HUBER

from Corollary 8.16. If Xis realisable, then the H∗Ap-module Xpis realisable by an

object of D(A)/Cp. But then the H∗A-module Xpis realisable by an object of D(A), by

the right hand diagram. 

Now our aim is to show that Xpbeing realisable for all graded primes pof H∗Aimplies

that Xis realisable. On that purpose, we study the behaviour of the local obstruction

κ(X)∈Ext3,−1

H∗A(X, X) under p-localisation. The following lemma is stated in a more

generally setting, but applies to this situation.

Lemma 11.4. Let T,Ube triangulated categories with arbitrary direct sums and let

N∈ T be a compact object. Assume that

F:T → U

is an exact functor such that FN is compact in U, and that

F: Modgr T(N, N)∗→Modgr U(F N, FN)∗

is an exact functor such that the diagram below commutes up to natural isomorphism.



T(N,−)∗

//Modgr T(N, N)∗



UU(FN,−)∗

//Modgr U(FN, FN)∗

Then we have for every graded T(N, N)∗-module X

F(κ(X)) = κ(e

FX)in Ext3,−1

U(FN,F N)∗(e

FX, e

FX).

Proof. Let (Σ−1Bδ

−→ R1α

−→ R0π

−→ B, ε:T(N, R0)∗→X) be an N-special T-presenta-

tion with associated extension κ(X)∈Ext3,−1

T(N,N)∗(X, X). Since Fis exact and preserves

direct sums, we obtain a triangle Σ−1FB Fδ

−−→ FR1F α

−−→ FR0F π

−−→ FB in U, where FR0

and FR1are FN-free. The exactness of ˜

Fand the commutativity of the diagram yield

an epimorphism ζ:U(FN, F R0)∗→e

FX. Consequently

(Σ−1FB Fδ

−−→ FR1F α

−−→ FR0F π

−−→ FB, ζ :U(F N, FR0)∗→e

FX)

is an FN-special U-presentation and the commutativity of the diagram

0//(e

FX)[−1] //

∼



U(FN, Σ−1F B)∗(F δ)∗

∼



U(FN, F R1)∗(F α)∗

∼



U(FN, F R0)∗ζ//

∼



FX //0

0//e

F(X[−1]) //e

F(T(N, Σ−1B)∗)

F(δ∗)

//e

F(T(N, R1)∗)

F(α∗)

//e

F(T(N, R0)∗)//e

FX //0

shows that κ(e

FX) = e

F(κ(X)) in Ext3,−1

U(FN,F N)∗(e

FX, e

FX). 

The next lemma is well-known for strictly commutative Noetherian rings (see for

example [9]), and it is easy to check that it also holds true for graded-commutative

coherent rings. Remember that a graded ring Ris called coherent if finitely generated

graded submodules of finitely presented graded R-modules are assumed to be finitely

presented.

REALISABILITY AND LOCALISATION 61

Lemma 11.5. Let Rbe a graded-commutative and coherent ring, and let pbe a graded

prime ideal of R. Let M, N be graded R-modules and assume that Mis finitely presented.

Then there is a natural isomorphism

Exti,∗

R(M, N)p∼

=Exti,∗

Rp(Mp, Np)

for all i≥0.

Theorem 11.6 (Local-global principle).Let Abe a dg algebra such that H∗Ais graded-

commutative and coherent. The following conditions are equivalent for a finitely pre-

sented graded H∗A-module X:

(1) Xis realisable.

(2) Xpis realisable for all graded prime ideals pof H∗A.

(3) Xmis realisable for all graded maximal ideals mof H∗A.

Proof. By Lemma 11.3, it suffices to show that Xis realisable if Xmis realisable by an

object in D(A)/Cmfor all graded maximal ideals mof H∗A.

The H∗A-module Xis realisable if and only if the class κ(X)∈Ext3,−1

H∗A(X, X) is

trivial (Theorem 10.3). By Proposition 3.6, the latter holds true if and only if the

fraction κ(X)

1in Ext3,−1

H∗A(X, X)mequals zero for all graded maximal ideals mof H∗A.

But since Xis finitely presented and H∗Ais assumed to be graded-commutative and

coherent, we may apply the natural isomorphism

(Ext3,∗

H∗A(X, X))m∼

=Ext3,∗

(H∗A)m(Xm, Xm)

from Lemma 11.5, which maps the fraction κ(X)

1to the extension κ(X)⊗H∗A(H∗A)m.

The object QA ∈ D(A)/Cmis compact (Lemma 9.3), hence we can apply Lemma 11.4

to the commutative diagram

D(A)



D(A)(A,−)∗

//Modgr H∗A

−⊗H∗A(H∗A)m



D(A)/Cm

D(A)/Cm(QA,−)∗

//Modgr H∗Am

and obtain

κ(X)⊗H∗A(H∗A)m=κ(X⊗H∗A(H∗A)m).

Altogether we infer that Xis realisable if and only if the class κ(Xm) is trivial for all

m, or equivalently, if Xmis realisable by an object in D(A)/Cmfor all graded maximal

ideals mof H∗A.

A graded H∗A-module is realisable if and only if all its direct summands are realisable.

Since p-localisation commutes with direct sums, this shows

Corollary 11.7. Let Abe a dg algebra such that H∗Ais graded-commutative and co-

herent. Let X∈Modgr H∗Abe an arbitrary direct sum of finitely presented graded

H∗A-modules. Then the following are equivalent:

(1) Xis realisable.

(2) Xpis realisable for all graded prime ideals pof H∗A.

(3) Xmis realisable for all graded maximal ideals mof H∗A.

62 BIRGIT HUBER

Theorem 11.6 and Corollary 11.7 apply in particular for realisability over the group

cohomology ring H∗(G, k), where Gis a finite group and kis a noetherian ring (see

Example 10.5). The finiteness of the group is not necessary as long as H∗(G, k) is still

coherent and kG is noetherian. The latter is to ensure that the category K(Inj kG) is

closed under taking arbitrary direct sums.

We also remark that in all results in this section, the realisability setting

D(A)H∗

−−→ Modgr H∗A

can be replaced by the more general setting

TT(N,−)∗

−−−−−→ Modgr T(N, N)∗,

where Tis a triangulated category which admits arbitrary direct sums and is generated

by compact objects, and N∈ T is a compact object such that T(N, N)∗is graded-

commutative and coherent.

