a Xi :1302.2442 4 [ma h.AG] 13 Sep 2014
GODEMENT RESOLUTIONS AND SHEAF HOMOTOPY THEORY
BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
Abs ac . The Godemen cosimplicial esolu ion is a ailable o a wide ange o ca ego ies o shea es.
In his pape we in es iga e unde which condi ions o he G o hendieck si e and he ca ego y o
coe icien s i can be used o ob ain ib an models and hence o do shea homo opy heo y. Fo
ins ance, o which G o hendieck si es and coe icien s we can de ine shea cohomology and de i ed
unc o s h ough i .
Con en s
1. In oduc ion 2
2. Homo opical p elimina ies 4
2.1. Descen ca ego ies 4
2.2. Ca an-Eilenbe g ca ego ies 7
3. Ca ego ies o shea es 8
3.1. Shea es o se s 8
3.2. Shea es wi h gene al coe icien s. 10
3.3. The cosimplicial Godemen esolu ion 11
4. Ca an-Eilenbe g ca ego ies o shea es 12
4.1. Ca an-Eilenbe g ib an shea es 12
4.2. The hype cohomology shea 13
4.3. Cha ac e iza ion 17
4.4. De i ed unc o s o shea es 20
5. Examples 21
5.1. Bounded complexes o shea es 22
Da e: Sep embe 16, 2014.
Fi s named au ho pa ially suppo ed by ERC S a ing G an p ojec TGASS and by con ac s SGR-119 and
FQM-218. Second named au ho pa ially suppo ed by p ojec s MTM2009-09557, 2009 SGR 119 and MTM2012-
38122-C03-01/FEDER. To appea in Collec anea Ma hema ica. The inal publica ion is a ailable a Sp inge ia
h p://dx.doi.o g/10.1007/s13348-014-0123-x.
1
2 BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
5.2. Unbounded complexes o shea es 22
5.3. Shea es o ib an simplicial se s 25
5.4. Shea es o ib an spec a 27
5.5. Shea es o il e ed complexes 27
6. Va ying Xand D28
6.1. Va ying X28
6.2. Va ying D31
Re e ences 32
1. In oduc ion
1.0.1. Godemen esolu ions ha e been an essen ial ool in shea homo opy heo y and i s
applica ions almos om he s a [Go] and keep c opping up in di e en con ex s: see o
ins ance [SGA4] o abelian shea es on a G o hendieck si e, [Th] o shea es o spec a on
a G o hendieck si e, [N] o shea es o ( il e ed) dg commu a i e algeb as o e opological
spaces, [MV] o simplicial shea es on a G o hendieck si e, [SdS] o shea es o OX-modules
o e schemes, o [GL] and [Ba] o shea es o DG-ca ego ies o e schemes..., o name bu a ew.
In pa icula , he g ea lexibili y o he cosimplicial Godemen esolu ion, oge he wi h i s
excellen unc o ial p ope ies, appea o accoun o i s omnip esence: in ac , in o de o
de ine i o a shea F:Xop −→ D on a G o hendieck si e wi h enough poin s Xand alues in
some ca ego y o coe icien s D, we only need D o ha e il e ed colimi s and a bi a y p oduc s.
In his si ua ion, we ob ain a unc o
G•:Sh(X,D)−→ ∆Sh(X,D)
om shea es on Xwi h alues in D o cosimplicial ones.
The ques ion we add ess in his pape is he ollowing: unde which condi ions o he G o hen-
dieck si e Xand he ca ego y o coe icien s Dcan he cosimplicial Godemen cons uc ion be
used o ans e homo opical s uc u e om D o he ca ego y o shea es Sh(X,D)?
1.0.2. Le us elabo a e a li le u he . Making use o a ( ealiza ion o he) homo opy limi
s: ∆D −→ D, which we call a simple unc o , we can “ eassemble” all he cosimplicial pieces o
GpFob aining a single shea which migh be en i led o be a “model” o F. To ge ancho age
o he ideas, he eade may hink o Das being he ca ego y o cochain complexes o abelian
g oups C∗(Ab) and s he o al complex o a double complex. In his way we ob ain a shea
oge he a uni e sal map
ρF:F −→ HX(F) = sG•(F),(1.0.1)
called he e he hype cohomology shea o F ollowing Thomason and Mi chell ([Th], [Mi ]).
GODEMENT RESOLUTIONS 3
So a pa icula ins ance o ou ini ial ques ion is he ollowing: assume ha Xhas a inal
objec X, when would i make sense o de ine shea cohomology o Xwi h coe icien s in Fas
Γ(X, HX(F))? Mo e p ecisely, we a e asking when his o mula would de ine a igh de i ed
unc o in he sense o Quillen [Q]; ha is, a le Kan ex ension.
