Universit¨at Paderborn
Fakult¨at f¨ur Elektrotechnik, Informatik und Mathematik
Subcategories of Triangulated Categories
and the Smashing Conjecture
Dissertation zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften (Dr. rer. nat.)
vorgelegt von
Kristian Br¨uning
Mai 2007
Abstract
In this thesis the global structure of three classes of algebraic triangulated categories
is investigated by describing their thick, localizing and smashing subcategories and by
analyzing the Smashing Conjecture. We show that the Smashing Conjecture for the stable
module category of a self-injective artin algebra Ais equivalent to the statement that a
class of model categories associated with Ais finitely generated. Smashing localizations
of the derived category of a differential graded algebra are realized by morphisms of
dg algebras. We use this theory to define a localization of a dg algebra with graded-
commutative cohomology at a prime ideal of the cohomology ring. For a hereditary
abelian category Awe classify the thick subcategories and the localizing subcategories of
the bounded and unbounded derived category of A, respectively. As an application we
prove that the Smashing Conjecture holds for the derived category of a hereditary artin
algebra of finite representation type.
Zusammenfassung
In dieser Arbeit wird die globale Struktur von drei Klassen algebraischer triangulierter
Kategorien untersucht. Daf¨ur werden dicke, lokalisierende und smashing Unterkate-
gorien beschrieben und die Smashing Conjecture wird analysiert. Wir zeigen, dass die
Smashing Conjecture f¨ur die stabile Modulkategorie einer selbstinjektiven Artin Alge-
bra A¨aquivalent dazu ist, dass eine Klasse von Modellkategorien, die zu Aassoziiert
ist, endlich erzeugt ist. Smashing Lokalisierungen der derivierten Kategorie einer dif-
ferentiell graduierten Algebra werden als Morphismen von dg Algebren realisiert. Diese
Theorie wird benutzt, um die Lokalisierung einer dg Algebra mit graduiert-kommutativer
Kohomologie an einem Primideal des Kohomologierings zu definieren. Des Weiteren wer-
den die dicken und lokalisierenden Unterkategorien der beschr¨ankten beziehungsweise der
unbeschr¨ankten derivierten Kategorie einer erblichen abelschen Kategorie klassifiziert.
Als eine Folgerung zeigen wir, dass die Smashing Conjecture f¨ur die derivierte Kategorie
einer erblichen Artin Algebra vom endlichen Darstellungstyp wahr ist.
Contents
1 Introduction 1
2 Triangulated categories and their localization 4
2.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Thick subcategories and Verdier-quotients . . . . . . . . . . . . . . . . . . 9
2.3 Localizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Smashing and finite localizations . . . . . . . . . . . . . . . . . . . . . . . 15
3 The Telescope Conjecture and the Smashing Conjecture 18
3.1 The Telescope Conjecture in stable homotopy theory . . . . . . . . . . . . 18
3.2 The Smashing Conjecture in commutative algebra . . . . . . . . . . . . . 21
3.3 Keller’s counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 The Smashing Conjecture for stable module categories . . . . . . . . . . . 23
3.5 The Smashing Conjecture for arbitrary triangulated categories . . . . . . 24
3.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Cotorsion pairs, model categories and finite generation 27
4.1 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Cotorsion pairs and model categories . . . . . . . . . . . . . . . . . . . . . 31
4.3 Finite generation and the Smashing Conjecture . . . . . . . . . . . . . . . 35
5 Realizing smashing localizations of differential graded algebras 40
5.1 Differential graded algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Cofibrant differential graded algebras . . . . . . . . . . . . . . . . . . . . . 42
5.3 Cohomological p-Localization . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.4 Smashing localizations of the derived category of a dg algebra . . . . . . . 43
5.5 Smashing subcategories of algebraic triangulated categories . . . . . . . . 50
5.6 The p-localization of a dg algebra . . . . . . . . . . . . . . . . . . . . . . . 50
6 Thick subcategories of the derived category of a hereditary algebra 54
6.1 Representation theory of hereditary algebras of finite representation type 54
6.2 The derived category of hereditary abelian categories . . . . . . . . . . . . 57
6.3 Thick subcategories of abelian categories . . . . . . . . . . . . . . . . . . . 58
6.4 Classification of thick subcategories . . . . . . . . . . . . . . . . . . . . . . 60
6.5 Classification of localizing subcategories . . . . . . . . . . . . . . . . . . . 64
References 66
1 Introduction
In this thesis we study subcategories of triangulated categories and their finiteness prop-
erties that are related to the Smashing Conjecture. In particular we reformulate this
conjecture for stable module categories, describe smashing subcategories of the derived
category of a differential graded algebra, classify the thick and localizing subcategories of
the derived category of a hereditary abelian category and prove the Smashing Conjecture
for the derived category of a hereditary artin algebra of finite representation type.
A ring is a fundamental object in mathematics. It is a set together with a rule how to
multiply and add elements. Interesting rings arise for instance from physics as endomor-
phism rings of Hilbert spaces, from number theory as rings of integers and from topology
as the ring of stable homotopy groups of spheres. Many of these examples are very large
and complicated. Since it is in general not possible to describe them easily as in terms
of generators and relations, tools are needed to extract information from a ring. These
tools are algebraic invariants as the center or more generally the Hochschild cohomology,
or the Grothendieck group and its generalization, the algebraic K-theory. The center of
the endomorphism ring of a configuration space contains the simultaneously measurable
observables. The Grothendieck group controls parts of the representation theory of the
ring and the higher algebraic K-groups contain deep number theoretic information.
Since we want to distinguish rings, the following question comes up: When do two
rings Rand Sshare the same properties, that is, agree in the algebraic invariants? To
a given ring Rit is possible to assign a triangulated category D(R), the derived category
of R, which can be used to give an answer to the question. Two rings share the same
properties if their derived categories are equivalent. So the derived category of a ring can
be thought of as a “higher invariant” and motivates the study of triangulated categories.
The concept of a triangulated category is ubiquitous. In ring theory and homological
algebra it arises as the derived category of a ring, in algebraic topology as the stable
homotopy category, in representation theory of groups as the stable module category of a
group algebra and in algebraic geometry as the derived category of sheaves on a scheme.
Classification is a main purpose of pure mathematics. For instance in representation
theory we are interested in classifying all modules over a fixed ring R. It is known that
the overwhelming part of rings is wild in the sense that it is not possible to determine all
modules simultaneously. Triangulated categories provide a framework to classify objects
in a weaker way. Two objects Xand Yin a small triangulated category Tare considered
to be related if they generate the same thick subcategory, that is the collection of objects
in Tthat can be constructed by the ambient structure of Tstarting with X. Since
R-modules can be considered as objects in the derived category of Ra classification
of the thick subcategories leads to a classification of the R-modules. In a triangulated
category with small coproducts the localizing subcategories are the analogs of the thick
subcategories and are hence worth classifying.
If a localizing subcategory Cof a triangulated category Tgives rise to a localization
functor L:T → T such that Cis the full subcategory of the objects that are annihilated
by Land if Lcommutes with small coproducts, then the category Cis called smashing.
It is known that if Cis of finite type, i.e., generated by compact objects in T, then it is
smashing. The Smashing Conjecture for a triangulated category with small coproducts
Tstates the other direction: every smashing subcategory of Tis of finite type. This
conjecture originates from topology. The Smashing Conjecture for the stable homotopy
1
category is a generalization of the Telescope Conjecture of Ravenel [Rav87b, 1.33] which
has important consequences for the computation of the stable homotopy groups of spheres.
Another reason for studying the Smashing Conjecture is its impact on non-commutative
localization of rings and algebraic K-theory.
The stable module category Mod(R) of a Frobenius algebra Ris triangulated and pos-
sesses small coproducts. If Ris artinian, then the Smashing Conjecture for Mod(R) is
equivalent to the statement that certain cotorsion pairs are of finite type [KS03]. Fur-
thermore there is a connection between cotorsion pairs in Mod(R) and associated model
structures on the module category [BR02, Hov02]. In fact every cotorsion pair Xgives
rise to an associated model category Mod(R)X. In this thesis we extend this connection
and find a reformulation of the Smashing Conjecture for the stable module category of a
self-injective artin algebra in terms of model categories.
Theorem 1. Let Rbe a self-injective artin algebra and Mod(R)be the category of ar-
bitrary R-modules. The Smashing Conjecture for the stable module category of Ris
equivalent to the statement that for all cotorsion pairs X= (C,F)such that Cand Fare
closed under filtered colimits the associated model category Mod(R)Xis finitely generated.
The derived category of a differential graded algebra (dg algebra) plays a central role in
the study of triangulated categories arising in algebra. Every algebraic triangulated cat-
egory with small coproducts that is generated by a compact object is triangle equivalent
to the derived category of a dg algebra [Kel94a].
Theorem 2. Let Abe a dg algebra and Cbe a smashing subcategory of D(A). If
L:D(A)→ D(A)is the localization functor associated with C, then there are dg alge-
bras ALand A0and a diagram of morphisms of dg algebras A←A0→ALthat induces
up to isomorphism the canonical map D(A)(A, A)→ D(A)(LA, LA)in cohomology. If A
is cofibrant, then the canonical map is induced by a morphism A→ALof dg algebras.
Using Keller’s result we deduce:
Corollary 1. Let Tbe an algebraic triangulated category Twith small coproducts that
is generated by a compact object. If Cis a smashing subcategory in Tand L:T → T is
the localization functor corresponding to C, then there is a dg algebra Aand a morphism
of dg algebras A→ALthat induces L.
Let Abe a cofibrant dg algebra with graded-commutative cohomology ring H∗Aand
let pbe a prime ideal in H∗Athat is generated by homogeneous elements. Since the full
subcategory Cpof dg A-modules with the property that H∗Mp= 0 is smashing, there
exists a localization functor Lp:D(A)→ D(A) that annihilates Cp. Define the localization
of the dg algebra Aat the prime by Ap:= ALp.
Corollary 2. The dg algebra morphism A→Apof Theorem 2 induces the canonical
map H∗A→(H∗A)p. Furthermore D(Ap)can be characterized by a universal property.
The derived category of a hereditary abelian category Ais strongly related to Aitself
and can be described in a combinatorial way if Ais the category of representations of a
quiver.
We enhance the theory of classifications of thick subcategories [DHS88, Nee92, BCR97]
to the field of representation theory of algebras:
2
Theorem 3. For a hereditary abelian category Athe zeroth homology group functor
induces a one to one correspondence between the thick subcategories of the bounded derived
category Db(A)and the thick subcategories in A.
In particular Theorem 3 holds for the bounded derived category Db(mod(A)) of the
category of finitely presented modules over a hereditary algebra A. As an application
we determine the thick subcategories of Db(mod(A)) in two examples combinatorially.
Furthermore Theorem 3 implies that the thick subcategories of the category of represen-
tations of a Dynkin quiver are independent of the orientation of the quiver.
An analogous result holds in the full derived category.
Theorem 4. Let Abe a hereditary Grothendieck category. The localizing subcategories
of D(A)correspond bijectively under the zeroth homology group functor to the thick sub-
categories that are closed under arbitrary direct sums in A.
As a consequence we are able to prove the Smashing Conjecture in the following case.
Theorem 5. Let Abe a hereditary artin algebra of finite representation type and let
D(Mod(A)) be the derived category of the category of all A-modules. The Smashing
Conjecture holds for D(Mod(A)).
Outline In Section 2 the background that is necessary to formulate the Smashing Con-
jecture is presented. Furthermore we point out the two faces of localization: endofunctors
and adjoint pairs of functors. Section 3 contains a historical overview on the Smashing
Conjecture and the Telescope Conjecture. We explain in what sense the Smashing Conjec-
ture is a generalization of the Telescope Conjecture. Furthermore results and applications
of the Smashing Conjecture are described. We recall the language of model categories
and the relation of abelian model categories with cotorsion pairs in Section 4. On the one
hand we enhance this relation by proving that cofibrantly generated model categories de-
termine cotorsion pairs that are cogenerated by a set and on the other hand we specialize
this relation and use our result to obtain Theorem 1. In Section 5 we recall differential
graded algebras and their derived category. We show Theorem 2 and draw the conse-
quences Corollary 1 and Corollary 2. Finally Section 6 contains the classifications stated
in Theorem 3 and Theorem 4, the illustration by two combinatorial examples and the
proof of Theorem 5.
Acknowledgements I would like to thank my advisor Henning Krause for his support
of this thesis. I had the freedom to follow my interests and in the moments of uncertainty
he helped me with knowledge and guidance. Special thanks to Apostolos Beligiannis and
Dirk Kussin for being my referees. I am grateful to Bernhard Keller and Dave Benson for
their ideas concerning my joint project with Birgit Huber that led to the results in Sec-
tion 5. Thanks to the Deutsche Forschungsgemeinschaft for their financial support that
in particular allowed me to attend very interesting conferences. I enjoyed the atmosphere
in our office: every question could be asked no matter how embarrassing or philosophi-
cally it was, thanks to Birgit Huber and Karsten Schmidt. It was a pleasure to be part
of the representation theory group that provided a friendly and helpful environment for
my research. Thanks to Steffen Sagave, Karsten Schmidt for proofreading parts of this
thesis. It was very productive to discuss various aspects of mathematics with Steffen
Sagave. His suggestions improved this thesis. Finally I thank my wife Mareke for her
love and for the energy that she is giving me.
3
2 Triangulated categories and their localization
In this section we study triangulated categories and various localizations of them. We
pursuit two purposes besides introducing notation. The first is to systematically point out
the parallels between subcategories and localizations of triangulated categories following
[Kra06]. At the end the framework to formulate the smashing conjecture is provided.
Verdier’s thesis [Ver96], the book of Weibel [Wei94] and the appendix in Margolis’ book
[Mar83] serve as references for triangulated categories as described in Paragraph 2.1. For
the theory of localizations of triangulated categories developed in Paragraph 2.2, 2.3 and
2.4 the reader is referred to [Ver96, HPS97, Kra06].
2.1 Definitions and examples
Starting with some basics we define the notion of a triangulated category, state some
properties and illustrate the concept with three examples that are important later on.
Let Tbe an additive category and Σ: T → T an additive endofunctor. A dia-
gram (a, b, c): Xa
−→ Yb
−→ Zc
−→ ΣXin Tis called triangle. A morphism of triangles
(a, b, c)→(a0, b0, c0) is given by the following commutative diagram in T:
Xa//
f1
Yb//
f2
Zc//
f3
ΣX
Σf1
X0a0
//Y0b0
//Z0c0
//ΣX0
Definition 2.1.1. A triple (T,Σ,∆) consisting of an additive category T, an additive
endo-equivalence Σ: T → T called the suspension functor and a class of triangles ∆
called the exact triangles is a triangulated category, if it satisfies the following axioms:
(TR1) A triangle isomorphic to an exact triangle is exact. The triangle 0 →X=
−→ X→0
is exact, and every map a:X→Ycan be completed to an exact triangle
Xa
−→ Y→Z→ΣX.
(TR2) A triangle (a, b, c) is exact, if and only if (b, c, −Σa) is exact.
(TR3) If (a, b, c) and (a0, b0, c0) are exact triangles, then morphisms f1and f2in Tsuch
that f2◦a=a0◦f1can be completed to a morphism of triangles
Xa//
f1
Yb//
f2
Zc//
f3
ΣX
Σf1
X0a0
//Y0b0
//Z0c0
//ΣX0.
(TR4) If (a1, a2, a3), (b1, b2, b3) and (c1, c2, c3) are exact triangles such that c1=b1◦a1, then
4
there is an exact triangle (d1, d2, d3) making the following diagram commutative
Xa1//Ya2//
b1
U
d1
a3//ΣX
Xc1//Z
b2
c2//V
d2
b3//ΣX
Σa1
W
b3
W
d3
Σb3//ΣY
ΣYΣa2//ΣU.
If Xa
−→ B→C→ΣXis an exact triangle, then Cis called the cone of a, and we
sometimes write cone(a) for C.
Remark 2.1.2. The concept of a triangulated category was discovered independently by
Verdier [Ver96] and Puppe [Pup62] who studied derived categories and stable homotopy
theory, respectively.
The homomorphisms in a triangulated category Tform a Z-graded abelian group by
setting
Tn(X, Y ) = T(X, ΣnY).
The graded abelian group T∗(X, Y ) is a graded right module over the graded ring
End∗
T(X) := T∗(X, X).
There are plenty of examples of triangulated categories. We will concentrate on three
types of triangulated categories which play a role in the Sections 6, 3 and 4: the derived
category, the stable homotopy category and the stable module category.
Example 2.1.3. [Ver96, II Theorem 2.2.6, III Theorem 1.2.2] Let Abe an abelian
category. Then Ch(A) denotes the category of Z-graded complexes in A. Formal inversion
of the quasi isomorphisms yields the derived category D(A). It can be constructed using
calculus of fractions [GZ67, Ver96]. In general the homomorphisms between two objects in
D(A) do not form a set but by adding the condition that Ais a Grothendieck category 1
this set-theoretic problem can be overcome [Bek00]. We will write D(R), if Ais the
module category of a ring R. Starting with the category of bounded complexes Chb(A)
in Awe obtain the bounded derived category Db(A). If Ais the category of finitely
presented modules mod(R) over a ring R, then the bounded derived category is denoted
by Db(R)2.