Remark. (1) One might want to have this local-global principle for arbitrary graded

H∗A-modules. It is well-known that every graded module is a direct limit of finitely

presented graded modules, but it is open whether a realisable finitely presented module

can be written as direct limit of realisable finitely presented modules. It is not known

either whether an arbitrary direct limit of realisable modules is realisable.

(2) It would be nice to have a local-global principle also for strict realisability (see

Remark 10.4). Let Xbe a graded H∗A-module, where Ais a dg algebra with graded-

commutative cohomology ring. If Xis strictly realisable, then so is Xp(Lemma 11.3).

If Xis finitely presented and H∗Acoherent, does Xpbeing strictly realisable for all

primes pimply that Xis strictly realisable?

The infinite sequence of obstructions deciding on strict realisability

κn(X)∈Extn,2−n

H∗A(X, X), n ≥3,

where κn(X) is defined provided that the previous one κn−1(X) vanishes, arises from an

A-exact ∞-Postnikov-System which in cohomology gives rise to a map having cokernel X

(see [5, App. A] for details). More precisely, an A-exact l-Postnikov system gives rise to

the obstructions κn(X),3≤n≤l, and can be extended to an A-exact (l+ 1)-Postnikov

system provided that the class κl(X) is trivial. If an A-exact ∞-Postnikov system exists,

then Xis strictly realisable [5, Prop. A.19].

Now one might want to prove iteratively κl(Xp)∼

=κl(X)pfor all l≥3 and use

the same methods as in the proof of Theorem 11.6. The problem is that all but the

first obstruction are not uniquely determined and consequently, the compatibility of the

obstructions for the realisability of Xand the realisability of Xpcannot be expected in

general.

12. Localising the global obstruction

Let Abe a dg algebra over a field kand assume that H∗Ais graded-commutative. We

have shown in Section 11.2 that realisability is a local property. The local-global principle

we have shown applies for finitely presented modules but does not yield information

on global realisability. In this chapter we develop a local-global principle for global

realisability.

REALISABILITY AND LOCALISATION 63

Applying our results from Chapter 9, we can state a global obstruction for the p-

local H∗A-modules: Let Apthe localisation of Aat a prime pin cohomology, defined in

Section 9.2. From Corollary 8.16, Proposition 9.4 and Theorem 9.12 we conclude that

the diagram

D(A)



D(A)(A,−)∗

//Modgr H∗A

−⊗H∗A(H∗A)p



D(Ap)D(Ap)(Ap,−)∗

//Modgr H∗(Ap)

commutes up to isomorphism. Since in particular H∗(Ap)∼

=(H∗A)p, we infer that the

canonical class µAp∈HH3,−1(H∗Ap) is a global obstruction for the p-local modules.

At first sight, it is not clear whether the Hochschild classes µA∈HH3,−1(H∗A) and

µAp∈HH3,−1(H∗Ap) are associated in some way. In order to relate them, we show in

Section 12.1 the existence of a map of Hochschild cohomology rings

Γ: HH∗,∗(H∗A)−→ HH∗,∗(H∗Ap)

which has the property Γ(µA) = µAp.

After discussing localisation of Hochschild cohomology groups of graded-commutative

algebras in Section 12.2, we are ready to prove a local-global principle for global reali-

sability.

12.1. A map of Hochschild cohomology rings. In general, a morphism of graded

algebras ϕ:R→Tdoes not induce a homomorphism HH∗,∗(R)→HH∗,∗(T). We show

in this section that such a map does exist whenever TRand RTare flat and ϕ:R→Tis

an epimorphism in the category of rings, i.e. for any ring T0and morphisms α, β :T→T0

with αϕ =βϕ, it follows α=β. Such a map can be characterised in the following way:

Lemma 12.1. [52, Ch. XI, Prop. 1.2] The following conditions are equivalent for a

morphism of graded rings ϕ:R→T:

(1) ϕis an epimorphism in the category of rings.

(2) The map R⊗TR→R, r ⊗r07→ rr0,is an isomorphism.

(3) The restriction functor Modgr R→Modgr Tis full.

We call a map of graded algebras ϕ:R→Taflat epimorphism if ϕis an epimorphism

in the category of rings and furthermore, the modules TRand RTare flat.

Example 12.2. If Ris a graded-commutative ring and S⊆Ra multiplicative subset

of homogeneous elements, then R→S−1Ris a flat epimorphism: S−1Ris flat as both

left and right R-module, and it is an epimorphism of rings by the universal property of

the ring of fractions.

By B(Λ) we denote the graded Bar resolution of a graded algebra Λ as introduced in

Chapter 5.2.

Lemma 12.3. Let ϕ:R→Tbe a map of graded algebras which is a flat epimorphism

of rings. Then the complex T⊗RB(R)⊗RTis a Te-projective resolution of Tand a

chain map η:T⊗RB(R)⊗RT→B(T)is given by

ηn:T⊗RB(R)n⊗RT−→ B(T)n

(t, r0,· · · , rn+1, t0)7−→ (tϕ(r0), ϕ(r1),· · · , ϕ(rn), ϕ(rn+1)t0).

64 BIRGIT HUBER

Proof. We first remark that T⊗RB(R)⊗RTis exact because T⊗R−and − ⊗RT

are exact. Since RTTand TTRare projective as T-modules and B(R)nis a projective

Re-module for all n≥0, we get indeed a Te-projective resolution of T⊗RR⊗RT. But

since ϕ:R→Tis an epimorphism of rings, the map

T⊗RR⊗RT→T, (t, r, t0)7→ tϕ(r)t0,

is an isomorphism of Te-modules. 

As a consequence we obtain

Proposition 12.4. The maps

Homl

Re(B(R)n, R)−→ Homl

Te(T⊗RB(R)n⊗RT, T),

ζ7−→ T⊗Rζ⊗RT

where l∈Zand n≥0, induce a homomorphism of bigraded algebras

Γ: HH∗,∗(R)−→ HH∗,∗(T).

Proof. Γ is obviously a morphism of graded vector spaces, and it is easy to check that

it commutes with Yoneda multiplication. 

Theorem 12.5. Let A, B dg algebras over a field kand suppose that ψ:A→Bis a

morphism of dg algebras inducing a flat epimorphism ψ∗:H∗A→H∗Bin cohomology.