1.0.3. In o de o alk abou de i ed unc o s and homo opy ca ego ies, we need o speci y
he class o mo phisms wi h espec o which we localize. In all he examples we a e awa e o ,
his is he class ha keeps ack o he opology o X, he one o local equi alences: we ha e a
dis inguished class o mo phisms E, o “equi alences”, in he ca ego y o coe icien s D; and, o
a mo phism o shea es ϕ:F −→ G o be called a local equi alence, we equi e e e y mo phism
induced on s alks ϕx:Fx−→ Gx o be in E. Le us no e his class o local equi alences as
W. Fo ins ance, o D=C∗(Ab) we could ake E o be he class o quasi-isomo phisms,quis,
mo phisms which induce isomo phisms in cohomology.
1.0.4. A i s app oach could be o s udy ou ques ion in he con ex o Quillen model ca e-
go ies. Tha is, o assume ha ou coe icien ca ego y (D,E) suppo s a Quillen model s uc u e
and ha i induces one on (Sh(X,D),W) in such a way ha he Godemen esolu ion becomes
a ib an model o e e y shea . As p o ed in [Be1] and [Be2], his is indeed possible unde
ce ain (non- i ial) hypo heses on he model ca ego y (D,E).
Ins ead, we op he e o keep o he minimum he amoun o s uc u e on (Sh(X,D),W) neces-
sa y o ha e he shea es HX(F) as ib an models. This allows us o 1) co e a mo e gene al class
o coe icien ca ego ies (e.g. il e ed complexes o e any (AB4)∗and (AB5) abelian ca ego y)
and 2) ha e mo e lexibili y in he ans e ence o he esul ing echnique o he mul iplica i e
se ing. This las poin is he subjec o a o hcoming sequel o his pape , whe e we ans e
he esul s ob ained he e o Sh(X,D) o shea o ope ads and algeb as (o e any ope ad) on
D, and hei co esponding il e ed e sions.
One such minimal amoun o s uc u e is a ained wi h Ca an-Eilenbe g ca ego ies, o CE-
ca ego ies, o sho : an app oach o homo opical algeb a s a ed in [GNPR1] and u he
de eloped in [P], [C1] and [C2]. A ( igh ) CE-ca ego y consis s o a ca ego y Cendowed wi h
wo classes o dis inguished mo phisms, s ong and weak equi alences, S ⊂ W, and a CE- ib an
model o each objec (see 2.2.2 o he p ecise de ini ion). The name o hese s uc u es comes
om he classic book [CE], whe e, in mode n pa lance, he homo opy heo y o he ca ego y
o cochain complexes C∗(Ab) is de eloped a ound wo classes o dis inguished mo phisms:
homo opy equi alences (S) and quis (W).
Bu CE-s uc u es allow mo e eedom o choice o classes Sand W han classical “homo opy
equi alences” and “weak equi alences”. This is pa icula ly in e es ing o ou ca ego ies o
shea es, o which he na u al choices a e:
•global equi alences, as S: hose mo phisms o shea es such ha ϕ(U) : F(U)−→ G(U)
belongs o he class o equi alences E in D o e e y objec (open se ) U∈ X , and
•local equi alences, as W: al eady men ioned abo e.
In o de o p o ide ou ca ego ies o shea es Sh(X,D) wi h a CE-s uc u e, we need e y ew
elemen s in ou ca ego y o coe icien s D: essen ially, ou needs educe o a class o equi alences
4 BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
E and a simple unc o s: ∆D −→ D which is a ealiza ion o he homo opy limi . This is
summa ized in he no ion o descen ca ego y (see [Rod1], [Rod2] and he second sec ion in his
pape ; c . also [GN]).
1.0.5. Ou main esul (Theo em 4.3.2) p o ides equi alen condi ions gua an eeing ha ou
ini ial ques ion has a posi i e answe :
Theo em 1.0.1. Le Xbe a G o hendieck si e and (D,E) a descen ca ego y sa is ying he
hypo heses (4.1.1). Then, he ollowing s a emen s a e equi alen :
(1) (Sh(X,D),S,W)is a igh Ca an-Eilenbe g ca ego y and o e e y shea F ∈ Sh(X,D),
ρF:F −→ HX(F)is a CE- ib an model.
(2) Fo e e y shea F ∈ Sh(X,D),ρF:F −→ HX(F)is in W.
(3) The simple unc o commu es weakly wi h s alks.
(4) Fo e e y shea F ∈ Sh(X,D),HX(F)sa is ies Thomason’s descen ; ha is, ρHX(F):
HX(F)−→ H2
X(F)is in S.