The suspension functor is given by the shift functor [1]. It maps a complex (C, dC)
to the complex (C[1], dC[1]) which is defined by C[1]n:= Cn+1 and dn
C[1] := −dn+1
C. Let
f:C→Dbe a map of complexes. The mapping cone cone(f) of fis the complex with
1An abelian category Ais called Grothendieck category if Ais cocomplete, direct limits are exact and
there is a generator in A[Ste75].
2If Ris right coherent, then the module category is abelian. Otherwise Db(R) can be defined by using
that the module category is exact in the sense of Quillen.
5
Cn+1 ⊕Dnin degree nand the differential
dn
cone(f)=−dn+1
C0
fn+1 dn
D.
Note that there is a canonical map cone(f)→C[1]. A triangle X→Y→Z→X[1] in
the derived category is said to be exact, if it is isomorphic to the diagram
Cf
−→ D→cone(f)→C[1] for some map f.
Example 2.1.4. [Hap88, I Theorem 2.6] Let Rbe a Frobenius ring, i.e., a ring with
enough projective and injective modules such that the projectives and injectives coincide.
Two morphisms fand gbetween R-modules are called stably equivalent, if their difference
f−gfactors through a projective. Define the stable module category Mod(R) to have the
same objects as Mod(R) and Mod(R)(M, N) to be the set of stable equivalence classes
of morphisms between the R-modules Mand N.
Let X∈Mod(R) and X→Ebe a monomorphism such that Eis injective. Choose a
module ΣMsuch that
0→M→E→ΣM→0
is exact. Then Σ: Mod(R)→Mod(R) is a well-defined equivalence of categories and
serves as the suspension functor for the stable module category. A triangle in Mod(R) is
exact, if it is isomorphic to (a, b, c) in the following commutative diagram:
0//Xa//Yb//
Z//
c
0
0//X//E//ΣX//0.
In the same way it is possible to construct a stable module category mod(R) starting
with the category of finitely presented modules mod(R).
Example 2.1.5. [Vog70, BF78, Hov99] A spectrum is a sequence of pointed spaces (sim-
plicial sets or topological spaces) E={En}n≥0together with structure maps
σn: ΣEn→En+1. Here, Σ is the (topological) suspension functor that is defined on
a pointed space Xby ΣX=X×[0,1]
X×{0,1}∪{∗}×[0,1] .
Spectra form a category by means of sequences of maps of spaces that commute with
the structure maps. An important spectrum is the sphere spectrum Swhich is the n-
dimensional sphere in degree n. More generally for a given space Xthe suspension
spectrum Σ∞Xis in degree nthe n-fold suspension ΣnXof X.
Let nbe an integer; the n-th homotopy group of a spectrum Eis defined as πn(E) :=
colimkπk+n(En). The n-th stable homotopy group πs
∗(X) of a space Xis defined to
be the n-the homotopy group of Σ∞X. Call a map of spectra stable equivalence, if it
induces an isomorphism in all homotopy groups. The stable homotopy category SHC is
the localization of the category of spectra with respect to stable equivalences (as for the
derived category, it is not a priori clear that the “Hom-sets” in SHC are sets, but with
the use of model categories [BF78] this can be shown).
The shift functor [1]: SHC → SHC maps a spectrum (En, σn) to (En+1, σn+1) where
E[1]0=∗is the one point space. It turns out that the shift functor is isomorphic to the
functor that suspends a spectrum level-wise. The shift functor serves as the suspension
functor on the triangulated structure on SHC. The exact triangles are the triangles
isomorphic in SHC to the homotopy cofiber sequences.
6
In the categories SHC, the derived category D(A) of an abelian category Athat admits
small coproducts and the stable module category Mod(R) small coproducts exist and the
coproduct can be used to define a finiteness condition.
Definition 2.1.6. Let Tbe a triangulated category with small coproducts. An object
Xin Tis compact, if for each family of objects {Yi}i∈Ithe canonical map
M
i∈I
T(X, Yi)→ T (X, M
i∈I
Yi)
is an isomorphism. We write Tcfor the full subcategory of compact objects in T.
The compact objects in SHC are the finite spectra. A spectrum Eis finite, if there
is an integer ksuch that the k-fold suspension ΣkEis homotopy equivalent to a suspen-
sion spectrum of a finite CW-complex (or finite simplicial set). A complex in D(R) is
compact, if it is isomorphic, in the derived category, to a bounded complex of finitely
generated projective modules. Such complexes are called perfect, and the full subcate-
gory of perfect complexes is abbreviated with Dper(R). Furthermore Mod(R)c= mod(R).
Now we turn to important properties of triangulated categories. The Hom-functor relates
exact triangles to long exact sequences in an abelian category.
Proposition 2.1.7. [Mar83, Appendix 2 Proposition 5] Let X→Y→Z→ΣXbe
an exact triangle and let Abe an object. Then there is a long exact sequence of abelian
groups
· · · → T (A, ΣnX)→ T (A, ΣnY)→ T (A, ΣnZ)→ T (A, Σn+1X)→. . . .
Analogous to the Five-Lemma in abelian categories we have:
Proposition 2.1.8 (Five-Lemma). [Mar83, Appendix 2 Proposition 6]
Let X→Y→Z→ΣXand X0→Y0→Z0→ΣX0be exact triangles and let
X//
Y//
Z//
ΣX
X0//Y0//Z0//ΣX0
be commutative. If two of the vertical arrows are isomorphisms, then so is the third.
As an easy consequence, we have
Corollary 2.1.9. A map f:X→Yin Tis an isomorphism, if and only if cone(f)∼
=0.
Proof. Let cone(f)∼
=0. The following diagram determines a map of triangles
Σ−1(cone(f)) //
∼
=
Xf//
f
Y//cone(f)
∼
=
0Y Y //0,
and the Five-Lemma yields that fis an isomorphism.
7
If f:A→Bis an isomorphism in T, then there is a commutative diagram
A A //0//ΣA
Af//B
∼
=f−1
OO
//cone(f)//
OO
ΣA
and by axiom (TR3) there is a fill-in map cone(f)→0 making the diagram commute.
Hence we get a map of triangles and by the Five-Lemma the fill-in map is an isomorphism.
Therefore cone(f)∼
=0.
Definition 2.1.10. A functor T → S between triangulated categories is called exact or
triangle functor, if it commutes with the suspensions and maps exact triangles to exact
triangles. A functor T → A from a triangulated category to an abelian category is called
cohomological, if it maps exact triangles to long exact sequences of abelian groups.
Proposition 2.1.7 tells us that the Hom-functor
T∗(A, −): T → Mod(T∗(A, A))
is cohomological.
Definition 2.1.11. (i) A small triangulated category Tis generated by a set of objects
S, if Tis the smallest subcategory containing Sthat is closed under triangles,
suspensions and direct summands.
(ii) A triangulated category with small coproducts Sis generated by a set S0of objects,
if Sis the smallest subcategory containing S0that is closed under triangles, shifts
and direct sums.
(iii) A triangulated category is compactly generated, if it is generated by a set of compact
objects.
The sphere spectrum Sgenerates SHCc. For a Frobenius algebra Aover a commutative
ring kthe category mod(A) is generated by k. The triangulated categories with small
coproducts D(R), SHC and Mod(A) are generated by the ring R, the sphere spectrum S
and the ring k, respectively.
The following theorem characterizes the Hom-functors among the cohomological func-
tors.
Brown Representability Theorem 2.1.12. [Nee96, Theorem 3.1] Let Tbe a trian-
gulated category with small coproducts which is compactly generated and let Abbe the
category of abelian groups. A functor F:Top → Abis cohomological and sends small
coproducts to small products, if and only if there is an object Xin Tsuch that Fis
isomorphic to HomT(−, X).
Corollary 2.1.13. Let Tbe a compactly generated triangulated category with small co-
products. An exact functor F:T → S has a right adjoint, if and only if it commutes with
arbitrary direct sums.
Proof. The right adjoint is defined to map an object Yin Sto the representing object of
the cohomological functor HomS(F(−), Y ): Top → Ab.
In the following three subsections, we return to our purpose and describe subcategories
and their relation to localization.
8
2.2 Thick subcategories and Verdier-quotients
Thick subcategories and the related Verdier-localizations are introduced in this part.
Throughout this subsection let Tbe a triangulated category.
Definition 2.2.1. A full subcategory C ⊂ T is called triangulated, if it is closed under
forming triangles and suspensions in T. If furthermore Cis closed under retracts, it is
called thick.
Lemma 2.2.2. The subcategory of compact objects in a triangulated category Twith
direct sums is thick.
Proof. The category Tcis obviously closed under retracts and suspensions. It is closed
under forming triangles in Tbecause of Proposition 2.1.7 and the Five-Lemma for abelian
categories.
Example 2.2.3. Let F:T → S be exact and let ker(F) be the full subcategory
{X∈ T | F(X) = 0}in T. Then ker(F) is thick. Also the kernel of a cohomological
functor H:T → A, which is defined as Tn∈Zker(H◦Σn), is thick.
Theorem 2.2.4. [Ver96, II.2.1.8] Let C ⊂ T be a triangulated subcategory. Then there
is a triangulated category T/Cand an exact functor Q:T → T /Cannihilating Cthat
satisfy:
(i) The kernel of Qis the smallest thick subcategory containing C.
(ii) The functor Qis universal among the exact functors annihilating C, i.e, if F:T →
Sis exact such that F(C) = 0 for all objects C∈ C, then there is a unique exact
functor Gsuch that
TF//
Q
S
T/C
G
==
commutes.
(iii) For every cohomological functor H:T → A annihilating Cthere is a unique coho-
mological functor H0:T/C → A such that H=H0◦Q.
The category T/Ccan be constructed as a category of fractions T[S−1] such that
S={σ|cone(σ)∈ C} is the class of morphisms which are inverted and the quotient
functor Qis the canonical functor T → T [S−1]. We call Q:T → T /CVerdier-localization
or quotient functor.
2.3 Localizations
In this part localizations of triangulated categories are defined, and we clarify the relations
between localization functors and localizing subcategories. As an example cohomological
localization is discussed.
Let Tbe a triangulated category with small coproducts.
9
Definition 2.3.1. A full triangulated subcategory Cin Tis called localizing, if it is closed
under direct sums.
Proposition 2.3.2. [Nee01, 1.6.8] Every localizing subcategory in a triangulated category
Twith all small coproducts is thick.
Let C ⊂ T be localizing. Since Cis also thick we can construct the Verdier-quotient
functor Q:T → T /C.A priori the category T/Cis a large category, that is the morphisms
between two objects do not necessarily form a set. The following lemma gives a necessary
and sufficient condition for T/Cto be a category. It can be proved by using the Brown
Representability Theorem 2.1.12.
Lemma 2.3.3. [Ric00][Theorem 5.1] Let Cbe a localizing subcategory in a compactly
generated triangulated category with small coproducts T. The following statements are
equivalent:
(i) The maps between two objects in T/Cform a set.
(ii) The quotient functor Q:T → T /Chas a right adjoint.
So it is natural to investigate the existence of a right adjoint of the quotient functor
T → T /C. The following lemmas provide the necessary background on adjoint functors
to address this question.
Let Cand Dbe categories and
CF//D
G
oo
be a pair of adjoint functors such that Fis the left adjoint. Let ψ: id →G◦Fbe the unit
and φ:F◦G→id the counit of the adjunction. Let S0={σ∈Mor(C)|F(σ) is invertible}.
Using calculus of fractions [GZ67] it is possible to construct a functor QS0:C → C[(S0)−1]
that is universal among the functors that invert elements of S0.
Lemma 2.3.4. [GZ67, I.1.3] The following assertions are equivalent:
(i) The functor ¯
F:C[(S0)−1]→ D with F=¯
F◦QS0is an equivalence.
(ii) The functor Gis fully faithful.
(iii) The counit φ:F◦G→idDis invertible.
Now let (F, G) be an adjoint pair satisfying the conditions of Lemma 2.3.4. Let L=
G◦F:C → C and ψ: idC→G◦Fbe the counit of the adjunction. The following lemma
clarifies when, starting with a pair (L, ψ), we can recover the pair of adjoint functors
(F, G).
Lemma 2.3.5. [Kra06, Lemma 2.2] Let L:C → C be a functor and ψ: idC→Lbe a
natural transformation. Then the following statement are equivalent:
(i) Lψ:L→L2is invertible and Lψ =ψL.
(ii) There is an adjoint pair of functors
CF//D
G
oo
such that Fis the left adjoint, Gis fully faithful, L=G◦Fand ψ: idC→G◦F
is the unit of adjunction.
10
Proof. We indicate how to translate the data of (i) in (ii) and vice versa. Starting with an
endofunctor L:C → C and a natural transformation ψ, define Dto be the full subcategory
with objects {X∈ C | ψX:X∼
=
−→ LX}. Let F:C → D be given by Land G:D → C be the
inclusion. Conversely, if an adjoint pair (F, G) is given, let L=G◦Fand ψ: idC→G◦F
be the unit of the adjunction.
Let Cbe a localizing subcategory in Tand Q:T → T /Cthe quotient functor. The
question concerning the existence of a right adjoint functor for Qcan now be answered
by specializing Lemma 2.3.4 and Lemma 2.3.5 to F=Q.
Corollary 2.3.6. The following statements are equivalent:
(i) The quotient functor Q:T → T /Chas a right adjoint R.
(ii) The quotient functor Q:T → T /Chas a right adjoint Rwhich is fully faithful.
(iii) The quotient functor Q:T → T /Chas a right adjoint R, and the unit φ:Q◦R→
idCis invertible.
If the right adjoint Rof Qexists, we set L:= R◦Qand let ψ: idC→Lbe the unit of
the adjunction. Then Lψ =ψL, and Lψ:L→L2is invertible.
Proof. We will show that the sets S={σ∈Mor(T)|cone(σ)∈ C} and S0={τ∈
Mor(T)|Q(τ) is invertible}are the same. Then Q=QS0and the equivalence of the three
assertions follows from Lemma 2.3.4. The last assertion is a consequence of Lemma 2.3.5.
The inclusion S⊂S0holds by definition. For the other inclusion, let τ:X→Ysuch
that Q(τ) is invertible. By Corollary 2.1.9 cone(Qτ)∼
=0. Since Qis an exact functor,
it follows that cone(Qτ)∼
=Q(cone(τ)). Therefore cone(τ) is in the kernel of Qwhich is
equal to C.
This corollary motivates
Definition 2.3.7. Let Tbe a triangulated category with small coproducts. Let in ad-
dition (L, ψ) be a pair consisting of an exact endofunctor L:T → T and a natural
transformation ψ: idT→L. The pair (L, ψ) is called localization functor or just local-
ization, if Lψ:L→L2is invertible, the natural transformation ψcommutes with the
suspension functor and Lψ =ψL.
Sometimes we suppress the natural transformation ψof a localization functor in our
notation. Two important classes of objects are associated with a localization.
Definition 2.3.8. Let L:T → T be a localization. An object X∈ T is called L-local,
if the localization morphism ψX:X→LX is an isomorphism. The object Xis called
L-acyclic, if LX = 0. The full subcategory of L-local objects is denoted by TLand ker(L)
is the full subcategory of L-acyclics.
Since the category ker(L) is localizing a localization functor determines a localizing
subcategory.
The following four assertions deal with the properties of localizations of triangulated
categories. They are well-known and were studied for instance in [HPS97]. We refer to
[Kra06] since we need slightly more general statements.
11
Lemma 2.3.9. [Kra06, Lemma 2.5] Let Xbe an object in Tand La localization. Then
the following assertions are equivalent:
(i) Xis L-local.
(ii) There is an object X0in Tsuch that X∼
=LX0.
(iii) For all f:Y→Zwith the property that L(f)is an isomorphism, the map
f∗: HomT(Z, X)→HomT(Y, X)
is an isomorphism of abelian groups.
Lemma 2.3.10. [Kra06, Proposition 2.7] The inclusion functor TL→ T is right adjoint
to the functor T → TLsending an object Xto LX.
Therefore the localization Lis determined by the category of L-local objects.
The L-acyclic and L-local objects are related by the following proposition.
Proposition 2.3.11. [Kra06, Lemma 2.8]
(i) An object X∈ T is L-acyclic, if and only if HomT(X, Y ) = 0 for all L-local objects
Y.
(ii) The functor Linduces an equivalence of triangulated categories
T/ker(L)≃
−→ TL.
The L-acyclic objects determine the L-locals up to equivalence. Therefore the local-
ization functor Lis determined by ker(L).
The converse is not known in general. If C ⊂ T is a localizing subcategory, then it is
not clear whether a localization functor L:T → T with C= ker(L) exists. Nevertheless
it is possible to make some assertions. Casacuberta, Guti´errez and Rosick´y [CGR06]
have shown that such a functor exists by adding Vopˇenka’s Principle to the axioms of
set theory. Miller gave a construction of a localization functor on the stable homotopy
category assuming that the given localizing subcategory is generated by objects which
are compact in SHC [Mil92]. Such localizations are called finite and will be studied in
the next section.
In the remaining part of this section we introduce localization functors that are induced
by localizations in the homology as an important example.
Let Tbe a triangulated category with small coproducts which is generated by a compact
object Aand let Γ := T∗(A, A) be the graded endomorphism ring. Denote by Modgr(Γ)
the category of graded Γ-modules and let H∗:T → Modgr(Λ) be the cohomological
functor T∗(C, −). The functor H∗relates the triangulated category Tto the abelian
category Modgr(Γ).