Then the map

Γ: HH∗,∗(H∗A)−→ HH∗,∗(H∗B)

satisfies Γ(µA) = µB.

Proof. We choose representing cocycles mA∈Hom−1

(H∗A)e(H∗A⊗5, H∗A) of µAand

mB∈Hom−1

(H∗B)e(H∗B⊗5, H∗B) of µBand show that the diagram

H∗B⊗H∗AH∗A⊗5⊗H∗AH∗B

η3



H∗B⊗mA⊗H∗B//H∗B⊗H∗AH∗A⊗H∗AH∗B

η0

∼



H∗B⊗5mB//H∗B

commutes up to coboundaries.

An element (s, x0, x1, x2, x3, x4, t)∈H∗B⊗H∗A⊗5⊗H∗Bis sent by mB◦η3to

(12.1) (−1)|s|smB(ψ∗(x0), ψ∗(x1), ψ∗(x2), ψ∗(x3), ψ∗(x4))t.

On the other hand, under η0◦(H∗B⊗mA⊗H∗B) our element maps to

(12.2) (−1)|s|sψ∗(mA(x0, x1, x2, x3, x4))t.

By Lemma 7.6, we have a (2,−1)-Hochschild cocycle u∈Hom−1

(H∗A)e(H∗A⊗4, H∗B) such

that

mB◦(ψ∗)⊗5−ψ∗◦mA=u◦d3,

where d3:H∗A⊗5→H∗A⊗4is the differential of the Bar resolution of H∗A. Hence the

difference of (12.2) and (12.1) equals

(H∗B⊗(u◦d3)⊗H∗B)(s, x0, x1, x2, x3, x4, t).

REALISABILITY AND LOCALISATION 65

But now we are done since H∗B⊗(u◦d3)⊗H∗Bis a coboundary in the complex

Hom−1

(H∗B)e(H∗B⊗H∗AB(H∗A)⊗H∗AH∗B, H∗B)

which computes the Hochschild cohomology HH∗,−1(H∗B). 

12.2. Local-global principle for the global obstruction. Throughout this section

let Abe a dg algebra over a field kand suppose that H∗Ais graded-commutative. Fix

a graded prime ideal pof H∗A. In Theorem 9.12 we have shown the existence of a dg

algebra A0quasi-isomorphic to Aand a zigzag of dg algebra maps

A∼

←− A0ϕ

−→ Ap

which induces the canonical map can: H∗A→(H∗A)pin cohomology. Now we use this

result to obtain more information about the global obstruction for the p-local modules.

Proposition 12.6. The map

Γ: HH∗,∗(H∗A0)−→ HH∗,∗(H∗Ap)

satisfies Γ(µA0) = µAp. Moreover, the composition

HH∗,∗(H∗A)∼

=//HH∗,∗(H∗A0)Γ//HH∗,∗(H∗Ap)

maps µAto µAp.

Proof. The dg algebra morphism ϕ:A0→Apinduces in cohomology the flat epimor-

phism of rings H∗A→(H∗A)p. The first claim then follows from Theorem 12.5. For

the second just note that by Lemma 7.6, the isomorphism HH∗,∗(H∗A)∼

−→ HH∗,∗(H∗A0)

induced by the quasi-isomorphism A0→Amaps µAto µA0.

Proposition 12.6 implies that if µAis trivial, then so is µApfor all graded primes p

of H∗A. Observe that this was not clear before: if all H∗A-modules are realisable, then

so are in particular all p-local modules. But this does not imply that µApis trivial, see

Remark 10.3.

Under an additional assumption for H∗Awe now show that if µApis trivial for all

prime ideals pof H∗A, then µAis trivial.

Lemma 12.7. Let Rbe a graded-commutative algebra over a commutative ring k. As-

sume that Mis a graded Re-module which admits a resolution of finitely generated

projective graded Re-modules. Let Nbe any graded Re-module. Then for all i≥0, there

is a natural isomorphism

Exti,∗

Re(M, N)⊗Re(Rp)e∼

=Exti,∗

(Rp)e(M⊗Re(Rp)e, N ⊗Re(Rp)e).

Proof. Note first that Reis graded-commutative and consequently, Exti,∗

Re(M, N) is in-

deed a graded Re-module.

The claim is immediately checked if Mis a finitely generated graded free Re-module

and i= 0.

Now assume that Madmits a resolution of finitely generated projective graded Re-

modules. Then Mis in particular finitely presented and there exists an exact sequence

−→ F0→M→0,

66 BIRGIT HUBER

where F0and F1are finitely generated graded free Re-modules. The commutative dia-

gram

Hom∗

Re(F0, N)⊗Re(Rp)e

f∗⊗Re(Rp)e//

∼



Hom∗

Re(F1, N)⊗Re(Rp)e

∼



Hom∗

(Rp)e(F0⊗Re(Rp)e, N ⊗Re(Rp)e)(f⊗Re(Rp)e)∗//Hom∗

(Rp)e(F1⊗Re(Rp)e, N ⊗Re(Rp)e)

and the exactness of − ⊗Re(Rp)e∼

=Rp⊗R− ⊗RRpgive the isomorphism

Hom∗

Re(M, N)⊗Re(Rp)e∼

=Hom∗

(Rp)e(M⊗Re(Rp)e, N ⊗Re(Rp)e).

The claim for the Ext-groups follows from the condition that Madmits a resolution

of finitely generated projective Re-modules. 

Remark 12.8. The assumptions of Lemma 12.7 are satisfied if Reis Noetherian and

Ma finitely generated Re-module. Note that it is not enough to assume that Ris

Noetherian. In general, we cannot expect a tensor product of two Noetherian algebras

to be Noetherian: If Fis a perfect field of characteristic p > 0 and kis an imperfect

subfield of F, then the tensor product F⊗kFis not Noetherian, see [39, Sect. 1].

Before we prove the local-global principle, we establish a nice relation between the

Hochschild groups of Rand Rp.

Proposition 12.9. Let Rbe a graded-commutative algebra over a field k. Suppose that

the enveloping algebra Reis Noetherian, and let pbe a graded prime ideal of R. For all

n≥0, we have

HHn,∗(Rp)∼

=HHn,∗(R)p

as graded R-modules. In particular, a Hochschild group HHn,∗(R)is trivial if and only

if HHn,∗(Rp)is trivial for all graded prime ideals pof R.

Proof. We first point out that

HHn,∗(Rp)∼

=Rp⊗RHHn,∗(R)⊗RRp

by Lemma 12.7. In order to show that Rp⊗RHHn,∗(R)⊗RRpis, in fact, just lo-

calisation of HHn,∗(R) at p, we apply our results from Chapter 5.3. The Hochschild

cohomology ring HH∗,∗(R) is bigraded-commutative by Theorem 5.3 and since Ris

graded-commutative, Remark 5.4 implies that

HH0,∗(R) = R.

Hence we have a well-defined R-linear map

ν:Rp⊗RHHn,∗(R)⊗RRp−→ HHn,∗(R)p,

s⊗ζ⊗r0

s07−→ (−1)|r||ζ|ζ·rr0

ss0

and it is easy to check that νis bijective by stating the obvious inverse map. 

Note that the denominators s, s0do not need sign adjustment because we have chosen

them to have even degree (see Section 3.1.2).

Now we are ready to prove

REALISABILITY AND LOCALISATION 67

Theorem 12.10 (Local-global principle).Let Abe a differential graded algebra over

a field ksuch that H∗Ais graded commutative. Assume that the enveloping algebra

(H∗A)eis Noetherian. Then the following conditions are equivalent:

(1) µA∈HH3,−1(H∗A)is trivial.

(2) µAp∈HH3,−1(H∗Ap)is trivial for all graded prime ideals pof H∗A.

(3) µAm∈HH3,−1(H∗Am)is trivial for all graded maximal ideals mof H∗A.

In particular, all graded H∗A-modules are realisable if the Hochschild class µApis trivial

for all graded prime ideals pof H∗A.

Proof. Fix a graded prime pof H∗A. We prove that under the isomorphism

HH3,∗(H∗Ap)∼

=HH3,∗(H∗A)p

of Proposition 12.9, the class µApis mapped to the fraction µA

1. This shows the claim.

Let A∼

←− A0ϕ

−→ Apbe a zigzag of dg algebra maps which induces the canonical map

H∗A→(H∗A)pin cohomology. Corollary 12.6 implies that in the Hochschild group

HH3,−1(H∗Ap), we have

µAp= [H∗Ap⊗H∗A0mA0⊗H∗A0H∗Ap],

where mA0is a representing cocycle for µA0∈HH3,−1(H∗A0). We conclude from

Lemma 12.7 and Proposition 7.6 that under the composition of isomorphisms

HH3,∗(H∗Ap)∼

=HH3,∗(H∗Ap⊗H∗A0H∗A0⊗H∗A0H∗Ap)

∼

=H∗Ap⊗H∗A0HH3,∗(H∗A0)⊗H∗A0H∗Ap

∼

=H∗Ap⊗H∗AHH3,∗(H∗A)⊗H∗AH∗Ap,

the Hochschild class µApmaps to 1

1⊗µA⊗1

1. But now we can apply the isomorphism

ν:H∗Ap⊗H∗AHH3,∗(H∗A)⊗H∗AH∗Ap−→ HH3,∗(H∗A)p

from the proof of Proposition 12.9. Since ν(1

1⊗µA⊗1

1) = µA

1,we have proved the

claim. 

Remark 12.11. The reader might have noticed that the dg algebra Apwas not shown

to be uniquely determined up to quasi-isomorphism. The universal property we have

proved in Section 9.2.1 only holds on the level of derived categories. Hence for another

dg algebra A0

psatisfying H∗(A0

p)∼

=(H∗A)p, we obtain a canonical class µA0

pwhich could

behave differently.

However, what we have actually shown in Theorem 12.10 is that for every dg al-

gebra A0

padmitting a zigzag of dg algebra morphisms A∼

←− A00 ϕ

−→ A0

pwhich induces

can: H∗A→(H∗A)pin cohomology, the canonical class µA0

pis the image of µAunder

the map

HH3,−1(H∗A)−→ HH3,−1(H∗A)p, ζ 7−→ ζ

So the choice of the dg algebra inducing (H∗A)pin cohomology is not relevant, as long

as it admits such a zigzag.

Remark. One might want to have Proposition 12.9 and with it Theorem 12.10 under

weaker assumptions. In Proposition 12.9 we have assumed that Reis Noetherian to

ensure that Radmits a resolution of finitely generated Re-projective modules. We do

not know whether there is a way to avoid the latter condition.

68 BIRGIT HUBER

13. Comparing realisability over group and Tate cohomology

Now we focus on realisability in group representation theory and compare realisability

over the group cohomology ring and the Tate cohomology ring.

The group cohomology ring H∗(G, k) has better properties than the Tate cohomology

ring ˆ

H∗(G, k) which, for instance, is not Noetherian in general. However, when it comes

to the source categories of realisability, the stable module category Mod kG is more

“handsome” than the homotopy category K(Inj kG). This is the reason why we are

interested in studying the relation of realisability over group and Tate cohomology.

The triangulated categories K(Inj kG) and Mod kG are related by a smashing locali-

sation

K(Inj kG)

Q//Mod kG

(see Proposition 8.11) and we are now concerned with finding a relation between realis-

ability and this localisation of triangulated categories.

Remember that both H∗(G, k) and ˆ

H∗(G, k) are the cohomology of a dg algebra (Ex-

ample 10.5) and thus, they admit an A∞-algebra structure yielding global obstructions

which we now denote by µG∈HH3,−1(H∗(G, k)) and ˆµG∈HH3,−1(ˆ

H∗(G, k)).

In the first section we study realisability of fixed modules. Then we focus on global

realisability. The canonical class ˆµGhas been computed for some groups Gby Benson,

Krause and Schwede [5], and by Langer [37]. We consider the same groups and compute

the global obstructions for the group cohomology rings. We will see in Section 13.2 that

in all but one case, the Hochschild classes µGand ˆµGturn out to behave surprisingly

similar. As a first explication for this similarity we show in Section 13.3 that the algebra

morphism H∗(G, k)→ˆ

H∗(G, k) is induced by a zigzag of dg algebra morphisms. The

main result of this chapter is stated in the last section and gives a complete explanation

for the relation between µGand ˆµGwe observed before in examples.

We like to thank Dave Benson for discussions that helped to improve this chapter and

in particular, for a contribution to the proof of Theorem 13.21.