Definition 2.3.12. Let Abe an abelian category. A pair (L, ψ) consisting of an exact
endofunctor L:A → A and a natural transformation ψ: idA→Lis called localization
functor, if Lψ =ψL, the natural transformation ψcommutes with the suspension functor
and Lψ:L→L2is an isomorphism.
12
Theorem 2.3.13. [Kra06, Theorem 3.1] Let (L, ψ)be a localization functor on Modgr(Γ).
Then there is a localization (˜
L, ˜
ψ)on Tsuch that
T˜
L//
H∗
T
H∗
Modgr(Γ) L//Modgr(Γ)
commutes up to natural isomorphism. More precisely, LH∗ψ,ψH∗˜
Land
LH∗LH∗˜
ψ
−−−−→ LH∗˜
L(ψH∗˜
L)−1
−−−−−−→ H∗˜
L
are invertible. Furthermore ˜
LX = 0, if and only if LH∗X= 0. If Xis ˜
L-local, then
H∗Xis L-local, and if H∗reflects isomorphisms, then the converse also holds.
Remark 2.3.14. [Kra06, Rem. 2.4] Let L: Modgr Γ→Modgr Γ be an exact localization
functor and denote by ˆ
L:T → T the exact localization functor which exists by The-
orem 2.3.13. Write Cfor the ˆ
L-acyclic objects. By Lemma 2.3.5 ˆ
Land Lgive rise to
adjoint pairs of functors
T
Q//T/C
R
ooand Modgr Γ
F//(Modgr Γ)L
G
oo
satisfying ˆ
L=R◦Qand L=G◦F. The diagram below commutes up to natural
isomorphism.
T
Q
T(A,−)∗
//Modgr Γ
F
T/C
R
T/C(QA,−)∗
//(Modgr Γ)L
G
TT(A,−)∗
//Modgr Γ
The following two results are joint work with Birgit Huber [BH07].
Proposition 2.3.15. Suppose that the ring T(A, A)∗is graded-commutative and let
L: Modgr T(A, A)∗→Modgr T(A, A)∗be a localization with respect to a multiplicatively
closed subset of homogeneous elements S⊆ T (A, A)∗. If C= ker ˆ
L, then the diagram
T
Q
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.
13
Proof. The diagram commutes by Remark 2.3.14. 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
LA
∼
=//LH∗ˆ
LA
in which the lower and the right hand side morphism are bijective by Theorem 2.3.13.
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 //
Q
S−1T(A, A)∗
S−1Q
∼
=
r
uu
T/C(QA, QA)∗ν
∼
=//S−1T/C(QA, QA)∗
commute.
The lower triangle commutes by the following argument. Since both the maps ν◦r
and S−1Qmake the following diagram of T(A, A)∗-modules
T(A, A)∗can //
ν◦Q
S−1T(A, A)∗
ν◦r
uullllllllllllll
S−1Q
uullllllllllllll
S−1T/C(QA, QA)∗
commute, the universal property of localization of modules implies that ν◦r=S−1Q.
Hence ris an isomorphism.
Proposition 2.3.16. Suppose that the ring T(A, A)∗is graded-commutative and let
L: Modgr T(A, A)∗→Modgr T∗(A, A)∗be localization 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, n ∈Z}. On this
purpose consider an arbitrary object M∈ C and prove that T(cone(σ), M)∗= 0 for all
σ∈Simplies M= 0.
Every triangle
Aσ
−→ A[n]→cone(σ)→A[1]
14
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)∗.
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 L-local. On the other hand,
T(A, M)∗is L-acyclic. It follows that T(A, M)∗= 0 and hence M= 0.
2.4 Smashing and finite localizations
In this part two classes of localizations are introduced: the smashing and the finite
localizations. We investigate their relations which lead to the Smashing Conjecture.
Let Tbe a triangulated category with small coproducts.
Definition 2.4.1. A localization L:T → T is called smashing, if Lcommutes with small
coproducts.
Let (Q, R) be the adjoint pair corresponding to Laccording to
Lemma 2.3.5. Then Lis smashing, if and only if Rcommutes with direct sums. Therefore
the following definition is sensible:
Definition 2.4.2. A localizing subcategory C ⊂ T is called smashing, if the quotient
functor T → T /Chas a right adjoint that commutes with direct sums.
In the following we describe the origin of the name “smashing”: let Tbe equipped with
a symmetric monoidal product compatible with the triangulation, i.e., a pair (− ∧ −, S)
such that −∧−:T ×T → T is a functor that is exact in both variables and that commutes
with direct sums. The unit Sis asked to be a compact generator of T. An example of
such a category is the stable homotopy category together with the smash product − ∧ −
and the sphere spectrum Sas the unit [Ada74, EKMM97, HSS00]. The derived category
D(R) of a commutative ring Rtogether with the derived tensor product over Rand the
ring R, considered as a complex, that is concentrated in degree 0 is an algebraic example.
Lemma 2.4.3. Let Cbe localizing in T. If Cis an object in C, then for all objects X∈ T
the product C∧Xis in C.
Proof. Consider the full subcategory CC:= {X∈ T | C∧X∈ C} in T. The unit Sis in
CC. Since the smash product is exact and preserves direct sums CCis also closed under
exact triangles and arbitrary direct sums. Therefore, CC⊂ T is localizing and contains
S. Since Sis a compact generator, CC=T.
If (L, ψ) is a localization on Tand Xis an object in T, then there is a canonical map
αX:LS ∧X→LX.
To define it, first consider the exact triangle CS →SψS
−−→ LS →ΣCS. Smashing with
Xyields the exact triangle
CS ∧X→S∧X→LS ∧X→ΣCS ∧X.
15
Now CS ∧Xis L-acyclic by Lemma 2.4.3 and by Corollary 2.1.9 it follows that
L(ψS∧X): LX →L(LS∧X) is an isomorphism in T. We define αXto be the composition
LS ∧XψLS∧X
−−−−→ L(LS ∧X)(L(ψS∧X))−1
−−−−−−−−−→ LX.
Now we are able to characterize smashing localizations:
Proposition 2.4.4. [HPS97, 3.3.2] The following assertions are equivalent:
(i) The functor L:T → T is a smashing localization.
(ii) The natural map αX:LS ∧X→LX is an isomorphism.
(iii) The category TLof L-local objects is localizing.
The name “smashing” originates from assertion (ii). For a smashing localization functor
Lapplying Lis the same as smashing with LS.
We give a class of examples of smashing localizations.
Proposition 2.4.5. Let L:T → T be a localization functor. If the L-acyclics ker(L)
are generated by a set of objects {Ci}i∈Iwhich are compact in T, then the localization L
is smashing.
Proof. Let C:= ker(L) be the category of L-acyclics, and let {Xα|α∈A}be a family of
objects in T. We first show that Lα∈ALXαis L-local. Since {Ci}i∈Iis a set of compact
generators of Cby Proposition 2.3.11 it is enough to show that T(Ci,Lα∈ALXα) = 0
for all i∈I. Since each Ciis compact there is an isomorphism
T(Ci,M
α∈A
LXα)∼
=M
α∈A
T(Ci, LXα).
The abelian group T(Ci, LXα) = 0 for all i∈Ibecause Ciis L-acyclic and LXαis
L-local. Hence LαLXαis L-local. Therefore the map
ψLi∈ILXi:M
i∈I
LXi→L(M
i∈I
LXi)
is an isomorphism.
To end the proof we show that the map
L(M
i∈I
ψXi): L(M
i∈I
Xi)→L(M
i∈I
LXi)
is an isomorphism. For each i∈Ithere is an exact triangle
CXi→Xi→LXi→ΣCXi
such that CXi∈ C. Since coproducts are exact, the following triangle
M
i∈I
CXi→M
i∈I
Xi→M
i∈I
LXi→M
i∈I
ΣCXi
16
is exact. The object LICXiis L-acyclic because Cis localizing. As Lannihilates
L-acyclic objects it follows that L(Li∈IψXi) is an isomorphism. Hence the composition
(ψLi∈ILXi)−1◦L(M
i∈I
ψXi): L(M
i∈I
Xi)→M
i∈I
LXi
is an isomorphism and Lcommutes with direct sums. Therefore Lis smashing.
Definition 2.4.6. A localization L:T → T is called a finite localization, if the category
of L-acyclic objects is generated as a triangulated category with small coproducts by a
set of objects that are compact in T. A localizing subcategory C ⊂ T is said to be of
finite type, if C= ker(L) for a finite localization functor L.
As we have seen in Proposition 2.4.5 every finite localization is smashing. We state the
converse in the following conjecture which was formulated for the first time by Neeman
[Nee92].
Smashing Conjecture 2.4.7. In a triangulated category with small coproducts, every
smashing localization is finite.
We consider the Smashing Conjecture as an assertion on a fixed triangulated category
with small coproducts Trather than a statement about all triangulated categories with
small coproducts. Therefore we use the terminology “the Smashing Conjecture for T” in
the sequel. The Smashing Conjecture is sometimes called Telescope Conjecture due to
its origin. We choose this name to avoid confusion.
In the next chapter we discuss the origin, examples, results and applications of the
Smashing Conjecture.
17
3 The Telescope Conjecture and the Smashing Conjecture
This section gives an overview on the history, results and applications of the Smashing
Conjecture 2.4.7. The aim of Paragraph 3.1 is to show that the Smashing Conjecture
is a generalization of Ravenel’s Telescope Conjecture of stable homotopy theory which
was the starting point of the investigation. In Paragraph 3.2 and 3.3 an example of a
triangulated category for which the Smashing Conjecture is valid and an example for
which it does not hold are introduced. In 3.4 Krause and Solberg’s reformulation of the
Smashing Conjecture for stable module categories and a result by Angeleri-H¨ugel, ˇ
Saroch
and Trlifaj are stated. At the end we describe general results on the Smashing Conjecture
in Paragraph 3.5 and point out applications to chain lifting problems, non-commutative
localization of rings and algebraic K-theory. During this section we give no proofs and
do not go into the details in order to present the topic streamlined and compact.
3.1 The Telescope Conjecture in stable homotopy theory
We recall basic notions of stable homotopy theory, in particular Bousfield localization,
the p-local stable homotopy category and two examples of spectra. Having these it is
possible to describe the Periodicity Theorem, define the mapping telescope and state the
Telescope Conjecture in its very first version. In addition we indicate a motivation for
the telescope conjecture.
Recall from the previous section that the stable homotopy category SHC is a triangu-
lated category with small coproducts that is compactly generated by the sphere spectrum
S. Furthermore it is symmetric monoidal by means of the smash product − ∧ − and the
unit S.
Spectra are strongly related to generalized cohomology theories. Given a generalized
cohomology theory
E∗:SHCop →(graded abelian groups)
by the Brown Representability Theorem 2.1.12 there is a spectrum Esuch that
En(−)∼
=HomSHC(−,ΣnE).
On the other hand a spectrum Egives rise to a generalized homology theory
E∗:SHC → (graded abelian groups)
by setting En(F) := πn(E∧F). For example, the Moore spectrum MZ(p), which is defined
as the cone of Sp·id
−−→ S, is the spectrum representing the generalized homology theory
π∗(−)⊗ZZ(p).
Bousfield [Bou79] showed that for a spectrum Ethere is a localization functor
LE:SHC → SHC such that the LE-acyclic objects are the E∗-acyclics. That is,
LE(F) = 0, if and only if E∗(F) = 0 for any spectrum F. It is obtained as the fi-
brant replacement in a model structure on the category of spectra in which the weak
equivalences are the isomorphisms in the represented homology theory E∗. We will call
LEsometimes E∗-localization.
For a prime pthe p-local stable homotopy category SHCpis defined to be the category
of LMZ(p)-local objects.
A monoid Rin SHC with respect to the smash product is called ring spectrum. The
multiplication on R
R∧R→R
18
induces a ring structure on π∗R. So a ring spectrum encodes a “blueprint” for a discrete
ring. The ring of homotopy groups π∗(R) is often called coefficients since R∗(S)∼
=π∗(R).
It plays an important role for the represented cohomology theory R∗.
One typical example for a ring spectrum arises from classical algebra. Given an asso-
ciative ring with unit, there is a ring spectrum HR, the Eilenberg-Mac Lane spectrum,
with the property that π0HR ∼
=R. It represents singular (co-)homology with coefficients
in R. Another example of a ring spectrum is the sphere spectrum S. The ring structure
S∧S→Sis the canonical isomorphism which exists since Sis the unit of the monoidal
structure. This spectrum represents the homology theory given by the stable homotopy
groups π∗(−).
From now on we fix a prime p∈Zand work in the p-local stable homotopy category.
We discuss two examples of spectra that play an important role in stable homotopy
theory, the Johnson-Wilson spectra E(n) and the Morava K-theories K(n) for n≥0.
Both are p-local spectra, and we follow the usual convention of suppressing the prime pin
the notation. We describe important properties of these spectra and their ring of stable
homotopy groups.
The Morava K-theory for n= 0 is K(0) = HQ. For n=∞we define K(∞) to be the
Eilenberg-Mac Lane spectrum HZ(p). The spectrum K(n) for n6= 0 has the coefficients
π∗(K(n)) = Fp[vn, v−1
n],
where vnis in degree 2pn−2. The Morava K-theories have the following remarkable
properties:
Proposition 3.1.1. [HS98, Propositions 1.4, 1.5, 1.9]
(i) K(n)is a ring spectrum and a skew field object, i.e., all modules over a K(n)are
free.
(ii) There is a K¨uneth-isomorphism
K(n)∗(X∧Y)∼
=K(n)∗(X)⊗K(n)∗(pt)K(n)∗(Y)
for two p-local spectra Xand Y.
(iii) A skew field object in SHCpis the direct sum of shifted copies of Morava K-theories.
The Morava K-theories are constructed from the cobordism spectrum MU by localizing
and taking quotients. Similarly the Johnson-Wilson spectrum E(n) for n > 0 can be
defined. Its homotopy is the following ring: π∗E(n) = Z(p)[v1, . . . , vn, v−1
n]. See [HS98],
[EKMM97, V.4] and [Wei05] for a construction.
The Johnson-Wilson spectra and the Morava K-theories are also related via their local-
ization functors. In fact, E(n)∗-localization is isomorphic to (Ln
i=1 K(i))∗-localization.
Let Lndenote the E(n)∗-localization. The functor Lf
nis the finite localization functor
associated with E(n). It can be constructed as finite localization in the sense of Miller
[Mil92] with respect to the set {X∈ SHCpfinite |E(n)∗(X) = 0}.
19
Periodicity and the mapping telescope
There are very readable papers [Rav92b, Rav93] and the book [Rav92a] by Ravenel which
can serve as introduction to this subject. In order to formulate the Telescope Conjecture
we need a result by Hopkins, Devinatz and Smith [DHS88] and some notation.
Again we fix a prime p.
Definition 3.1.2. A finite p-local spectrum Eis called of type n, if nis the smallest
integer such that K(n)∗(E)6= 0. A map ΣdE→Eis called vn-map, if it induces an
isomorphism in the n-th Morava K-theory and the zero-map in all other Morava K-
theories.
The mod-nMoore spectrum is of type n. Examples of vn-maps arise as self maps of
Moore spectra. Recall that π∗K(n) = Fp[v±1
n]. A vn-map induces the multiplication by
some power of vnin the n-th Morava K-theory, whence its name.
Let i≥0. Then fidenotes the iteration of icopies of a map f:X→X.
Theorem 3.1.3 (Periodicity Theorem). [HS98, Theorem 9] Let Ebe a spectrum of
type n. Then there is a non null-homotopic vn-map ΣdE→E.
Furthermore given two spectra E1and E2with vn-maps f1and f2then for every map
g:E1→E2there is a commutative square in SHC:
Σid1E1
Σid1g//
fi
1
Σjd2E2
fj
2
E1
g//E2.
The second part of the Periodicity Theorem implies that a vn-map is unique in the
sense that some iterations of two vn-maps are homotopic.
Definition 3.1.4. Let Ebe a finite p-local spectrum of type n. Then the mapping tele-
scope ˆ
Eis defined to be the homotopy colimit of the iteration of the vn-map Σ−df:E→
Σ−dE
EΣ−df
−−−→ Σ−dEΣ−2df
−−−−→ Σ−2dE→. . . .
The mapping telescope is well-defined because two choices of a vn-map are homotopic
up to respective iterates.
The Telescope Conjecture and some applications
We state the telescope conjecture of Ravenel in four equivalent formulations and discuss
its relevance.
Recall that two spectra Eand Fare Bousfield equivalent, if for all spectra Xthe
equation E∗X= 0 holds, if and only if F∗X= 0, i.e., the E∗-localizations and F∗-
localization coincide. The Bousfield class hEiof a spectrum Eis the equivalence class of
Ewith respect to this equivalence relation.
Telescope Conjecture 3.1.5. [Rav87b, 1.33] Let Ebe a p-local spectrum of type n.
Then hˆ
Ei=hK(n)i. That is, localization with respect to the mapping telescope of a
p-local spectrum of type nis the same as K(n)∗-localization for the Morava K-theory
K(n).
20
The Telescope Conjecture describes the behavior of localization with respect to the
mapping telescope of a spectrum. This is the origin of its name.