13.1. Local realisability. Let kbe a field of characteristic p > 0 and Gbe a finite

group such that pdivides the order of G. In this section we discuss ways to construct a

realisable H∗(G, k)-module from a realisable module over H∗(G, k), and vice versa.

The following proposition will play an important role in the next section.

Proposition 13.1. Assume that the p-rank of Gequals one.

(1) If X∈Modgr H∗(G, k)is realisable, then X⊗H∗(G,k)ˆ

H∗(G, k)is a realisable

H∗(G, k)-module.

(2) A graded ˆ

H∗(G, k)-module Yis realisable if and only if its restriction to H∗(G, k)

is a realisable H∗(G, k)-module.

Proof. By Lemma 4.4 and Theorem 4.5, there exists a multiplicative subset S⊆H∗(G, k)

such that ˆ

H∗(G, k) = S−1H∗(G, k). Hence we may apply Theorem 8.12 and Remark

8.13 and obtain commutative diagrams

REALISABILITY AND LOCALISATION 69

K(Inj kG)



HomK(ik,−)∗

//Modgr H∗(G, k)

−⊗H∗(G,k)ˆ

H∗(G,k)



K(Inj kG)HomK(ik,−)∗

//Modgr H∗(G, k)

Mod kG HomkG(k,−)∗

//Modgr ˆ

H∗(G, k)Mod kG

HomkG(k,−)∗

//Modgr ˆ

H∗(G, k)

res

which yield the claim. 

If the p-rank of Gis at least two, then we cannot expect to obtain realisable modules

from realisable modules by induction or restriction as above. The reason for this is that

in general, the induction functor − ⊗H∗(G,k)ˆ

H∗(G, k) is not given by localisation with

respect to a multiplicatively closed subset.

However, there is another possibility to obtain a realisable H∗(G, k)-module from a

realisable ˆ

H∗(G, k)-module. This construction also works in the general case.

Lemma 13.2. Let X∈Modgr ˆ

H∗(G, k)realisable. Then its truncation to non-negative

degrees X⩾0is a realisable H∗(G, k)-module.

Proof. Let Xbe a direct summand of ˆ

H∗(G, M), where Mis a kG-module. Without

loss of generality, we may assume that Mdoes not have any projective direct summands.

Then ˆ

H0(G, M)∼

=H0(G, M) and consequently, ˆ

H∗(G, M)⩾0∼

=H∗(G, M). It follows

that X⩾0is a direct summand of H∗(G, M). 

13.2. Examples for the global obstruction. Let kbe a field of characteristic p > 0.

We study the group and Tate cohomology rings of cyclic p-groups and the Quaternion

group, and focus on the global obstructions µGand ˆµG.

Theorem 13.3. [5, Thm. 7.1] Let Gbe a cyclic group of order pn, where n≥1. If

pn= 2, then the Tate cohomology ring is a Laurent polynomial ring k[X, X−1]on a

1-dimensional class X. If pn≥3, then the Tate cohomology ring is a truncated Laurent

polynomial ring in two variables,

H∗(G, k) = k[X, Y, Y −1]/(X2),

with deg(X) = 1 and deg(Y) = 2. The secondary multiplication m3of the A∞-algebra

H∗(G, k)and thus ˆµG∈HH3,−1(ˆ

H∗(G, k)) is trivial except when p= 3 and n= 1. In

this case, the secondary multiplication is given by

m3(XY i, XY j, XY l) = Yi+j+l+1, i, j, l ∈Z,

and vanishes on all other tensor products of monomials. Furthermore, its Hochschild

class ˆµZ/(3) is non-trivial.

Note that strictly speaking, one can make choices to obtain m3in the shape as stated

above. With other choices of f1and f2in Construction 7.1 we could obtain a different

map. However, the Hochschild class ˆµGof m3is independent of all choices.

Now we consider the group cohomology ring H∗(G, k) of a cyclic group Gas above. It

identifies with the subring of non-negative degrees of ˆ

H∗(G, k). The global obstruction

µG∈HH3,−1(H∗(G, k)) turns out to behave very similarly:

70 BIRGIT HUBER

Proposition 13.4. Let Gbe a cyclic group of order pn, where n≥1. If pn= 2, then

the group cohomology ring is a polynomial ring k[X], where Xhas degree 1. If pn≥3,

then H∗(G, k)is a truncated polynomial ring in two variables,

H∗(G, k) = k[X, Y ]/(X2),

with deg(X) = 1 and deg(Y) = 2. The secondary multiplication m3of the A∞-algebra

H∗(G, k)and thus µG∈HH3,−1(H∗(G, k)) is trivial except when p= 3 and n= 1. In

this case, it satisfies

m3(XY i, XY j, XY l) = Yi+j+l+1, i, j, l ≥0,

and vanishes on all other tensor products of monomials. Its Hochschild class µZ3is

non-trivial.

Proof. Since the characteristic of kis positive, we may identify kG with the truncated

polynomial ring K[T]/(Tr), where r=pn. An injective resolution of kwhich is moreover

2-periodic is given by

ik:· · · //0//0//I0T//I1

−Tr−1

//I2T//I3

−Tr−1

//· · ·

Let x:ik →Σik be the degree-one chain map



· · · //0//



0//



I0T//



−Tr−1

Tr−2



I2T//



Tr−2



−Tr−1

//· · ·

Σik · · · //0//I0−T//I1Tr−1//I2−T//I3Tr−1//I4−T//· · ·

and y:ik →Σ2ik the degree-two chain map



· · · //0//



0//



I0T//



−Tr−1



I2T//



· · ·

Σ2ik · · · //I0T//I1−Tr−1//I2T//I3−Tr−1//I4T//· · ·

One easily checks that xy =yx. If r≥3, then x2is nullhomotopic by the homotopy

q, given by multiplication with Tr−3in odd degrees, and the zero map in even degrees.

If r= 2, then obviously x2=y. We infer that xand yare cycles of End(ik) representing

the classes X∈H1(G, k) and Y∈H2(G, k), respectively.

In order to compute the secondary multiplication, we define a cycle selection homo-

morphism

f1:H∗End(ik)→ End(ik)

on the k-basis {XYi|∈ {0,1}, i ≥0}of H∗(G, k), given by

f1(XYi) = xyi.