Let Ln:SHCp→ SHCpbe the localization functor associated with the Johnson-Wilson
spectrum E(n). Let X∈ SHCpbe of type n. The canonical map X→LnXfactors
through the telescope X→ˆ
Xλ
−→ LnX.
Proposition 3.1.6. [MRS01, 1.13] The Telescope Conjecture is equivalent to the state-
ment that for all X∈ SHCpthe map λ:ˆ
X→LnXis an isomorphism in the stable
homotopy category.
A major problem in algebraic topology is the computation of stable homotopy groups
of spaces and in particular the stable homotopy groups of spheres. For example there is
not a single non-contractible finite CW-complex for which the stable homotopy groups
are entirely known. A motivation for Ravenel to state the Telescope Conjecture was
its consequences on computation: One device to organize a computation is the Adams-
Novikov spectral sequence (ANSS). It exists for every space Xand may or may not
converge to the stable homotopy groups of this space. It is possible to compute the E2-
term of this spectral sequence for the mapping telescope of a type nspectrum [Dav95,
Rav86]. The ANSS collapses, but it is not clear, if the spectral sequence converges. It does
always converge for the localized spectra LnX[Rav87a]. So the Telescope Conjecture
would imply that the ANSS of ˆ
Xconverges to πs
∗ˆ
Xand would yield a possibility to
compute some stable homotopy groups.
Proposition 3.1.7. [MRS01, 1.13] Let Xbe a spectrum of type n. The Telescope Con-
jecture is equivalent to the assertion that the ANSS for ˆ
Xconverges to π∗ˆ
X.
The Telescope Conjecture was verified for n= 1 and p= 2 by Mahowald [Mah82] and
for n= 1 at all odd primes by Miller [Mil81]. Ravenel announced a counterexample for
n= 2 [Rav92b] but withdrew it later on [MRS01]. Despite the lack of counterexamples,
the Telescope Conjecture is commonly not expected to hold for all nand p.
In order to relate the Telescope Conjecture 3.1.5 to the Smashing Conjecture 2.4.7 we
discuss some properties of the Johnson-Wilson spectra. Let E(n) be the Johnson-Wilson
spectrum and Lf
nthe finite localization with respect to E(n). It is known by [Rav92a,
Theorem 7.5.6] that the functor Ln:SHC → SHC is smashing in the sense of Defini-
tion 2.4.1. In [MRS01, 1.13] there is a reformulation of the Telescope Conjecture 3.1.5 in
terms of these localizations:
Proposition 3.1.8. [MRS01, 1.13] The Telescope Conjecture is equivalent to the state-
ment that Ln=Lf
n, if Ln−1=Lf
n−1.
In particular if Ln=Lf
nfor all n, then the Telescope Conjecture is true for all n.
So the Smashing Conjecture 2.4.7 for the p-local stable homotopy category implies the
Telescope Conjecture 3.1.5.
3.2 The Smashing Conjecture in commutative algebra
The first triangulated category for which the Smashing Conjecture 2.4.7 was verified is the
derived category of a commutative noetherian ring R[Nee92, Hop87]. Neeman classified
the thick subcategories in Dper(R) as well as the localizing and smashing subcategories
21
in D(R) to deduce the Smashing Conjecture. See [ATJLSS04] for a generalization to
schemes.
Let Rbe a commutative noetherian ring and p∈SpecRbe a prime ideal. Define κ(p)
to be the residue field, i.e., the quotient of Rpby its maximal ideal pRp. For a complex
Xlet Supp(X) = {p∈Spec R|Xp6= 0}denote the support of X.
Theorem 3.2.1. [Nee92, Theorem 1.5, Theorem 3.3]
(i) The following maps are mutually inverse to each other
{C ⊂ Dper(R)| C thick}f//{P⊂Spec(R)|Pclosed under specialization}
g
oo
where f(C) = {p| ∃X∈ C :p∈Supp(X)}and g(P)is the full subcategory given by
{X∈ Dper(R)|Supp(X)⊂P}.
(ii) There are mutually inverse bijections
{C ⊂ D(R)| C smashing}f//{P⊂Spec(R)|Pclosed under specialization}
g
oo
where f(C) = {p|X⊗k(p)6= 0 ∀X∈ C} and g(P)is the smallest localizing subcat-
egory in D(R)that contains k(p)for all p∈P.
Corollary 3.2.2. [Nee92, Corollary 3.4] The Smashing Conjecture 2.4.7 is true for the
derived category of a commutative noetherian ring.
3.3 Keller’s counterexample
From the perspective of stable homotopy theory commutative non-noetherian rings are
more interesting because the ring π∗(S) of stable homotopy groups of spheres is not
noetherian. So it is natural to ask, if the Smashing Conjecture 2.4.7 remains true for the
derived category of an arbitrary commutative ring R. Keller gave an example [Kel94b]
of a non-noetherian commutative ring for which the Smashing Conjecture is not true.
Let kbe a field and l≥2 be an integer. Define B:= k[t, tl−1, tl−2, . . . ]. Its augmentation
ideal Jis generated by t, tl−1, tl−2,.... Let Abe the localization of Bat the ideal Jand
Ibe the Jacobson radical of A.
Theorem 3.3.1. [Kel94b] The localizing subcategory R ⊂ D(A)generated by the ideal Iis
smashing and contains no compact object of D(A). Hence the Smashing Conjecture 2.4.7
for D(A)is not true.
The key property of the pair (A, I) is its homological behavior. Wodzicki showed that
TorA
i(A/I, A/I) = 0 for i≥1 [Wod89]. Recall that a B´ezout domain is a domain for
which every finitely generated ideal is principal. If Ris a B´ezout domain, then we have
that for every ideal athe equations TorR
1(R/a, R/a) = a/a2and TorR
i(R/a,−) = 0 for
i > 1 hold. This forces that the Smashing Conjecture for D(R) is not true [Kra05,
Section 15].
22
3.4 The Smashing Conjecture for stable module categories
We describe Krause and Solberg’s characterization of the Smashing Conjecture for stable
module categories in terms of cotorsion pairs [KS03, Conjecture 7.9] and discuss Angeleri-
H¨ugel, Trlifaj and ˇ
Saroch’s result [AHST06, Corollary 4.10] concerning a generalization
of the Smashing Conjecture.
Recall that for a Frobenius ring Rthere are enough projective and injective modules
and that these concepts coincide. Furthermore the stable module categories Mod(R)
and mod(R) are triangulated (Example 2.1.4), and the compact objects in Mod(R) are
mod(R).
The first step toward the reformulation is to understand the localizing subcategories
in Mod(R). In fact, they arise as pairs of subcategories.
A localizing subcategory Cin a triangulated category Tdetermines another subcate-
gory, the C-local objects,
Cloc ={Y∈ T | T (X, Y ) = 0 ∀X∈ C}.
A pair (C,Cloc) of subcategories is called localizing pair. It turns out that localizing pairs
are related to the following data in the module category.
Definition 3.4.1. Let Abe an abelian category. A pair of subcategories X= (C,F) is
called cotorsion pair, if the following axioms hold:
(i) Ext1(X, Y ) = 0 for all X∈ C, if and only if Y∈ F.
(ii) Ext1(X, Y ) = 0 for all Y∈ F, if and only if X∈ C.
(iii) Every object A∈ A has a special right C-approximation, i.e., there is a short exact
sequence
0→Y→X→A→0
with X∈ C and Y∈ F.
(vi) Every object A∈ A has a special left F-approximation, i.e., there is a short exact
sequence
0→A→Y→X→0
with X∈ C and Y∈ F.
A cotorsion pair (C,F) is called thick, if C ⊂ A is closed under kernels of epimorphisms,
cokernels of monomorphisms. It is called hereditary, if Exti(X, Y ) = 0 for i≥2 and all
X∈ C and Y∈ F. A cotorsion pair (C,F) is cogenerated by a set G ⊆ C, if the following
holds: Ext1(G, Y ) = 0 for all G∈ G if and only if Y∈ F. A cotorsion pair in Mod(R) is
of finite type, if it is cogenerated by a set G ⊆ C ∩ mod(R).
A ring Ris by definition self-injective if Ris injective as a right R-module. Note that
an artin algebra Ais self-injective, if and only if it is a Frobenius ring. Krause and Solberg
translate localizing pairs in the stable category into thick cotorsion pairs in the module
category.
Theorem 3.4.2. [KS03, 6.3] Let Abe a self-injective artin algebra. There is a one to
one correspondence between the thick cotorsion pairs in Mod(A)and the localizing pairs
in Mod(A)given by (C,F)7→ (C,F).
23
Here Cdenotes the image of Cunder the canonical functor Mod(A)→Mod(A). If
furthermore Cis smashing, then the corresponding cotorsion pair is of a special form:
Theorem 3.4.3. [KS03, 7.6, 7.7] Let Abe a self-injective artin algebra and let (C,F)be
a cotorsion pair in Mod(A). The category Cis smashing, if and only if Fis closed under
filtered colimits. In that case Cis also closed under filtered colimits.
Let Abe a set of objects in Mod(A). Then colim(A) denotes the full subcategory of
all A-modules that are filtered colimits of modules in A. If Cis of finite type in the sense
of Definition 2.4.6, then the cotorsion pair fulfills an additional finiteness property.
Theorem 3.4.4. [KS03, 7.7] Let Abe a self-injective artin algebra and let (C,F)be a
cotorsion pair in Mod(A). Then Cis of finite type, if and only if C= colim(C ∩mod(A)),
i.e., the modules in Care precisely the filtered colimits of finitely presented objects in C.
The preceding considerations motivate the following conjecture.
Conjecture 3.4.5. [KS03, Conjecture 7.9] Let Abe an artin algebra and (C,F) a
cotorsion pair in Mod(A). If Cand Fare closed under filtered colimits, then C=
colim(C ∩ mod(A)).
Note that if Ais self-injective, this conjecture is equivalent to the Smashing Conjec-
ture for Mod(A). Angeleri-H¨ugel, ˇ
Saroch and Trlifaj proved the Conjecture 3.4.5 in the
following case:
Theorem 3.4.6. [AHST06, 3.3,4.10] Let Rbe a noetherian ring and (C,F)be a hered-
itary cotorsion pair in Mod(R)such that Cand Fare closed under filtered colimits. If C
consists of modules of bounded projective dimension or Bconsists of modules of bounded
injective dimension, then (C,F)is of finite type.
The idea of the proof is based on the following. If Cconsists of modules of bounded
projective dimension, then Xis a tilting cotorsion pair which are known to be of finite
type. In the other case Xis known to be countably generated and by using methods from
set theory it is shown to be finitely generated.
3.5 The Smashing Conjecture for arbitrary triangulated categories
The Smashing Conjecture 2.4.7 cannot be true for arbitrary compactly generated trian-
gulated categories with direct sums as shown by Paragraph 3.3. Nevertheless Krause
has shown a generalization of the Smashing Conjecture 2.4.7 in [Kra00] and discovered a
relation to cohomological quotients [Kra05].
Let Tbe a compactly generated triangulated category with small coproducts. A local-
izing subcategory Cin Tis said to be generated by a class Iof maps in T, if every map
in Ifactors through an object in C. Note that Cis generated by the set of identity maps
{idXi}i∈J, if and only if Cis generated by the objects {Xi|i∈J}. In that the following
is a generalization of the Smashing Conjecture 2.4.7:
Theorem 3.5.1. [Kra00, Corollary A] Every smashing subcategory is generated by a set
of maps between compact objects.
The Smashing Conjecture 2.4.7 is related to cohomological quotients and the theory of
rings with several objects [Kra05].
24
Definition 3.5.2. Let F:S → T be an exact functor of triangulated categories. The
annihilator of Fis defined as Ann(F) := {f∈Mor(S)|F(f) = 0}. An exact functor
F:S → T is called a cohomological quotient functor, if for any cohomological functor
H:S → A to an abelian category Awith Ann(F)⊂Ann(H), there is a unique cohomo-
logical functor making the following triangle commutative
SF//
H
T
∃!
~~
A.
The annihilator Ann(F) of a cohomological quotient functor F:S → T is called exact
ideal.
The following theorem establishes a connection between the Smashing Conjecture,
cohomological quotients and flat epimorphisms between rings with several objects.
Theorem 3.5.3. [Kra05, 13.4] The Smashing Conjecture 2.4.7 for Sis equivalent to
each of the following statements
(i) Every exact ideal in Scis generated by idempotent elements.
(ii) Every cohomological functor F:Sc→ T induces an equivalence up to direct factors
Sc/ker(F)→ T .
(iii) Every two-sided flat epimorphism F:Sc→ T satisfying Σ(Ann(F)) = Ann(F)is
an Ore-localization.
In particular (i) shows that the Smashing Conjecture can be reduced to an assertion
on the compact objects in the triangulated category.
3.6 Applications
We discuss two applications that were discovered by Krause. The first deals with the
relation of non-commutative localization and the Smashing Conjecture. In the second
application the validity of the Smashing Conjecture implies the existence of a long exact
sequence in algebraic K-theory for certain rings [Kra05]. Both themes were originally
studied in [NR04].
Let Rbe a ring and R→Sbe a ring homomorphism. Consider the problem of lifting
a complex of S-modules or a map between complexes of S-modules along the map R→S
to a chain complex/map of R-modules up to homotopy. To be precise, given a complex
Yin Kb(S), the homotopy category of bounded complexes of S-modules, we are looking
for a complex X∈Kb(R) such that X⊗RS∼
=Yin the homotopy category. Similarly, if
we are given a map α:X⊗RS→X0⊗RS, we seek for maps φ:Y→Xand α0:Y→X0
such that φ⊗RSis invertible and α= (α0⊗RS)◦(φ⊗RS)−1.
If R→Sis a commutative localization, then both lifting problems can be solved.
The question is now, to what extend this is true in a non-commutative situation. So
let R→Sbe a homological epimorphism in the sense of [GL91], i.e., S⊗RS∼
=Sand
TorR
i(S, S) = 0.
25
Proposition 3.6.1. [Kra05, Corollary 14.7] The map R→Sis a homological epimor-
phism, if and only if
− ⊗RS:Kb(R)→Kb(S)
is a cohomological quotient in the sense of Definition 3.5.2.
Since cohomological quotients are related to the Smashing Conjecture by Theorem 3.5.3(ii)
this proposition indicates that the Smashing Conjecture should affect the lifting problem
for homological epimorphisms:
Theorem 3.6.2. [Kra05, Corollary 14.7] If the Smashing Conjecture 2.4.7 is true for
D(R), then both lifting problems can be solved, if and only if R→Sis a homological
epimorphism.
For a commutative localization R→Sthere is a long exact sequence in algebraic K-
groups [NR04]. If the Smashing Conjecture for the derived category of the ring Ris true,
we obtain a generalization:
Theorem 3.6.3. [Kra05, Theorem 2] If the Smashing Conjecture 2.4.7 is true for D(R)
and R→Sis a homological epimorphism, then the following maps
K(R, f)→K(R)→K(S)
form a homotopy fiber sequence, apart from the surjectivity of K0(R)→K0(S). In
particular there is a long exact sequence of algebraic K-groups
· · · → Kn(R)→Kn(S)→Kn−1(R, f)→ · · · → K0(R)→K0(S).
Here K(R, f)is the Waldhausen K-theory of a suitable bi-complicial Waldhausen category
in the sense of Thomason [TT90].
The preceding two theorems show that the Smashing Conjecture effects non-commutative
algebra. Besides its original consequences on the Telescope Conjecture and with it on
algebraic topology, these theorems foreshadow applications of the Smashing Conjecture
in other areas.
26
4 Cotorsion pairs, model categories and finite generation
The Smashing Conjecture 2.4.7 for the stable module category of a self-injective artin
algebra Ais equivalent to the assertion that a class of cotorsion pairs in Mod(A) is of
finite type by Paragraph 3.4. There is a strong connection between cotorsion pairs and
model categories as was independently shown by Beligiannis-Reiten [BR02] and Hovey
[Hov02]. Here we investigate how the finite type of cotorsion pairs is reflected in model
category theory. We continue and deepen Hovey’s study and prove a reformulation of the
Smashing Conjecture in terms of model categories.
4.1 Model categories
In this paragraph the language of model categories is recalled. In particular we discuss
cofibrantly and finitely generated model categories that become important later on. The
concepts are illustrated with the model category of modules over a Frobenius ring R.
Definition 4.1.1. Let Cbe a cocomplete category. An object Xin Cis finite if for every
sequence Y0→Y1→ · · · → Yn→..., the canonical map
colimn∈NHomC(X, Yn)→HomC(X, colimn∈NYn)
is bijective.
For example the sets with finitely many elements are finite, and the finitely presented
modules over a ring Rare finite in the category Mod(R).
Lemma 4.1.2. In a cocomplete category pushouts of finite objects are finite.
Proof. Since filtered colimits commute with finite limits [ML98, IX.2 Theorem 1] the
finite objects are closed under finite colimits and in particular under pushouts.
Definition 4.1.3. Amodel category consists of a complete and cocomplete category
Ctogether with three nonempty classes of morphisms weq(C), cof(C) and fib(C) that
are called weak equivalences,cofibrations, and fibrations, respectively. The elements of
weq(C)∩cof(C) are named acyclic cofibrations and the morphisms in weq(C)∩fib(C) are
called acyclic fibrations. These maps are subject to the following conditions.