If r= 2, then this map is multiplicative and it follows that f2and with it m3can be

chosen to be trivial. If r≥3, then f1is multiplicative except on two odd dimensional

classes. The product f1(XY i)f1(XY j) = x2yi+jis only nullhomotopic, by the homotopy

qyi+j.

REALISABILITY AND LOCALISATION 71

We define

f2:H∗End(ik)⊗H∗End(ik)→ End(ik)

to be trivial except on two odd dimensional classes: In this case, we set

f2(XY i, XY j) = qyi+j.

Since m3maps (A, B, C) to the cohomology class of the expression

(13.1) (−1)|A|f1(A)f2(B, C)−f2(A, B)f1(C)−f2(AB, C) + f2(A, BC)

(see (7.6)), we infer that m3vanishes on all tensor product of monomials with at least

one monomial having even degree. Using the fact that the homotopy qcommutes with

y, one checks that the tensor product of three odd degree monomials (XY i, XY j, XY l)

is mapped under (13.1) to

(qx +xq)yi+j+l.

The chain map qx +xq :ik →Σ2ik is given by multiplication with Tr−3in each

degree. Now if r > 3, then qx +xq is nullhomotopic via the homotopy given by the zero

map in even degrees, and by multiplication with Tr−4in odd degrees. Thus m3= 0.

But if r= 3, then qx +xq =yand we conclude

m3(XY i, XY j, XY l) = Yi+j+l+1.

It remains to show that also the Hochschild class µZ3is non-trivial. But this is is a

consequence of the following remark. 

Remark 13.5. Using Massey Products, Benson, Krause and Schwede proved that

H∗(Z3, k)/(X) is a non-realisable ˆ

H∗(Z3, k)-module [5, Exm. 7.6]. From Proposition 13.1

we conclude that ˆ

H∗(Z3, k)/(X), viewed as H∗(Z3, k)-module, is not realisable either.

Theorem 13.6. [37, Satz 2.10] Denote by Q8be the quaternion group and let kbe a

field of characteristic two. Then the Tate cohomology ring ˆ

H∗(Q8, k)is given by

k[X, Y, S±1]/(X2+Y2=XY, X3=Y3= 0, X2Y=XY 2),

with |X|=|Y|= 1,|S|= 4. The canonical class ˆµQ8is non-trivial.

Furthermore, Langer shows the existence of a non-realisable module [37, Lemma 2.23]:

Write ˆ

Hfor ˆ

H∗(Q8, k). The cokernel of the map

g:ˆ

H[−1] ⊕ˆ

H[−1] hY X+Y

X Y i

−−−−−−−→ ˆ

H⊕ˆ

is not a realisable module.

Since r2(Q8) = 1 (see Theorem 4.6), it follows from Proposition 13.1 that Coker g,

viewed as module over H∗(Q8, k), is not realisable. We obtain

Corollary 13.7. Denote by Q8be the quaternion group and let kbe a field of charac-

teristic two. Then

H∗(Q8, k) = k[X, Y, S]/(X2+Y2=XY, X3=Y3= 0, X2Y=XY 2),

with |X|=|Y|= 1,|S|= 4. The canonical class µQ8is non-trivial. 

One might wonder whether it always holds true that µGis trivial if and only if ˆµG

is trivial. But in general, this is not the case. For any finite abelian 2-group G, the

class µGis trivial by Proposition 13.4 and Remark 7.8. However, the situation for Tate

cohomology is entirely different when it comes to the Klein four group:

72 BIRGIT HUBER

Theorem 13.8. [5, Exm. 7.7] [37, Thm. 3.1] Let G=Z2×Z2and kbe a field with

Char(k) = 2. Then ˆµZ2is non-trivial. For any other finite abelian 2-group, the canonical

class is trivial.

13.2.1. Reduction to Sylow subgroups. Let kbe a field of characteristic p > 0 and Gbe a

finite group such that pdivides the order of G. Let Pap-Sylow subgroup of G. Benson,

Krause and Schwede [5] have shown that the canonical class ˆµGis already determined

by ˆµP. In order to compare µGand ˆµGin more cases, we now briefly explain why the

same holds true for µGand µP.

Let Pbe a subgroup of G. If M, N are kG-modules, we define the transfer or trace

map

tr: HomkP (M, N)→HomkG(M, N)

as follows:

trP,G(Φ)(m) = X

i∈I

giΦ(g−1

im),

where {gi|i∈I}is any choice of left coset representatives of Pin G. See [2, Ch. 3.6]

for details. The transfer tr induces a well-defined map

tr: ˆ

Hn(P, k)→ˆ

Hn(G, k)

for n∈Z, independent on the choice of the resolution [2, Lemma 3.6.16].

By considering a kG-Tate resolution of the trivial module as kP-Tate resolution of k

regarded as kP-module, we obtain for n∈Za restriction map

res: ˆ

Hn(G, k)→ˆ

Hn(P, k).

Lemma 13.9. [2, Ch. 3.6] The composition tr ◦res: ˆ

Hn(G, k)→ˆ

Hn(G, k)is given by

multiplication with [G:P].

Note that the restriction map induces morphisms of graded algebras res: H∗(G, k)→

H∗(P, k) and res: ˆ

H∗(G, k)→ˆ

H∗(P, k). Moreover, if Pis a p-Sylow subgroup of G,

then these algebra morphisms are split monomorphisms by Lemma 13.9.

Theorem 13.10. [5, Thm. 8.3] Let kbe a field of characteristic p > 0,Ga finite group

and Pap-Sylow subgroup of G. Then the canonical class ˆµG∈HH3,−1(ˆ

H∗(G, k)) is

determined by the canonical class ˆµP∈HH3,−1(ˆ

H∗(P, k)) of the Sylow subgroup, and is

given by the formula

ˆµG=tr ◦ˆµP◦res⊗3

[G:P].

The key point of the proof is that the restriction map in Tate cohomology is induced

by a morphism of dg algebras. Since similarly, res: H∗(G, k)→H∗(P, k) is induced by

the inclusion dg algebras

EndkG(ik)→ EndkP (ik)

given by viewing the kG-injective resolution ik as injective resolution over kP, we obtain

Lemma 13.11. Let kbe a field of characteristic p > 0,Ga finite group and Pa

p-Sylow subgroup of G. Then for the canonical classes µG∈HH3,−1(H∗(G, k)) and

µP∈HH3,−1(H∗(P, k)), it holds

µG=tr ◦µP◦res⊗3

[G:P].

REALISABILITY AND LOCALISATION 73

Benson, Krause and Schwede concluded in particular that if Gis a group whose p-

Sylow group is cyclic with order different from 3, then ˆµGis trivial [5, Cor. 8.4]. We

infer from Lemma 13.11 and Proposition 13.4 that the same holds true for µG.

Corollary 13.12. Let kbe a field of characteristic p > 0and Gbe a group whose

p-Sylow subgroup is cyclic of order pn, n ≥1. Suppose that n≥2if p= 3. Then both

the Hochschild classes µGand ˆµGare trivial.

In order to investigate the case P=Z/(3), one needs to distinguish whether Gis

3-nilpotent or not:

Definition 13.13. Let Gbe a finite group and let Pbe a p-Sylow subgroup. Gis called

p-nilpotent if there exists a normal subgroup UEGsuch that the composite

Pincl

−−→ Gproj

−−→ G/U

is an isomorphism.

Proposition 13.14. [5, Prop. 8.5] Let kbe a field of characteristic p > 0and Gbe a

finite group with p-Sylow subgroup P. The following conditions are equivalent:

(1) res: ˆ

H∗(G, k)→ˆ

H∗(P, k)is an isomorphism.

(2) res: H∗(G, k)→H∗(P, k)is an isomorphism.

(3) res: H1(G, k)→H1(P, k)is an isomorphism.

(4) The group Gis p-nilpotent.

Since both the restriction maps ˆ

H∗(G, k)→ˆ

H∗(P, k) and H∗(G, k)→H∗(P, k) are

induced by a morphism of dg algebras, we obtain as a consequence of Proposition 7.6

Corollary 13.15. Let kbe a field of characteristic 3and Ga finite, 3-nilpotent group

whose 3-Sylow group is cyclic of order 3. Then both the Hochschild classes µGand ˆµG

are non-trivial.

For ˆµG, the result above is stated in [5, Sect. 8].

Theorem 13.16. [5, Thm. 8.6] Let kbe a field of characteristic 3and Ga finite group

whose 3-Sylow group is cyclic of order 3. Assume that Gis not 3-nilpotent. Then

H∗(G, k) = k[V, W, W−1]/(V2),

where Vis of degree 3and Wof degree 4. The canonical class ˆµGis represented by the

(3,−1)-cocycle mgiven by

m(V Wi, V Wj, V Wl) = Wi+j+l+2, i, j, l ∈Z,

and vanishes on all other tensor products of monomials in Vand W. The Hochschild

class ˆµGis non-trivial.

In [5], it is further shown that ˆ

H∗(G, k)/(V) is not realisable. We conclude from

Proposition 13.1 that Vviewed as module over H∗(G, k) is not realisable either. In

particular, the canonical class µGmust be non-trivial. Using Lemma 13.11, we can

compute a representing cocycle for the canonical class, using the same methods as in [5,

Thm. 8.6].

74 BIRGIT HUBER

Proposition 13.17. Let kbe a field of characteristic 3and Ga finite group whose

3-Sylow group is cyclic of order 3. Assume that Gis not 3-nilpotent. Then

H∗(G, k) = k[V, W]/(V2)

where Vis of degree 3and Wof degree 4. A representing (3,−1)-cocycle mfor the

canonical class µGis given by

m(V Wi, V Wj, V W l) = Wi+j+l+2, i, j, l ≥0,

and vanishes on all other tensor products of monomials in Vand W. Its Hochschild

class ˆµGis non-trivial. 

13.3. Lifting H∗(G, k)→ˆ

H∗(G, k)to a morphism of dg algebras. Let Gbe a finite

group and ka field. In the examples we have considered in the last section, the class µG

always arises as restriction of ˆµGto non-negative degrees. Now we give a explanation

for this fact: We show that the canonical inclusion

ι:H∗(G, k)→ˆ

H∗(G, k)

is induced by a zigzag of dg algebra morphisms. The construction is analogous to the

one in Section 9. However, for the convenience of the reader we sketch it briefly. Let

η: id →RQ be the unit and ε:QR →id the counit of the adjunction

K(Inj kG)

Q//Kac(Inj kG),

where Q=− ⊗ tk and Ris the inclusion. This is a smashing localisation by Proposi-

tion 8.11.

The map

ηik :ik →RQ(ik)

is up to natural isomorphism just the canonical inclusion ik →tk, see Proposition 8.11.

Therefore the map

K(Inj kG)(ik, ik)∗→K(Inj kG)(RQ(ik), RQ(ik))∗, f 7→ RQ(f)

is up to isomorphism the canonical inclusion

ι:H∗(G, k)→ˆ

H∗(G, k).

We have K(Inj kG)(ik, RQ(ik)) ∼

=H0(Hom(ik, RQ(ik)). For any representing cocycle

¯ηik ∈Z0(Hom(ik, RQ(ik)) of ηik we obtain as in Lemma 9.5 a quasi-isomorphism

¯η∗

ik :End(RQ(ik)) → Hom(ik, RQ(ik)), f 7→ f◦¯ηik.

We choose for ¯ηik the inclusion of complexes ik →tk which is a degree-wise split

monomorphism of complexes. Thus ¯η∗

ik is surjective.

Theorem 13.18. There exists a dg algebra End(ik)0quasi-isomorphic to End(ik)and

a zigzag of dg algebra morphisms

End(ik)End(ik)0

∼

ooϕ//End(tk)

REALISABILITY AND LOCALISATION 75

inducing the canonical inclusion ι:H∗(G, k)→ˆ

H∗(G, k)in cohomology. That is, we

have diagrams

End(ik)0

∼



KH∗(G, k)

∼

H∗ρ



H∗ϕ

End(ik)End(tk)H∗(G, k)ι//ˆ

H∗(G, k)

where right hand diagram is commutative and identifies with the cohomology of the left

hand diagram.

Proof. We form the pullback diagram

End(ik)0p2//



End(RQ(ik))



∼¯η∗



End(ik)¯ηik ∗//Hom(ik, RQ(ik))

Since ¯η∗

ik is a surjective quasi-isomorphism, it follows from Lemma 9.1 that End(ik)0is

a dg algebra quasi-isomorphic to End(ik).