MC1 The weak equivalences satisfy the“two out of three axiom”, i.e., let fand gbe
morphisms in Csuch that the composition f◦gexists; if two of the three morphisms
f,gand f◦gare weak equivalences, then so is the third.
MC2 Cofibrations, fibrations and weak equivalences are stable under retracts, that is, if
there is a commutative diagram
A//
f
C//
g
A
f
B//D//B
such that both compositions of the horizontal maps are the identity and gis a weak
equivalence, cofibration or fibration, then so is f.
27
MC3 The cofibrations have the left lifting property (LLP) with respect to the acyclic
fibrations. That is, for every commutative diagram
A//
i
X
p
∼
B//Y
in which iis a cofibration, and pis an acyclic fibration, there is a lift B→X
making the diagram commute.
The fibrations have the right lifting property (RLP) with respect to the acyclic
cofibrations, that means, if pis a fibration and iis an acyclic cofibration, that both
fit into a commutative diagram
A//
i∼
X
p
B//Y
then there is a lift B→Xmaking both triangles commute.
MC4 A morphism f:X→Ycan be factored into a cofibration followed by an acyclic
fibration
Xf//
=
=
=
=
=
=
=Y
Z
∼@@@@
and an acyclic cofibration followed by a fibration
Xf//
∼
>
>
>
>
>
>
>Y
Z.
@@@@
Remark 4.1.4. The triple (weq(C),cof(C),fib(C)) is called a model structure on C. We
use ∼//as a symbol for a weak equivalence, //(∼)//for (acyclic) cofibrations and
(∼)////for (acyclic) fibrations like in axiom MC3. Since Cis complete and cocomplete
it has an initial object ∅and a terminal object ∗. A model category is called pointed
if the initial and terminal objects are isomorphic. An object Xis called fibrant if the
unique morphism X→ ∗ is a fibration and cofibrant if the unique morphism ∅ → Xis
a cofibration. For an arbitrary object Xin C, the morphism ∅ → Xcan be factored
by axiom MC4 as a cofibration ∅ → Xcfollowed by an acyclic fibration Xc→X. So
the object Xcis cofibrant. Since the factorizations can be chosen functorially in all
examples that arise the assignment X7→ Xcis an endofunctor on C. It is called cofibrant
replacement functor and is equipped with a natural transformation qfrom the identity
on Cto (−)c. Dually, there is a fibrant replacement functor X7→ (X)fthat is obtained
by factoring X→ ∗ into an acyclic cofibration followed by a fibration. An object X
in a pointed model category is called acyclic if the canonical map X→0 is a weak
equivalence.
28
Example 4.1.5. Let Rbe Frobenius ring, that is, Mod(R) has enough projectives and
injectives and they coincide. Two maps f, g:M→Nbetween R-modules are called stably
equivalent if the difference f−gfactors through a projective. A map f:M→Nbetween
R-modules is defined to be a stable equivalence if there is a map g:N→Msuch that
f◦gand g◦fare stably equivalent to idNand idM, respectively. By definition a stable
equivalence becomes an isomorphism in the stable category Mod(R). The cofibrations
are the injective maps and the fibrations are the surjective maps. These three classes
of maps specify a model structure on Mod(R) [Hov99, Theorem 2.2.12]. Originally this
structure was discovered by Pirashvili [Pir86]. This model category has the very special
property that all objects are fibrant and cofibrant.
By axiom MC3 fibrations have the RLP with respect to acyclic cofibrations, and acyclic
fibrations have the RLP with respect to cofibrations. They are in fact characterized by
this property.
Proposition 4.1.6. [Hov99, Lemma 1.1.10] Let Cbe a model category.
(i) A map is a fibration if and only if it has the RLP with respect to all acyclic cofi-
brations.
(ii) A map is an acyclic fibration if and only if it has the RLP with respect to all
cofibrations.
Corollary 4.1.7. [Hov99, Corollary 1.1.11] Let Cbe a model category. Then pushouts
along (acyclic) cofibrations are (acyclic) cofibrations, i.e., let
A//
f
C
g
B//D
be a pushout diagram. If fis a cofibration or acyclic cofibration, then gis also a cofibra-
tion or an acyclic cofibration. Dually, fibrations and acyclic fibrations are closed under
pullbacks.
Remark 4.1.8. In a model category Cit is possible to define when two maps are ho-
motopic [Hov99, Definition 1.2.4]. Let Ccf denote the full subcategory of cofibrant and
fibrant objects. The homotopy relation on the morphisms of Ccf defines an equivalence
relation that is compatible with the composition [Hov99, Corollary 1.2.7]. Then it is pos-
sible to define the homotopy category Ho(C). Its objects are the cofibrant fibrant objects
of C, and the morphisms between objects Xand Yare defined as the set of equivalence
classes of morphisms in Cwith respect to the homotopy relation.
In Example 4.1.5 the two maps fand gare homotopic if and only if they are stably
equivalent. The homotopy category is the stable module category.
In many examples the fibrations are determined by having the RLP with respect to
only a subset Iof the class of acyclic cofibrations. In the same way the acyclic fibrations
are determined by a set Jof cofibrations. We illustrate this property with the category
of modules over a Frobenius ring Rdescribed in Example 4.1.5. Recall that a fibration
is a surjective map M→Nbetween R-modules. The following lemma is immediate.
29
Lemma 4.1.9. Let Rbe an arbitrary ring. A map M→Nin Mod(R)is surjective, if
and only if it has the RLP with respect to the trivial map 0→R.
So the fibrations are determined by having the RLP with respect to a single map or
in other words with respect to the set J:= {0→R}. An acyclic fibration can be
characterized as follows.
Proposition 4.1.10. [Hov99, Lemma 2.2.7, Theorem 2.2.12] Let Rbe a Frobenius ring.
A map in the module category Mod(R)is an acyclic fibration if and only if it is a surjective
map with injective kernel.
Recall that a module Qis injective if and only if Q→0 has the RLP with respect to
all injective maps M→Nof R-modules. Baer’s criterion says that not all these maps
are necessary.
Proposition 4.1.11. [Jac89, Proposition 3.15] Let Rbe an arbitrary ring. An R- module
Qis injective, if and only if Q→0has the RLP with respect to all a//Rwhere ais
an ideal in R.
This criterion motivates and is the key to the following.
Proposition 4.1.12. [Hov99, Proposition 2.2.9] Let Rbe a Frobenius ring. A map in
Mod(R)is an acyclic fibration if and only if it has the RLP with respect to the set
I:= {a//R|aideal in R}.
If Iis a set of morphisms, then let RLP(I) denote the class of maps which have the
RLP with respect to all maps in I. Now we introduce the underlying concept.
Definition 4.1.13. A model category Cis cofibrantly generated if there are sets Iand J
such that
(i) the class of fibrations is the class RLP(J).
(ii) The class of acyclic fibrations is the class RLP(I).
The elements in Iare called generating cofibrations and the elements in Jgenerating
acyclic cofibrations. A cofibrantly generated model category Cis called finitely generated
if the domains and codomains of the maps in Iand Jare finite.
Remark 4.1.14. In the original definition of a cofibrantly generated model category
the sources and targets of generating cofibrations and generating acyclic cofibrations
are asked to be small. The notion of smallness involves cardinal numbers and is quite
technical. It is suppressed from the notation since all examples of cofibrantly generated
model categories we consider in this section are module categories. In a module category
every object is small since it is a Grothendieck category [Hov01b, Proposition A.2].
Note that by Proposition 4.1.6 the elements in I(and J) are cofibrations (and acyclic
cofibrations). If Ris a Frobenius ring then the model structure specified in Example 4.1.5
is cofibrantly generated:
30
Theorem 4.1.15. [Hov99, Theorem 2.2.12] Let Rbe a Frobenius ring. Then there is
a finitely generated model structure on Mod(R)where the cofibrations are the injections,
the fibrations are the surjections, the weak equivalences are the stable equivalences, I:=
{0→R}is a set of generating cofibrations and J:= {a//R|aideal in R}is a set
of generating acyclic cofibrations.
Remark 4.1.16. Nearly all model categories that occur in nature are cofibrantly gen-
erated and most of them are finitely generated. It requires some work to construct ex-
amples of non cofibrantly generated model categories. Nevertheless, there are examples
[Isa01, AHRT02, CH02, Cho03].
4.2 Cotorsion pairs and model categories
Beligiannis-Reiten and Hovey have shown a remarkable theorem which relates model
structures to cotorsion pairs. In this paragraph we recall it and prove a specialization of
this result.
Let Abe a bicomplete abelian category and a model category. Furthermore assume
that the model structure and the abelian structure are compatible in the following sense:
•Every cofibration i:A////Bin Ais a monomorphism.
•A map p:B→Cis a fibration if and only if pis an epimorphism and the kernel
ker(p) in the short exact sequence
0→ker(p)→B→C→0
is fibrant. A fibration is acyclic, if and only if its kernel is acyclic.
We call such a category Aabelian model category.
Theorem 4.2.1. [BR02, Theorem 5.3] , [Hov02, Theorem 2.2] Let Abe a bicomplete
abelian category. If Ais equipped with a compatible model structure, let C,Fand W
denote the full subcategories of cofibrant, fibrant, and acyclic objects in A, respectively.
Then
(i) Wis thick, i.e., closed under extensions and retracts, and
(ii) (C,F ∩ W)and (C ∩ W,F)are cotorsion pairs.
Conversely, classes C,Fand Win Asatisfying (i) and (ii) determine a unique model
structure on Athat is compatible with the abelian structure.
We now specialize to the case where W=A.
Theorem 4.2.2. Let (C,F)be a cotorsion pair in a bicomplete abelian category A. Then
there is an abelian model structure on Asuch that all maps are weak equivalences, the
fibrations are epimorphisms with kernel in Fand the cofibrations are the monomorphisms
with cokernel in C.
31
Proof. The proof agrees in most instances with the proof of 4.2.1 applied to the special
situation of this theorem. We include it for the convenience of the reader.
The “2 out of 3” property for weak equivalences is evident. For the retract axiom,
assume that we are given a diagram
A//
f
C//
g
A
f
B//D//B
such that A→C→Aand B→D→Bare identity maps. We show that if gis a
monomorphism with cokernel in Cthen so is f. By a diagram chase one can prove that fis
a monomorphism. Passing to cokernels, we obtain maps coker(f)→coker(g)→coker(f)
whose composition is idcoker(f). Therefore coker(f) is a direct summand of coker(g) and
since Ext1
A(X, coker(g)) = 0 for all X∈ F, we can conclude that Ext1
A(X, coker(f)) = 0
for all X∈ F. Hence coker(f)∈ C.
Note that the notions of “(co-)fibration” and “acyclic (co-)fibration” coincide because
of the choice of the class of weak equivalences. Therefore to prove the lifting axiom, we
only have to construct a lift lin the commutative diagram
0
0
F
Af//
i
X
p
Bg//
l
>>
j
Y
C
0
0
where the vertical rows are exact and C∈ C and F∈ F. The construction of lwill be
done in two steps following [Hov01a, 4.2]. First we find a map h:B→Xthat makes
the upper triangle commute and secondly, we subtract a map such that in addition the
lower triangle commutes.
Consider the following commutative diagram with exact columns and exact lower row:
A(B, X)p∗//
i∗
A(B, Y )
i∗
A(A, X)p∗//
δ
A(A, Y )
δ
Ext1
A(C, F)//Ext1
A(C, X)p∗//Ext1
A(C, Y ).
32
The map f∈ A(A, X) is sent to p◦fin A(A, Y ). Since i∗(g) = g◦i=p◦fthe morphism
p∗(f) is mapped to 0 by δ. Since Ext1(C, F) = 0 (because C∈ C and F∈ F) the
map p∗: Ext1(C, X)→Ext1(C, Y ) is injective. Since p∗(δ(f)) = δ(p∗(f)) = 0, we can
therefore conclude that δ(f) = 0. By the exactness of the left column in the diagram we
find a map h:B→Xsuch that h◦i=f.
For the second step consider the exact sequence
A(C, Y )j∗
−→ A(B, Y )i∗
−→ A(A, Y ).
The map p◦h−gis sent to 0 by i∗. Therefore there is a morphism F:C→Ysuch that
F◦j=p◦h−g. In the exact sequence
A(C, X)p∗
−→ A(C, Y )→Ext1
A(C, F)
the end term is 0 and hence we find a map G:C→Xsuch that p◦G=F. The map
l:= h−G◦jis the desired lift.
Following [Hov01a, 5.4] the factorization axiom can be proved in two steps. First a
map f:A→Bcan be factored into a monomorphism followed by an epimorphism in the
following way:
Af//
(1A,0) ##
G
G
G
G
G
G
G
G
GB
A⊕B.
f⊕1B
;;
w
w
w
w
w
w
w
w
w
We have no reason to expect that the kernel of the epimorphism is in For that the cokernel
of the monomorphism is in C. But we show in the next step that every monomorphism
or epimorphism can be factored into a monomorphism with cokernel in Cfollowed by an
epimorphism with kernel in F. Therefore we can find a monomorphism i1with cokernel
in Cand an epimorphism p1with kernel in Fsuch that (1A,0) = p1◦i1. The composition
(f⊕1B)◦p1is an epimorphism and can be factored (f⊕1B)◦p1=p2◦i2, where
p2is an epimorphism with kernel in Fand i2is a monomorphism with cokernel in C.
Lemma 4.2.3, which is shown after this proof, tells us that the kernel of a composition of
two monomorphisms with kernel in Cis again contained in C. Therefore f=p2◦i2◦i1
is the desired factorization.
Modulo the lemma we only have to show that a monomorphism and an epimorphism
can be factored into a monomorphism with cokernel in Cfollowed by an epimorphism
with kernel in F. So let i:A→Bbe a monomorphism with cokernel X. Choose an
approximation
0→FX→CX→X→0
33
such that FX∈ F and CX∈ C. Consider the following diagram
0
0
FX
FX
0//Aj//B0//
q
CX//
0
0//Ai//B//
X//
0
0 0
with exact rows and columns in which B0is a pullback. Then i=q◦jis the desired
factorization.
Now let p:X→Ybe an epimorphism with kernel K. Choose an approximation
0→K→FK→CK→0
with FK∈ F and CK∈ C. Consider the diagram with exact rows:
0
0
0//K//
X
j
p//Y//0
0//FK//
X0q//
Y//0
CK
CK
0 0
such that X0is a pushout. Then p=q◦jis the wanted factorization. With the following
lemma we are done.
Lemma 4.2.3. Let (C,F)be a cotorsion pair in A. If f:A→Band g:B→Care
epimorphisms with kernel in Fthen the kernel of the composition g◦fis in F. The
monomorphisms with cokernel in Care also stable under composition of maps.
Proof. We prove the first statement, and the second assertion follows dually.
Let fand gbe epimorphisms such that their kernel is in F. Let F= ker(f), F0= ker(g)
34
and F00 = ker(g◦f). An application of the snake lemma to the following diagram
0
0//F//
Af//B//
g
0
0//F00 //Ag◦f//C//0
shows that F00/F ∼
=F0. Therefore the sequence
0→F→F00 →F0→0
is exact. Furthermore Fand F0are in Fby assumption. For any X∈ C, we obtain an
exact sequence
Ext1
A(F0, X)→Ext1
A(F00, X)→Ext1
A(F, X)
in which the first and the last term vanish. Therefore, Ext1
A(F00, X) = 0 for all X∈ C
and hence F00 ∈ F.
Definition 4.2.4. Let Abe an abelian category and let X= (C,F) be a cotorsion pair
in A. The category Atogether with the unique model structure of Theorem 4.2.2 on A
is called model category associated with Xand is abbreviated with AX.
Remark 4.2.5. This model category structure is quite unusual from the perspective of
model category theory since all maps are weak equivalences. Therefore in the homo-
topy category all maps get inverted, and the homotopy category is a the trivial additive
category.
Note that the concepts of cofibration and acyclic cofibration and the notions of fibra-
tions and acyclic cofibration coincide here.
4.3 Finite generation and the Smashing Conjecture
In this part the relation between cogeneration of a cotorsion pair and cofibrant generation
of the associated model structure is studied. As a consequence we obtain a reformulation
of the Smashing Conjecture in terms of finite generation of this model structure.
Nearly all model categories that occur in nature are cofibrantly generated [Hov99]. So
it is natural to ask how this property is resembled in the theory of cotorsion pairs. Hovey
showed in a slightly more general setup the following
Proposition 4.3.1. [Hov02, Lemma 6.7] Let Rbe a ring and X= (C,F)be a cotorsion
pair in Mod(R)that is cogenerated by a set Gof R-modules. Choose for every Gin Ga
free module FGthat is of finite rank if Gis finitely generated together with an epimorphism
FG
pG
−−→ G→0.Then the model category (Mod(R))Xis cofibrantly generated by the set
IG:= {ker(pG)→FG|G∈ G} ∪ {0→R}.
If furthermore Ris noetherian, and Gconsists of finitely generated modules then Mod(R)X
is finitely generated.
35
Proof. Since every map in IGis a monomorphism and G ⊂ C by definition we conclude
that all maps in IGare cofibrations. Therefore the class of fibrations of (Mod(R))Xis
contained in the class RLP(IG).
For the other inclusion assume p:X→Yhas the RLP(IG). In particular there is a
lift lin the following diagram
0
//X
p
R
l
>>
//Y
which is equivalent to the surjectivity of p. The map ker(p)→0 has the RLP with
respect to all morphisms in IGas a pullback of the map pwhich has this lifting property.
Therefore, we find for each G∈ G a lift in
MG
//ker(p)
FG
;;
//0,
where MGis the kernel of the epimorphism FG→G→0. Hence the map
HomR(FG,ker(p)) →HomR(MG,ker(p))
is surjective for all G∈ G. Applying Hom(−,ker(p)) to
0→MG
iG
−→ FG
pG
−−→ G→0
yields the exact sequence
HomR(FG,ker(p)) i∗
G
−→ HomR(MG,ker(p)) ∂
−→ Ext1
R(G, ker(p)) →Ext1
R(FG,ker(p))
in which i∗
Gis surjective and Ext1(FG,ker(p)) = 0 because FGis free. Therefore the
connecting homomorphism ∂is surjective and trivial, and hence Ext1
R(G, ker(p)) = 0 for
all G∈ G. As Gcogenerates Xthe module ker(p) is in Fand p:X→Yis a fibration.
If Ris noetherian, then submodules of finitely generated modules are finitely generated.
Therefore if Gand FGare finitely generated then so is ker(pG) as a submodule of FG.
Now assume conversely that we are given a finitely generated, abelian model category.
What can we say about the associated cotorsion pairs (C,F ∩ W) and (C ∩ W,F) of
Theorem 4.2.1? In [Hov02] Hovey did not find an answer but proclaimed later on [Hov07]
that the converse of Proposition 4.3.1 is easy to show. Here you find the precise statement
and a proof.
Proposition 4.3.2. Let Abe an abelian model category that is cofibrantly generated.
Then the cotorsion pairs (C,F∩W)and (C∩W,F)are cogenerated by a set. If furthermore
Ais finitely generated then (C,F ∩ W)and (C ∩ W,F)are of finite type.
Proof. Let Ibe the set of generating cofibrations and Jbe the set of generating acyclic
cofibrations. Then (C,F ∩ W) is cogenerated by G:= {coker(i)|i∈I}and (C ∩ W,F)
is cogenerated by {coker(j)|j∈J}. We show the first statement. The second assertion
36
can be proved similarly. We have to show that Ext1
A(G, X) = 0 for all G∈ G if and only
if Xis fibrant and acyclic.
Assume that Ext1
A(G, X) = 0 for all G∈ G. By definition for every i:A→Bin Ithe
object Gi:= coker(i) is in G, and the following sequence is exact:
0→Ai
−→ B→Gi→0.
The map i∗:A(B, X)→ A(A, X) is surjective for all i∈Isince Ext1
A(Gi, X) vanishes
in the long exact Ext-sequence. But this means that for all generating cofibrations i∈I
there is a lift lin
A//
i
X
B
l
>>
//0.
Therefore X→0 is an acyclic fibration.
Conversely, let Xbe acyclic and fibrant. Let Gbe an arbitrary element in G. An element
in Ext1
A(G, X) is represented by a short exact sequence
0→X→Mp
−→ G→0.(1)
Since p:M→Gis an epimorphism with acyclic, fibrant kernel, it is an acyclic fibration.
By definition, there is a map i:A→Bsuch that Gis the cokernel of i, or in other words
the pushout in the following pushout diagram:
A////
B
0//G.
Since pushouts along cofibrations are again cofibrations by Corollary 4.1.7, the object G
is cofibrant. Therefore we find a lift sin the following diagram
0//
M
p∼
G
s
>>
G
that splits the sequence (1). Hence Ext1
A(G, X) = 0.
If Ais finitely generated, then the domains and codomains of the generating cofibrations
and generating acyclic cofibrations are finite. The set of cogenerators is obtained by taking
pushouts of generating cofibrations and generating acyclic cofibrations with the zero map,
respectively. Since pushouts of finite objects are finite by Lemma 4.1.2, we know that all
elements in the set of cogenerators are finite. Therefore (C,F ∩ W) and (C ∩ W,F) are
of finite type.
Corollary 4.3.3. Let X= (C,F)be a cotorsion pair in A. If the associated model
category AXis finitely generated then the cotorsion pair is of finite type.
Note that the concepts of the set of generating acyclic cofibrations and generating
cofibrations coincide for a model category associated with a cotorsion pair.
37
Example 4.3.4. Let Rbe a ring, and let Proj(R) be the class of projective modules.
The model structure associated with the cotorsion pair (Proj(R),Mod(R)) is finitely
generated by J:= {0→R}by [Hov99, 2.2.5, 2.2.11]. In more detail, the cofibrations are
the injections with projective cokernel and the surjections are the fibrations. Indeed a
map of R-modules is a surjection, if it has the right lifting property (RLP) with respect
to 0 →R.
Example 4.3.5. Let Rbe a noetherian ring and let Inj(R) be the class of injective mod-
ules. Dually, the model structure associated with (Mod(R),Inj(R)) is finitely generated
by the set
I:= {a//R|aideal in R}
because of [Hov99, 2.2.9, 2.2.10]. In this model structure the cofibrations are the injective
maps and the fibrations are the surjective maps with injective kernel. The fibrations are
the maps with the RLP with respect to I.
It is quite interesting that Iand Jare the generating cofibrations respectively the
generating acyclic cofibrations of the model structure of the module category over a
Frobenius ring described in Paragraph 4.2.
Theorem 4.3.6. Let Rbe a self-injective artin algebra. Then the Smashing Conjec-
ture 2.4.7 is equivalent to the statement that for all cotorsion pairs X= (C,F)such
that Cand Fare closed under filtered colimits the associated model category Mod(R)Xis
finitely generated.
Proof. By Theorem 3.4.4 and Theorem 3.4.3 the Smashing Conjecture 2.4.7 is equivalent
to Conjecture 3.4.5 which says that all cotorsion pairs (C,F) such that Cand Fare closed
under filtered colimits are of finite type.
Fix a cotorsion pair X= (C,F) with the property that Cand Fare closed under filtered
colimits. Since an artin ring is also noetherian, we can use Proposition 4.3.1 to conclude
that Conjecture 3.4.5 implies that the model category Mod(R)Xis finitely generated.
Conversely, if the model category Mod(R)Xis finitely generated, then Proposition 4.3.2
implies that the cotorsion pair Xis of finite type.
Philosophically, this result is satisfactory since nearly all model categories are finitely
generated and it is quite hard to find examples that are not. The same is commonly be-
lieved for the Smashing Conjecture: it is hard to find examples of smashing subcategories
(or cotorsion pairs) that are not of finite type.
Given an arbitrary cotorsion pair X= (C,F) in a module category over a ring R, and
assume we have a set of generating cofibrations IXfor the associated model structure.
Since every fibration is surjective it is natural to ask that the map 0 →Rmust belong
to IX. The condition IX⊂ {a//R|aideal in R}forces that the surjections with
injective kernel are a subset of the fibrations. Since all injective R-modules are in Fthis
is a sensible requirement.
Conjecture 4.3.7. Let Rbe a noetherian ring and X= (C,F) be a cotorsion pair such
that Cand Fare closed under filtered colimits. The model structure associated with the
cotorsion pair Xis cofibrantly generated by the set
{a//R|aideal in Rsuch that R/a∈ C}.
38
5 Realizing smashing localizations of differential graded algebras
The results in this chapter are joint work with Birgit Huber [BH07] which have been
achieved with substantial contributions of Bernhard Keller. We show that every smashing
localization of a derived category of a differential graded algebra (or shortly dg algebra)
can be realized by a morphism of dg algebras. More precisely if Ais a dg algebra and
L:D(A)→ D(A) a smashing localization, we prove the existence of a dg algebra ALwith
the property that D(A)/ker L≃ D(AL). Furthermore we show that there is a dg algebra
A0quasi-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 D(A)(A, A)∗→ D(A)(LA, LA)∗is
induced by a morphism A→AL, and the quotient functor is naturally isomorphic to the
left derived functor − ⊗L
AAL.
As a direct consequence we are able to show that every smashing localization functor
L:T → T on an algebraic triangulated category that is generated by one compact object
is induced by a morphism of dg algebras.
As an application, in Section 5.6 we consider dg algebras with graded-commutative
cohomology ring. For such a dg algebra A, we introduce the localization of Aat a prime
pin 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 morphism
H∗A→(H∗A)pis induced by a zigzag of dg algebra morphisms.
5.1 Differential graded algebras
In this section we review dg algebras and the derived category of a dg algebra. We refer
the reader to [Kel94a, Sch04, Kra04] for more details.
Fix a commutative ring k.
Definition 5.1.1. Adg algebra Aover kis a Z-graded k-algebra together with a k-linear
differential d:An→An+1 that interacts with the multiplication according to the Leibniz
rule
d(xa) = d(x)a+ (−1)nxd(a) (2)
for all x∈Anand all a∈A. A (right) dg A-module is a Z-graded module Mwith a
differential d:Mn→Mn+1 such that equation (2) holds for all x∈Mnand all a∈A.
A morphism of a dg algebra or of a dg module is a map of the underlying graded algebra
or module that commutes with the differential.
Cohomology of a dg A-module, the notion of a quasi-isomorphism and the shift are
defined on the underlying chain complexes. The cohomology of a dg algebra Ais a
graded ring and the cohomology of every dg A-module becomes a graded module over
H∗(A). Denote by Moddg Athe category of dg A-modules and by dga/kthe category
of dg algebras over k. A homotopy between morphisms of dg modules is a map of the
underlying graded modules that is also a chain homotopy. The homotopy category K(A)
is the quotient of Moddg Aby the ideal of null-homotopic maps.
40
Example 5.1.2. If X, Y are dg A-modules, then the homomorphism complex HomA(X, Y )
in degree nis defined by:
HomA(X, Y )n= HomA(X, Y [n]).
The differential dn: HomA(X, Y [n]) →HomA(X, Y [n+ 1]) is defined to be
dn(f) = dY◦f−(−1)nf◦dX.
There is an isomorphism
HnHomA(X, Y )∼
=HomK(A)(X, Y [n]).
The endomorphism ring EndA(X) = HomA(X, X) is a dg algebra and HomA(X, Y )
becomes a dg module over EndA(X) by composition of graded maps.
Let Aand Bbe dg algebras over k. A dg A-B-bimodule Xis 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. Fix an A-B-bimodule X. There is an internal Hom-functor
and a tensor product that form an adjoint pair
Moddg A
−⊗AX//Moddg B.
HomB(X,−)
oo
The derived category D(A) of Acan then be defined as the localization of the homotopy
category with respect to the quasi-isomorphisms. The homotopy category and the derived
category are triangulated and the canonical functor K(A)→ D(A) is exact.
The following well-known lemma collects basic results on the derived category of a dg
algebra
Lemma 5.1.3. [Kel94a, 3.1, 5.3] Let Abe a dg algebra over kand Mbe a dg A-module.
(i) The evaluation map
D(A)(A, M)∗→H∗M, 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.
(ii) A dg module is compact in D(A), if and only if it is contained in the full sub-
category of perfect complexes Dper(A), i.e., the smallest thick subcategory of D(A)
containing A.
A consequence of this lemma is that D(A) is compactly generated by the dg algebra A.
Definition 5.1.4. A dg A-module Mis homotopically projective if K(A)(M, N) = 0 for
all acyclic dg A-modules N.Homotopically injective modules are defined dually.
41
Up to homotopy the homotopically projective modules are the cofibrant objects in the
model category Moddg Adescribed in [SS00, Theorem 4.1].
For every dg A-module Mthere is a homotopically projective module pM and a quasi-
isomorphism pM →M. Dually, there is a homotopically injective module iM and a quasi
isomorphism M→iM, see [Kel98, 8.2.4].
Let Aand Bbe dg algebras over a commutative ring kand Xbe a dg A-B-bimodule.
The internal Hom-functor and the tensor product descend to an adjoint pair in the derived
category [Kel98, 8.2.6]
D(A)
−⊗L
AX
//D(B).
RHomB(X,−)
oo
Here the derived tensor product maps a dg A-module Mto pM ⊗L
AX.
5.2 Cofibrant differential graded algebras
The category of dg algebras dga/k over a commutative ring kadmits a model cate-
gory structure [SS00] in which the fibrations are the degree-wise surjective dg algebra
morphisms and the weak equivalences equal the quasi-isomorphisms. Recall that a dg
algebra is cofibrant if for any morphism of dg algebras f:A→Cand every surjective
quasi-isomorphism of dg algebras g:B→C, there exists a lift h:A→B. That is, we
have a commutative diagram
B
g
∼
Af//
h
>>
C.
There is a cofibrant replacement functor dga/k →dga/k that sends a dg algebra Ato
Ac. Furthermore there is a surjective quasi-isomorphism Ac→A.
Example 5.2.1. A class of cofibrant dg algebras arises from the tensor algebra functor
T: Ch(k)→Moddg Awhich is left adjoint to the functor that forgets the multiplicative
structure and only remembers the chain complex over k. If Cis a cofibrant chain complex,
i.e., a homotopically projective dg module over the ground ring k, then the algebra T(C)
is cofibrant for all n.
We are interested in dg algebras with graded-commutative cohomology. The following
example provides a class of such dg algebras that are in addition cofibrant.
Example 5.2.2. Let kbe a field and Va positively graded k-vectorspace. The tensor
algebra TV is defined by TV =
∞
L
q=0
TqV, where TqVis the tensor product of qcopies
of V. It becomes an algebra via the multiplication ab := a⊗b. Note that qis not the
degree. The degree of v1⊗ · · · ⊗ vq∈TqVis the sum of the degrees of the elements vi.
Let Ibe the ideal in TV generated by the elements v⊗w−(−1)nmw⊗v, where v∈Vn
and w∈Vm. The free graded-commutative algebra ΛVis defined as TV/I.
ASullivan algebra is a dg algebra (ΛV, d), whose underlying graded algebra is the free
graded-commutative algebra of a vectorspace Vthat is graded in positive degrees, and
such that there is an increasing exhaustive filtration
V(0) ⊂V(1) ⊂ · · · ⊂ V(k)⊂ · · · ⊂ V
42
of graded subspaces such that d|V(0) = 0 and im(d|V(k))⊂ΛV(k−1) for k≥1. The
Sullivan algebras are cofibrant [FHT01, Lemma 12.4].
5.3 Cohomological p-Localization
Assume that Ais a dg algebra such that H∗Ais graded-commutative. Let pbe a graded
prime ideal of H∗A, i.e., a prime ideal which is generated by homogeneous elements. Let
Cpdenote the 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 Theorem 2.3.13, Remark 2.3.14 and Proposition 2.3.15 we deduce:
Corollary 5.3.1. The localization
D(A)
Q//D(A)/Cp
R
oo
is smashing, and there is an isomorphism r:D(A)(A, A)∗
p
∼
=
−→ D(A)/Cp(QA, QA)∗of
graded rings making the diagram
D(A)(A, A)∗can //
Q
D(A)(A, A)∗
p
∼
=
r
vvlllllllllllll
D(A)/Cp(QA, QA)∗
commutative. Furthermore the squares
D(A)
Q
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
R
OO
D(A)/Cp(QA,−)∗
//Modgr H∗Ap
inc
OO
commute up to natural isomorphism.
5.4 Smashing localizations of the derived category of a dg algebra
Let Abe a differential graded algebra over some commutative ring kand let
L:D(A)→ D(A)
be a smashing localization. If Cdenotes the category of L-acyclic objects, then we have
an adjoint pair of functors
D(A)
Q//D(A)/C
R
oo
satisfying R◦Q=L. The right adjoint Ris fully faithful and commutes with arbitrary
direct sums.
43
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 the functors
H∗:D(A)→Modgr H∗A
and
D(A)(A, −)∗:D(A)→Modgr H∗A
by the natural evaluation isomorphism D(A)(A, X)∗→H∗X, f 7→ f(1).
The following lemma which we learned from Dave Benson is the key to our construction.
Lemma 5.4.1. Let A,Bbe dg algebras and Mbe a dg (B, A)-bimodule. Let α:A→M
and 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//
p1
B
β
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 consequently,
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 pair (a, b +b0) satisfies α(a) = β(b+b0) and thus
(a, b) = (a, b +b0)∈H∗X.
The following lemmas ensure that the cohomology of the dg algebra ALwhich we
construct below is independent of all choices that we will make.
Recall that Moddg Ais an exact category (in the sense of Quillen, see [Qui73]) 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. That is, there are enough projective and injective modules in
Moddg Aand the projective and injective modules coincide [Kel98, 8.3.3]. Since the maps
44
factoring through an injective object are precisely the null-homotopic maps, the associated
stable category coincides with the homotopy category K(A). We refer to [Kel98, Sect.
8.2.3] and [Kel94a, Sect. 2.2] for more details.
For a given map X→Ythere is an injective envelope Iof Xsuch that the canonical
map X→Y⊕Irepresents the same map in K(A) and is a split monomorphism. Hence
we obtain
Lemma 5.4.2. Let ϕ:X→Ybe any morphism in K(A). Then ϕcan be represented
by a morphism in Moddg Awhich is a split monomorphism in the category of graded
A-modules.
Lemma 5.4.3. Let X, Y be dg A-modules and let ν:X→Ybe an isomorphism in K(A).
(1) Let X→I(X)denote 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 5.4.2 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 5.4.1, the pullback diagram
Sp2//
p1
EndA(Y⊕I(X))
∼
¯ν∗
EndA(X)¯ν∗
∼//HomA(X, Y ⊕I(X))
yields the claim.