In cohomology, we obtain a commutative diagram

H∗End(ik)0H∗(p2)//

H∗(p1)∼



H∗End(RQ(ik))

∼

=H∗(¯η∗

ik)



H∗End(ik)H∗(¯ηik ∗)

//H∗(Hom(ik, RQ(ik))

where the composition

H∗(¯η∗

ik)−1H∗(¯ηik ∗): K(Inj kG)(ik, ik)∗→K(Inj kG)(RQ(ik), RQ(ik))∗

is given by applying the functor RQ to a map f∈K(Inj kG)(ik, ik)∗. But this is up to

isomorphism just the inclusion ι:H∗(G, k)→ˆ

H∗(G, k). 

Observe that the map ϕ:End(ik)0→ End(tk) is a monomorphism which moreover

induces a monomorphism in cohomology.

Now we show that the relation between µGand ˆµGwhich we have observed in the

examples is true in general.

Proposition 13.19. In the Hochschild group HH3,−1(H∗(G, k),ˆ

H∗(G, k)), it holds

ι◦µG= ˆµG◦ι⊗3,

where ι:H∗(G, k)→ˆ

H∗(G, k)is the canonical inclusion.

Proof. We use the zigzag

End(ik)End(ik)0

∼

ooϕ//End(tk)

of Theorem 13.18. By Proposition 7.6, we have

(13.2) H∗ρ◦µEnd(ik)0=µEnd (ik)◦(H∗ρ)⊗3in HH3,−1(H∗End(ik)0, H∗End(ik))

76 BIRGIT HUBER

and

(13.3) H∗ϕ◦µEnd(ik)0=µEnd (tk)◦(H∗ϕ)⊗3in HH3,−1(H∗End(ik)0, H∗End(tk)).

Hence in the Hochschild group HH3,−1(H∗(G, k),ˆ

H∗(G, k)), it holds

ι◦µEnd(ik)=µEnd (tk)◦ι⊗3.

Remark 13.20. Even more is true: Since both ρand ϕinduce monomorphisms in

cohomology, there exist choices in defining the secondary multiplications of H∗End(ik)

and H∗End(tk) to obtain the equations (13.2) and (13.3) even on the level of k-linear

maps, see Proposition 7.7. With these choices, we have

ι◦mH∗(G,k)

3=mˆ

H∗(G,k)

3◦ι⊗3in Hom−1

k(H∗(G, k)⊗3,ˆ

H∗(G, k)).

The secondary multiplication mH∗(Z3,k)

3we computed in Proposition 13.4 and mˆ

H∗(Z3,k)

of Theorem 13.3 satisfy this equation. However, we cannot just restrict a fixed mˆ

H∗(G,k)

to non-negative degrees to obtain mH∗(G,k)

13.4. Relating the global obstructions of H∗(G, k)and ˆ

H∗(G, k).In all but one

example we have considered in the first section, the Hochschild classes µGand ˆµGwere

either both trivial or both non-trivial. Now we are ready to give a general explanation

for this fact. The main result of this chapter is

Theorem 13.21. Let Gbe finite group, ka field of characteristic p > 0and assume

that pdivides the order of G. If the Hochschild class ˆµG∈HH3,−1(ˆ

H∗(G, k)) is trivial,

then so is the Hochschild class µG∈HH3,−1(H∗(G, k)). If the p-rank of Gequals one,

then ˆµGis trivial if and only if µGis trivial.

Proof. Let m:H∗(G, k)⊗3→H∗(G, k) be any (3,−1)-cocycle representing µG. In the

Hochschild group HH3,−1(H∗(G, k),ˆ

H∗(G, k)), we have the equation

ι◦µG= ˆµG◦ι⊗3

by Proposition 13.19. Hence if the Hochschild class ˆµGis trivial, then the k-linear map

ι◦mis a (3,−1)-Hochschild coboundary, say

ι◦m=δg,

where g∈Hom−1

k(H∗(G, k)⊗2,ˆ

H∗(G, k)).

If g(1,1) = 0, then we actually have that g∈Hom−1

k(H∗(G, k)⊗2, H∗(G, k)) and

m=δg.

Thus µGis trivial in this case.

If g(1,1) 6= 0, then we choose any map u∈Hom−1

k(H∗(G, k),ˆ

H∗(G, k)) satisfying

u(1) 6= 0. Since ˆ

H−1(G, k) is one-dimensional by Tate duality (see Proposition 4.2),

there exists an element κ∈ksuch that g(1,1) = κ·u(1).Setting

g0=g−κ·δu,

we obtain

ι◦m=δg0,

and since g0(1,1) = 0, we conclude that µGis trivial.

REALISABILITY AND LOCALISATION 77

If the p-rank of Gequals one, then the inclusion ι:H∗(G, k)→ˆ

H∗(G, k) is a flat

epimorphism of rings by Lemma 4.4. Moreover, it is induced by a zigzag of dg algebra

morphisms by Theorem 13.18. Thus we can apply Theorem 12.5 and conclude that the

map Γ: HH∗,∗(H∗(G, k)) →HH∗,∗(ˆ

H∗(G, k)) satisfies Γ(µG) = ˆµG. In particular, µG

being trivial implies that ˆµGis trivial. 

Note that the second statement of the theorem is not true if rp(G)≥2: If Gis the

Klein four group, then µGis trivial but ˆµGis not, see Section 13.2.

Remark. In order to prove that µGis non-trivial in Proposition 13.4, Corollary 13.7

and Proposition 13.17, we have shown the existence of a non-realisable module over

H∗(G, k). But in general, one cannot expect to have a non-realisable module whenever

the global obstruction is non-trivial: Benson, Krause and Schwede provide an example

of a dg algebra Asuch that the canonical class µA∈HH3,−1(H∗A) is non-trivial, but

such that all H∗A-modules are realisable [5, Prop. 5.16].

However, in all known examples of non-trivial global obstructions for group or Tate

cohomology, the existence of a non-realisable module could always be shown. Benson,

Krause and Schwede use Massey products to show the existence of a non-realisable

module over ˆ

H∗(Z3, k) and ˆ

H∗(Z2×Z2, k) [5, Exm. 6.7, Exm. 7.7]. Langer has used

Matrix Massey products [37, Lemma 2.23] to construct a non-realisable module over

H∗(Q8, k).

It is an open question whether for group and Tate cohomology, one can expect to

have a non-realisable module whenever the global obstruction is non-trivial.

78 BIRGIT HUBER

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