(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).
45
The proof of the following lemma is immediate.
Lemma 5.4.4. 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.
Proposition 5.4.5. The functor RHom(RQA, R−): D(A)/C → D(EndA(RQA)) is an
equivalence of triangulated categories.
Proof. Note that RHom(RQA, R−) preserves arbitrary direct sums because for any fam-
ily (Xi)i∈Iin D(A)/C, the map
a
i∈I
RHom(RQA, RXi)→RHom(RQA, R a
i∈I
Xi)
is a quasi-isomorphism.
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))
∼
=
OO
is commutative. The claim now follows from a version of ‘Beilinson’s Lemma’ [Bei78]
which is stated in [Sch04, Prop. 3.10].
Hence we have shown that the quotient category D(A)/Cis equivalent to the derived
category of the dg algebra EndA(LA), where LA was chosen to be homotopically pro-
jective. Note that Lemma 5.4.3 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 for 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.
R
oo
Since Ais homotopically projective, we can view ηAas an element in K(A)(A, RQA).
46
Lemma 5.4.6. 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.
Proof. Since Ris fully faithful, the usual adjunction isomorphism (see [ML98, 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 5.4.7. By Lemma 5.4.2 we may represent ηA:A→LA by a monomorphism of
dg A-modules
¯ηA:A→c
LA,
which is 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 that LA
was already chosen to be homotopically projective. By Lemma 5.4.3 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 5.4.3 and Proposition 5.4.5 it follows that
D(AL)≃ D(A)/C.
Theorem 5.4.8. 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 Aand a morphism of dg algebras
ϕ:A0→ALsuch that we have the commutative diagram
H∗A0
∼
=
H∗ϕ
((
R
R
R
R
R
R
R
R
R
R
R
R
R
R
D(A)(A, A)∗L//D(A)(LA, LA)∗
in cohomology.
47
Proof. We identify the dg algebras EndA(A) and Athrough the isomorphism given by
evaluation at 1. Let
A0p2//
p1∼
AL
∼¯η∗
A
EndA(A)¯ηA∗//HomA(A, LA)
be a pullback diagram.
The map ¯ηA∗is a quasi-isomorphism (Lemma 5.4.6) and surjective since ¯ηAis a split
monomorphism of graded A-modules (Remark 5.4.7). We infer from Lemma 5.4.1 that
A0is a dg algebra quasi-isomorphic to A, and we set ϕ=p2.
In cohomology we obtain the commutative diagram
H∗A0H∗(p2)//
H∗(p1)∼
=
H∗EndA(LA)
∼
=H∗(¯η∗
A)
H∗EndA(A)H∗(¯ηA∗)//H∗(HomA(A, LA))
and thus it only 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).
As it is well-known that εQA ◦Q(ηA)∼
=idQA (see [ML98, Ch. IV.1]), the claim follows.
If we assume in addition that Ais a cofibrant dg algebra (see Section 5.2), then the
map p1:A0→Ain the pullback diagram above splits. In particular 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 5.4.9. 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.
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 5.4.10. 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.
48
Proof. By Lemma 5.4.6 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 5.4.9. In addition RHomA(LA, LM) becomes an object in
D(A) through the dg algebra morphism ψ.
Proposition 5.4.11. Suppose that Ais a cofibrant dg algebra. Then the diagram
D(A)
Q
−⊗L
AAL//D(AL)
D(A)/C
RHomA(RQA,R−)
≃
66
m
m
m
m
m
m
m
m
m
m
m
m
m
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 the 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
OO
where canMis the canonical identification and νMis defined by
νM:RHomA(LA, LM)⊗L
AEndA(LA)→RHomA(LA, LM).
f⊗g7→ f◦g
In order to prove that τis an isomorphism, one checks that the full subcategory
{M∈ D(A)|τMis an isomorphism}of D(A) is a localizing subcategory containing A.
Note that τcommutes with the suspension functor since this holds for λby Lemma 5.4.10.
49
5.5 Smashing subcategories of algebraic triangulated categories
In this paragraph we will use a result of Keller to expand Theorem 5.4.8 to a broader
class of triangulated categories.
Definition 5.5.1. A triangulated category is algebraic if it is triangle equivalent to the
stable category of a Frobenius category.
The derived category of a ring or a dg algebra, the stable module category of a Frobenius
ring and the homotopy category of a ring or a dg algebra are algebraic. Triangulated
categories that are non-algebraic arise from topology, for instance the stable homotopy
category of Example 2.1.5.
Theorem 5.5.2. [Kel94a, 4.3] If Tis an algebraic triangulated category with small
coproducts that is generated by a compact object, then there is a dg algebra Asuch that
Tis triangle equivalent to D(A).
Using this theorem we obtain:
Corollary 5.5.3. Let Tbe as in Theorem 5.5.2. Let L:T → T be a smashing localization
and let Q:T → T /ker(L)be the corresponding quotient functor. There is a dg algebra
Aand a morphism of dg algebras A→ALsuch that Qis induced by A→AL.
Proof. By Theorem 5.5.2 there is a dg algebra Asuch that T ≃ D(A). Without loss of
generality we may assume that Ais cofibrant since there is a quasi isomorphism from
the cofibrant replacement of Ato A. Therefore by Corollary 5.4.9 we obtain a morphism
A→AL. By Proposition 5.4.11 we know that Q∼
=− ⊗L
AAL. Therefore, the morphism
A→ALinduces Q.
5.6 The p-localization of a dg algebra
Let Abe a dg algebra over a commutative ring kand assume throughout this paragraph
that the cohomology algebra H∗Ais graded-commutative. We fix a prime ideal pof H∗A
that is generated by homogeneous elements. By Cpwe denote the full subcategory of
objects Min D(A) such that (H∗M)p= 0. The localization Lp:D(A)→ D(A),given by
the adjoint pair
D(A)
Q//D(A)/Cp,
R
oo
is smashing by Corollary 5.3.1. Now we apply the results of Section 5.4 to this special
case. We define
Ap=ALp,
and we call Aplocalization of Aat a prime pin cohomology.
From Lemma 5.4.3 and Proposition 5.4.5 we infer that D(A)/Cp≃ D(Ap). For this
special smashing localization we have
Theorem 5.6.1. 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
50
is induced by a zigzag of dg algebra maps
A∼
←− A0ϕ
−→ Ap.
That is, we have a commutative diagram
H∗A0
∼
=
H∗ϕ
**
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
H∗Acan //(H∗A)p∼
=//H∗(Ap).
Proof. Since D(A)(A, A)p∼
=D(A)(LpA, LpA) by Corollary 5.3.1, the dg algebra Apsatis-
fies H∗(Ap)∼
=(H∗A)p.Theorem 5.4.8 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: H∗A→H∗(Ap) and Lpby
Corollary 5.3.1.
The following result is an immediate consequence of Corollary 5.4.9 and Theorem 5.6.1.
Corollary 5.6.2. 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 introduced in Example 5.2.2.
Let Abe a dg algebra over a commutative ring ksuch that H∗Ais graded-commutative
and pbe a prime ideal in H∗Athat is generated by homogeneous elements.
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 the map
can: H∗A→(H∗A)pis induced by a morphism of dg algebras ψ:A→Ap, and 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)
g
::
The dg algebra morphisms β:A→Band ψ:A→Apgive rise to functors
Fβ:D(A)−⊗L
AB
−−−−→ D(B) and Fψ:D(A)−⊗L
AAp
−−−−−→ D(Ap).
51
Now we prove a universal property on the level of derived categories.
Proposition 5.6.3. There is a unique functor G:D(Ap)→ D(B)making the following
diagram commute:
D(A)Fβ//
Fψ
D(B)
D(Ap)
G
::
Proof. We first note that by Proposition 5.4.11 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 2.3.16 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 5.6.4. There is a unique functor G:Dper(Ap)→ Dper(B)which makes the
following diagram commute:
Dper(A)Fβ//
Fψ
Dper(B)
Dper(Ap)
G
99
52
Remark 5.6.5. 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 5.4 does not apply since
in general, we cannot expect that Gis a smashing localization. It remains to enlighten
the relation of our construction with DG quotients, which have a universal property and
were introduced by Drinfeld [Dri04].
There is also a construction by To¨en [Toe] (see also [Kel06]) which seems to be related:
Let dgcatkbe the category of small dg categories over a commutative ring k. The local-
ization of dgcatkwith respect to the quasi-equivalences is denoted by Hqe. If Ais a small
dg category and if Sis a 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 S
invertible. This morphism has a universal property: Each morphism in Hqe making S
invertible factors uniquely through Q.
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.
53
6 Thick subcategories of the derived category of a hereditary
algebra
A full subcategory of a triangulated category is thick if it is closed under forming sus-
pensions, triangles and retracts. Thick subcategories were studied in stable homotopy
theory, commutative algebra and representation theory of groups. The first classification
theorem was obtained by Hopkins and Smith for the p-local finite stable homotopy cate-
gory [HS98]. They showed that any thick subcategory is equivalent to the K(n)∗-acyclics
of the cohomology theory represented by a Morava K-theory spectrum K(n). Hopkins
and Neeman showed that the thick subcategories in the category of perfect complexes
Dper(R) of a commutative noetherian ring Rcorrespond to the specialization closed sub-
sets of the prime ideal spectrum of R[Hop87, Nee92]. There also exists a generalization of
this result to schemes [Tho97]. Benson, Carlson and Rickard classified the thick subcat-
egories of the stable module category of the group algebra kG of a p-group Gin terms of
closed subvarieties of the maximal ideal spectrum of the group cohomology ring H∗(G;k)
[BCR97].
In the main theorem of this section, we classify the thick subcategories of the bounded
derived category Db(A) of a hereditary abelian category A. This result includes for in-
stance the bounded derived category of finitely presented right modules Db(modA) for
a finite dimensional algebra over a field kand enhances therefore the study of thick
subcategories to the field of representation theory of algebras. We determine the thick
subcategories explicitly in two examples. Furthermore we classify the localizing subcat-
egories in the full derived category D(A) in a similar way. At the end we show that the
Smashing Conjecture holds for the derived category of a hereditary artin algebra of finite
representation type.
6.1 Representation theory of hereditary algebras of finite representation type
In this paragraph we describe the structure theory for the module category of a hereditary
finite dimensional algebra of finite representation type via Auslander-Reiten theory.
Fix an algebraically closed field k. All our quivers are finite and acyclic.
Definition 6.1.1. Ak-algebra Ais hereditary if every ideal of Ais projective as an
A-module.
There are several characterizations of the notion a hereditary algebra. Recall that
the global dimension of an algebra is the supremum of the projective dimensions of all
A-modules.
Theorem 6.1.2. [ASS06, VII.Theorem 1.4] Let Abe a k-algebra. The following asser-
tions are equivalent:
(i) Ais hereditary.
(ii) Submodules of projective A-modules are projective.
(iii) The global dimension of Ais at most one.
(iv) Exti
A(M, N) = 0 for all A-modules Mand Nand for all i≥2.
54
Example 6.1.3. Let Qbe a quiver. The path algebra kQ is generated as a k-vector space
by the paths in Qand the multiplication is defined by concatenation of paths. The path
algebra of any quiver is hereditary [ARS97, III.Proposition 1.4].
In certain cases the module category of a hereditary algebra is determined by a path
algebra.
Theorem 6.1.4. [ARS97, III Corollary 1.10, Proposition 1.13] Let Abe a finite di-
mensional hereditary k-algebra. Then there is a quiver QAsuch that kQAand Aare
Morita-equivalent, i.e., mod(A)and mod(kQA)are equivalent.
Recall that a module Mis indecomposable, if the existence of a decomposition M=
N⊕Limplies that Nor Lare trivial.
Definition 6.1.5. Ak-algebra Ais representation finite, if there are only a finite number
of non-isomorphic, finitely generated indecomposable A-modules.
The representation finite path algebras are classified. Therefor recall the notion of a
Dynkin graph [ASS06, VII.2].
Theorem 6.1.6 (Gabriel). Let Qbe a connected quiver. Then the path algebra kQ is
representation finite, if and only if the underlying graph of Qis Dynkin of type Anfor
n≥1,Dnfor n≥4or of type E6,E7,E8.
Now we turn to the structure theory of the module category. Fix a finite dimensional
hereditary k-algebra Awhich is representation finite.
The following fundamental theorem reduces the study of modules to indecomposable
modules.
Theorem 6.1.7 (Krull-Remak-Schmidt). Let Abe a finite dimensional k-algebra.
For every finitely generated A-module Mthere are indecomposable A-modules M1,...,Mn
such that M∼
=Ln
i=1 Mi. Furthermore the modules M1,...,Mnare unique up to permu-
tation.
The following notion is central in the classification of morphisms.
Definition 6.1.8. Let Abe a finite dimensional k-algebra. A morphism of A-modules
f:M→Nis an irreducible morphism, if
(i) fis neither a section nor a retraction and
(ii) if f=f1◦f2, then either f1is a retraction or f2is a section.
Denote by Irr(M, N) the k-vectorspace of irreducible morphisms from Mto N.
As for objects the study of morphisms is reduced to the study of irreducible ones.
Theorem 6.1.9. [ARS97, V.Theorem 7.8] Let Abe a finite dimensional k-algebra of
finite representation type. Every morphism between finitely presented indecomposable A-
modules that is not invertible is a finite sum of finite compositions of irreducible maps.
The information of mod(A) can be collected in a combinatorial object.
55
Definition 6.1.10. The Auslander-Reiten quiver (AR-quiver) Γ(A) of the algebra Ahas
as vertices the isomorphism classes of indecomposable modules. The arrows from [M] to
[N] correspond bijectively to a k-basis of the vector space of irreducible maps Irr(M, N).
The quiver Γ(A) is locally finite in the sense that every vertex has only finitely many
neighbors. The Auslander-Reiten quiver is equipped with an extra structure: the trans-
late. It is a bijective map
τ: Γ(A)\Proj(A)→Γ(A)\Inj(A),
where Proj(A) and Inj(A) denote the sets of isomorphism classes of indecomposable
projective and injective modules, respectively. The following notion is central for the
structure of the AR-quiver.
Definition 6.1.11. A short exact sequence
0→L→M→N→0
is called almost split or an Auslander-Reiten sequence (AR-sequence), if Land Nare
indecomposable and the maps L→Mand M→Nare irreducible.
The following theorem describes the relation between an indecomposable module N
and its translate τ(N).
Theorem 6.1.12. [ASS06, IV.Theorem 4.4, Theorem 3.1] Let Abe a finite dimensional
k-algebra. For every indecomposable non-projective A-module Nthere is an AR-sequence
0→τN →
n
M
i=1
Mni
i→N→0
in which ni≥0and the modules Miare pairwise non-isomorphic indecomposable. Fur-
thermore ni= dimkIrr(Mi, N) = dimkIrr(τN, Mi).
If Nis an indecomposable non-projective module, then the theorem tells us that there
is the same number of arrows in Γ(A) ending in [N] as the number of arrows starting in
τ[N]. Therefore a typical part of the AR-quiver can be visualized as follows:
[M1]
""
E
E
E
E
E
E
E
E
[τN]
;;
w
w
w
w
w
w
w
w
##
G
G
G
G
G
G
G
G[N]
τ
oo________
[Mn].
<<
y
y
y
y
y
y
y
y
Here we assume that ni= 1 for all i= 1,...,n. The subquiver starting at a vertex [τN]
and ending in [N] is called mesh. The following example illustrates the structure of the
AR-quiver.
56
Example 6.1.13. Let Qbe the A3-quiver 1 2
oo3.
ooThe Auslander-Reiten quiver
of kQ looks as follows:
P3
?
?
?
?
?
?
?
?
P2
??
=
=
=
=
=
=
=
=
P3
P1
τ
oo_ _ _ _ _ _ _
?
?
?
?
?
?
?
?
P1
@@
P2
P1
@@
τ
oo_______ P3
P2.
τ
oo_ _ _ _ _ _ _
The indecomposable projective modules are P1,P2and P3.
Every finite dimensional hereditary k-algebra Aof finite representation type is Morita-
equivalent to the path algebra of a quiver Q. The quiver Qcan be obtained from the AR-
quiver Γ(A) as the opposite of the full subquiver with the indecomposable projectives as
vertices. Since the category mod(A) is equivalent to mod(kQ), the AR-quiver determines
the module category combinatorially.
The whole module category Mod(A) is determined by the category of finitely generated
modules.
Theorem 6.1.14. [Aus74, RT74] Let Abe an artin algebra of finite representation type.
Then every module is a direct sum of finitely generated indecomposable modules.
Therefore Auslander-Reiten theory provides a complete picture of both mod(A) and
Mod(A).
6.2 The derived category of hereditary abelian categories
Here we describe the structure of the derived category of a hereditary abelian category
which serves as the main tool to obtain the classification result in Paragraph 6.4.
Definition 6.2.1. An abelian category Ais called hereditary if Exti
A(M, N) vanishes for
all M, N ∈ A and all i≥2.
Throughout this paragraph let Abe a hereditary abelian category.
Example 6.2.2. A module category over a hereditary ring Ais hereditary.
There is a canonical embedding i:A//D(A) which sends an object Mto the stalk
complex · · · → 0→M→0→... which is concentrated in degree zero. By abuse of
notation we do not distinguish between objects in Aand im(i).
The derived category D(A) of a hereditary abelian category Ais closely related to A
itself since every complex of D(A) is isomorphic to a direct sum (and direct product) of
stalk complexes:
Lemma 6.2.3. For every X∈ D(A)there are isomorphisms in D(A)
Y
n∈Z
HnX[−n]∼
=X∼
=M
m∈Z
HmX[−m].
57
A proof of this well known lemma can be found in [Kra04]. The homomorphisms in
D(A) therefore reduce to
Hom∗
D(A)(M, N)∼
=HomA(M, N)⊕Ext1
A(M, N)
for M, N ∈ A. So the derived category consists of shifted copies of A, and the morphisms
are given by extensions and homomorphisms in A. This structure is visualized in Figure 1.
D(A)
A[−1] A[0] A[1]
Ext
Ext
Ext
Ext
...
...
Figure 1
Non-equivalent hereditary abelian categories can give rise to the same derived category:
Theorem 6.2.4. [Hap88, I.5.5, 5.6] Let kbe an algebraically closed field. If Qand Q0
are Dynkin quivers of the same type but of different orientation then D(mod(kQ)) and
D(mod(kQ0)) are equivalent as triangulated categories.
This result suggests that for representation finite path algebras the Dynkin type plays
an essential role.
The structure of the derived category motivates why the thick subcategories in D(A)
should be determined by data in A. If in addition A= mod(kQ) is the module cate-
gory of a path algebra of a Dynkin quiver then we should be able to describe the thick
subcategories combinatorially.
6.3 Thick subcategories of abelian categories
We define and investigate thick subcategories of an abelian category Aand discuss Hovey’s
classification of the thick subcategories in the category of modules over a regular coherent
commutative ring.
Throughout this paragraph let Abe an abelian category.
Definition 6.3.1. A full subcategory Mof Ais called thick if for every exact sequence
M1→M2→M3→M4→M5
the object M3is in Mif the objects M1, M2, M4, M5are in M.
Hovey calls these subcategories “wide” [Hov01a]. In the following two lemmas some
easy properties of thick subcategories are deduced. For the convenience of the reader the
proof [Hov01a] is reproduced here.
Lemma 6.3.2. A full subcategory Min Ais thick, if and only if it is closed under
forming of extensions, kernels and cokernels.
58
Proof. Let M ⊂ A be thick and
M1→M2→M3→M4→M5
be exact in A. If in the exact sequence above M1=M5= 0 and M2and M4are in
M, then M3is in Msince Mis thick. Therefore Mis closed under extensions. If we
set M1=M2= 0, respectively M4=M5= 0 it follows that Mis closed under kernels,
respectively cokernels.
Conversely, let M ⊂ A be closed under extensions, kernels and cokernels and let
M1→M2→M3→M4→M5
be exact with M1, M2, M4, M5∈ M. Since Mis closed under cokernels and kernels,
C:= coker(M1→M2) and K:= ker(M4→M5) are in M. Hence we obtain a diagram:
M1//M2//
B
B
B
B
B
B
B
BM3//
!!
B
B
B
B
B
B
B
BM4//M5
C
>>
|
|
|
|
|
|
|
|
!!
C
C
C
C
C
C
C
CK
==
|
|
|
|
|
|
|
|
!!
C
C
C
C
C
C
C
C
0
==
{
{
{
{
{
{
{
{0
==
{
{
{
{
{
{
{
{0.
Therefore M3is an extension of Cand Kand hence it is in M.
As an additional property we have
Lemma 6.3.3. A thick category in Ais closed under direct summands.
Proof. Let M⊕Nbe in the thick category M. The kernel of the map M⊕N→M⊕N
which sends (m, n) to (0, n) is M.
So a thick subcategory in Ais an abelian subcategory in Athat is closed under retracts
such that the inclusion functor is exact. This property motivates its name.
There are geometric examples of thick subcategories.
Example 6.3.4. The category of coherent modules over the structure sheaf OXof a
scheme Xis thick [Gro60, 5.3.5].
Other examples of thick subcategories arise from the category add(M) of direct sums
of direct summands of M.
Lemma 6.3.5. Let kbe a field.
(i) Let Abe an arbitrary k-algebra. If Mis an indecomposable finitely presented A-
module with HomA(M, M) = kand Ext1
A(M, M) = 0, then add(M)is thick.
(ii) If Ais a finite dimensional hereditary k-algebra of finite representation type and M
is an indecomposable A-module, then add(M)is thick.
59
Proof. Since Mis indecomposable the equation add(M) = {Ln
i=1 M|n≥0}holds. The
functor Ext1
A(−,−) is additive in both variables. Therefore add(M) is closed under
extensions because Mhas no non-trivial self-extensions. For positive integers nand m
let f:Mn→Mmbe A-linear. Every non-trivial component of fis of the form x·idMfor
some x∈k\ {0}because HomA(M, M) = k. Since kis a field every element x∈k\ {0}
is invertible. Therefore the kernel and the cokernel of x·idMare trivial and the kernel
and the cokernel of fare in add(M). Therefore (i) follows.
If Mis an indecomposable module over a finite dimensional hereditary k-algebra Aof
finite type, then by [ASS06, VII 5.14] HomA(M, M) = kand Ext1
A(M, M) = 0. Hence
(ii) follows from (i).
Hovey proved a classification result in a commutative situation. Note that if the global
dimension of a ring coherent Ris finite, then Db(mod(R)) is equivalent to the category
of perfect complexes Dper(R).
Theorem 6.3.6. [Hov01a, Theorem 3.6] Let Rbe a commutative regular coherent ring.
There is a one-to-one correspondence between the thick subcategories in Db(mod(R)) and
the thick subcategories of mod(R).
If Ris regular noetherian, then a thick subcategory is also closed under subobjects,
quotient-objects and extensions [Hov01a, 3.7] and is therefore a Serre subcategory. Gar-
kusha and Prest generalized Theorem 6.3.6 in the following way: if Ris a commutative
coherent ring, then the thick subcategories in Dper(R) correspond bijectively to the Serre
subcategories in mod(R) [GP07, Theorem C].
In these theorems the classifications [Nee92, Tho97] of the thick subcategories of
Dper(R) are used to determine the thick subcategories of mod(R). We go the other
way around and describe thick subcategories of the triangulated category in terms of the
abelian category.
6.4 Classification of thick subcategories
In this section we prove the classification result and determine all thick subcategories in
two examples combinatorially.
Theorem 6.4.1. Let Abe a hereditary abelian category. The assignments
f:C 7→ {H0C|C∈ C} and g:M 7→ {C∈ Db(A)|HnC∈ M ∀ n∈Z}
induce mutually inverse bijections between
•the class of thick subcategories in Db(A)and
•the class of thick subcategories in A.
Proof. The proof mainly uses Lemma 6.2.3. First note that gis well defined because M
is thick and closed under direct summands by Lemma 6.3.3. The map fis well defined
because of the following lemma:
Lemma 6.4.2. Let C ⊂ Db(A)be thick. The full subcategory f(C)⊂ A is thick.
60
It remains to show that fand gare mutually inverse. The inclusion f(g(M)) ⊂ M
is obvious. Any object M∈ M is in f(g(M)) since the stalk complex · · · → 0→M→
0→... is in g(M). Since a complex is determined by its homology (Lemma 6.2.3) the
equality g(f(C)) = Cholds.
In order to prove Lemma 6.4.2 we need the following
Lemma 6.4.3. If g:C→Dis a map of complexes such that the differentials of Cand D
are zero and gm= 0 for all m6=n, then ker(g)and coker(g)are retracts of H∗(cone(g)).
Proof. The only non-zero differential in cone(g) is cone(g)n−1→cone(g)n:
cone(g)n−2
d
Cn−1
0
0
##
H
H
H
H
H
H
H
H
HLDn−2
0
cone(g)n−1
d
Cn
0
g
##
H
H
H
H
H
H
H
H
H
HLDn−1
0
cone(g)n
d
Cn+1
0
0
##
H
H
H
H
H
H
H
H
HLDn
0
cone(g)n+1 Cn+2 LDn+1.
Thus we can compute the homology:
Hm(cone(g)) =
Cm+1 ⊕Dmm≤n−2 or m≥n+ 1
ker(g)⊕Dn−1m=n−1
Cn+1 ⊕coker(g)m=n.
Proof of Lemma 6.4.2. We show that f(C) is closed under extensions, kernels and
cokernels. So let C1, C2be in Cand M∈ A such that there is a short exact sequence
0→H0C1→M→H0C2→0.
This sequence corresponds to a triangle
H0C1→M→H0C2→ΣH0C1
in Db(A). Here we consider an object of Aas a complex in Db(A) by means of the
inclusion A → Db(A) which sends the object Mto the complex concentrated in degree
0. By abuse of notation we call it again M. Each homology group of a complex C∈ C is
again contained in Csince by Lemma 6.2.3 HnCis a retract of Cup to isomorphism and C
is thick. Therefore H0C1and H0C2are in C, and because Cis closed under suspensions,
ΣH0C1∈ C. Since Cis closed under extensions, we conclude that Mis in C. Hence
M∈f(C) because the zeroth homology of · · · → 0→M→0→... is M.
So it only remains to show that f(C) is closed under kernels and cokernels. Let C1, C2
be in Cand fbe a morphism in the exact sequence in A:
0→ker(f)→H0C1
f
−→ H0C2→coker(f)→0.
61
Now extend fto a map of complexes
M
n∈Z
HnC1[−n]→M
m∈Z
HmC2[−m]
which is fin degree 0 and zero in all other degrees. We call it again f. Since Ci∼
=
Ln∈ZHnCi[−n] for i= 1,2, the map fbelongs to C. The cone of fis in C. By
Lemma 6.4.3 ker(f) and coker(f) are retracts of H0(cone(f)) and are hence (considered
as stalk complexes) in C. Therefore the kernel and cokernel of f, considered as objects in
A, are in f(C). 2
Corollary 6.4.4. Let Aand A0be hereditary abelian categories. If Db(A)and Db(A0)
are triangle equivalent, then there is a one-to-one correspondence between the thick sub-
categories in Aand the thick subcategories in A0.
With this theorem we have reduced the classification of thick subcategories in the
triangulated category Db(A) to the task of understanding thick subcategories in A. In
easy examples it is possible to determine them combinatorially. Let kbe a field and A
be a representation finite hereditary k-algebra. As a consequence of Lemma 6.3.5 there
are examples of thick subcategories of the category of finitely presented modules mod(A).
As an immediate consequence we are able to determine the thick subcategories of finite
dimensional representations of an A2- and an A3-quiver. For the two examples let k
be an algebraically closed field, Qthe respective quiver, A=kQ the path algebra and
A= mod(kQ) the category of finitely presented modules over A. We use the Auslander-
Reiten quiver to describe the category Acombinatorially.
Example 6.4.5. Let Qbe the quiver 1 2.
ooThe Auslander-Reiten quiver ΓkQ is the
following graph:
P2
@
@
@
@
@
@
@
P1
??
P2
P1.
There are exactly four non-trivial thick subcategories: add(P1), add(P2), add(P2
P1) and
mod(kQ).
Example 6.4.6. Let Qbe the quiver 1 2
oo3.
ooThe Auslander-Reiten quiver has
the following shape:
P3
?
?
?
?
?
?
?
?
P2
??
=
=
=
=
=
=
=
=
P3
P1
?
?
?
?
?
?
?
?
P1
@@
P2
P1
@@
P3
P2.
Lemma 6.3.5 tells us that there are six thick subcategories containing exactly one inde-
composable. Furthermore there are two thick subcategories that contain two indecom-
posable modules, four with three indecomposables and the whole module category with
six indecomposables.
62
The left column of Table 1 shows the thick subcategories in terms of the contained
indecomposable modules. E.g. hP1, P3iis the smallest thick subcategory containing P1
and P3. The right column displays the part of the corresponding Auslander-Reiten quiver
that is contained in the thick subcategory C. Modules in Care labelled with fat bullets
and morphisms in Cwith full arrows.
hP1i,...,hP3P1i•
hP1, P3P2i·
·
??
·
•
??
·
??
•
hP3, P2P1i•
·
??
·
·
??
•
??
·
hP1, P2, P2P1i·
•
??
@
@
@
@
@
@
@·
•
??
~
~
~
~
~
~
~•
??
·
hP2P1, P3P1, P3P2i·
·
??
•
@
@
@
@
@
@
@
·
??
•
??
~
~
~
~
~
~
~•
hP2, P3, P3P2i•
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
•
??
~
~
~
~
~
~
~
·
·
??
·
??
•
hP1, P3, P3P1i•
@
@
@
@
@
@
@
·
•
•
??
·
??
·
hP1, P2, P3imod(A)
Table 1
63
The thick subcategories are symmetric with respect to reflection at the axis going through
P3and P2
P1in the Auslander-Reiten quiver. The categories add(P3), add(P2
P1), hP3,P2
P1i,
hP1,P3
P2iand mod(kQ) are invariant under the reflection. Under the reflection add(P3)
corresponds to add(P3
P1), add(P1) corresponds to add(P3
P2), hP1, P2,P2
P1icorresponds to
hP3
P1,P2
P1,P3
P2iand hP1, P3,P3
P1icorresponds to hP2, P3,P3
P2i.
It would be interesting to work out all thick subcategories for all representation finite
algebras.
In fact, the orientation of the quiver does not play a role.
Corollary 6.4.7. Let kbe an algebraically closed field. Let Qand Q0be quivers whose
underlying graph is Dynkin of the same type but whose orientation is different. Then
mod(kQ)contains the same number of thick subcategories as mod(kQ0).
Proof. By Theorem 6.2.4 mod(kQ) and mod(kQ0) are derived equivalent and Corol-
lary 6.4.4 yields the assertion.
If the algebra Ais not of global dimension one, then Lemma 6.2.3 does not remain true.
But if the global dimension of Ais finite the Happel functor Db(mod(A)) →mod( ˆ
A) is an
equivalence [Hap88, II.4.9]. Here, ˆ
Adenotes the repetitive algebra of A. A generalization
of the classification Theorem 6.4.1 may possibly be achieved by characterizing the thick
subcategories of mod(ˆ
A) in terms of the thick subcategories of mod(A).
6.5 Classification of localizing subcategories
In this section we use the strategy of Theorem 6.4.1 to classify the localizing subcategories
of the full derived category of a hereditary Grothendieck category. As an application we
prove that the Smashing Conjecture is true for D(A) for a hereditary artin algebra Aof
finite representation type.
Recall that a full subcategory of a triangulated category with arbitrary direct sums is
called localizing if it is thick and closed under arbitrary direct sums. These categories are
the unbounded analogues of the thick subcategories. Recall from Example 2.1.3 that for
a Grothendieck category Athe unbounded derived category exists.
Theorem 6.5.1. Let Abe a hereditary Grothendieck category. The assignments
f:C 7→ {H0C|C∈ C} and g:M 7→ {C∈ D(A)|HnC∈ M ∀ n∈Z}
induce mutually inverse bijections between
•the class of localizing subcategories in D(A)and
•the class of thick subcategories in Athat are closed under small coproducts.
Proof. Adding the following comments the proof of Theorem 6.4.1 applies. Lemma 6.2.3
is not limited to the bounded derived category, and hence can be used here. The map
gis well-defined, since the homology functor commutes with infinite direct sums. And
finally if Cis localizing, then f(C) is closed under direct sums for the same reason.
64
Since the module category of a representation finite algebra is determined by the finitely
generated modules by Theorem 6.1.14, we can show the following
Corollary 6.5.2. Let Abe a hereditary artin algebra of finite representation type.
(i) Every thick subcategory M ⊂ Mod(A)that is closed under direct sums is the small-
est thick subcategory that contains M ∩ mod(A)and is closed under direct sums.
(ii) Every localizing subcategory C ⊂ D(A)is determined by its intersection with the
perfect complexes: C=hC ∩ Dper(A)iloc.
Proof. By Theorem 6.1.14, (i) is true. For the assertion (ii) let C ⊂ D(A) be localizing
and C∈ C be an object. By Lemma 6.2.3 it suffices to show that H0Cis contained
in hC ∩ Dper(A)iloc. Because of Theorem 6.1.14 there are a set Iand finitely generated
modules {Mi|i∈I}such that H0C∼
=Li∈IMi. Since Cis thick, it follows that Mi∈ C.
For every Michoose a projective resolution
0→P0
i→P1
i→Mi→0
such that P0
i, P1
iare finitely generated. The complex Pi: 0 →P0
i→P1
i→0 is perfect
and hence in Dperf(A). Since Pi→Miis a quasi isomorphism and Mi∈ C we can conclude
that Pi∈ C ∩ Dper(A). Hence H0Cis a direct sum of perfect complexes in C.
Since the perfect complexes form precisely the compact objects in D(A), Corollary 6.5.2(ii)
shows:
Corollary 6.5.3. The Smashing Conjecture 2.4.7 is true for the derived category of a
hereditary artin algebra of finite representation type.
In fact, even all localizing subcategories are determined by the intersection with the
compact objects.
If Ais not of finite type the Smashing Conjecture is possibly also true since every
module over Ais a filtered colimit of finitely presented modules. Choosing a clever
indexing category may lead to a proof of the Smashing Conjecture for arbitrary hereditary
algebras.
65
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[AHST06] L. Angeleri-H¨ugel, J. Saroch, and J. Trlifaj. On the telescope conjecture for
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