This heo em shows, i s , ha he exis ence o a CE-s uc u e on he ca ego y o shea es
Sh(X,D) boils down o he p ope y ha o e e y shea F he uni e sal a ow ρF:F −→
HX(F) is a local equi alence (condi ion (2)). Hence, we need no hing else ha his CE-s uc u e
o answe ou p oblem; i.e., he ac ha he Godemen cons uc ion can be used o ans e
homo opical s uc u e om D o he ca ego y o shea es Sh(X,D) is equi alen o he exis ence
o his CE-s uc u e.
The heo em also shows ha he ac o he Godemen esolu ion being a CE- ib an model is
equi alen o Thomason’s classic descen ( o shea es o spec a [Th], condi ion (4); see also
Co olla y 4.3.5). So being CE- ib an is qui e a na u al and cen al no ion o shea es.
Finally, he heo em gi es a down- o-ea h equi alen condi ion o all his o happen, which
will be he one we will use in p ac ice: condi ion (3) says ha he simple unc o smus
commu e wi h s alks up o local equi alence. Fo ins ance, o bounded cochain complexes his
is a consequence o he commu a ion o he o al complex unc o To wi h il e ed colimi s.
1.0.6. Acknowledgemen s. This pape de elops an idea sugges ed o us by Vicen e Na a o. We
owe him a deb o g a i ude o sha ing i wi h us. The second named au ho also bene i ed
om many ui ul con e sa ions wi h Pe e Pascual. We a e indeb ed o F ancisco Guill´en,
Fe nando Mu o, Luis Na ´aez and Abd´o Roig o hei commen s. People a sci.ma h. esea ch
and Ma ho e low made use ul sugges ions kindly answe ing ou ques ions he e.
2. Homo opical p elimina ies
We in oduce he e he de ini ions and esul s conce ning descen and CE-ca ego ies necessa y
o ou pape . The in e es ed eade may consul [Rod1], [Rod2] and [GNPR1] o u he
de ails.
2.1. Descen ca ego ies.
GODEMENT RESOLUTIONS 5
2.1.1. No a ions. By ∆we mean he simplicial ca ego y. We deno e by ∆D( esp. ∆opD) he
ca ego y o cosimplicial ( esp. simplicial) objec s in a ixed ca ego y D. The diagonal unc o
D : ∆∆D −→ ∆Dis gi en by D({Zn,m}n,m≥0) = {Zn,n}n≥0. The cons an simplicial objec
de ined by A∈ D will be deno ed by c(A) o by A×∆.
2.1.2. A (cosimplicial) descen ca ego y consis s, oughly, o a ca ego y Dendowed wi h a class
E o ‘weak equi alences’ and wi h a ‘simple’ unc o s:∆D −→ D subjec o he axioms below.
These axioms ensu e ha sis a ealiza ion o he homo opy limi o cosimplicial objec s, and
ha he localized ca ego y D[E−1] possesses a ich homo opical s uc u e.
De ini ion 2.1.1. [Rod1, 1.1] A (cosimplicial)descen ca ego y is he da a (D,E,s, µ, λ) whe e
Dis a ca ego y closed unde ini e p oduc s and E is a sa u a ed class o mo phisms o D, closed
unde ini e p oduc s, called weak equi alences. The iple (s, µ, λ) is subjec o he ollowing
axioms:
(S1) The simple unc o s:∆D −→ D commu es wi h ini e p oduc s up o equi alence.
Tha is, he canonical mo phism s(X×Y)−→ s(X)×s(Y) is in E o all X,Yin ∆D.
(S2) µ:ss 99K sD is a zigzag o na u al weak equi alences. Recall ha sDZdeno es he
simple o he diagonal o Z, while ssZ=s(n−→ s(m→Zn,m)).
(S3) λ: idD99K s(− × ∆) is a zigzag o na u al weak equi alences, which is assumed o be
compa ible wi h µin he sense o op.ci ..
(S4) I :X−→ Yis a mo phism in ∆Dwi h n∈E o all n, hen s( )∈E.
(S5) The image unde he simple unc o o he cosimplicial map Ad0:A∆[1] −→ Ais a weak
equi alence o each objec Ao D.
Fo he sake o b e i y, we will also deno e a descen ca ego y by (D,E).
Rema k 2.1.2. The p esence o zigzags in he de ini ion o descen ca ego y is needed o
ensu e i s homo opy in a iance (see [Rod1], P oposi ion 1.8). Howe e , e e y example used
in his pape has bo h µand λas ac ual na u al ans o ma ions (see Examples 2.1.4 - 2.1.6).
Since his signi ican ly simpli ies exposi ion, we will assume hey a e so o he descen ca ego ies
conside ed h oughou he pape . We will also assume ha simple unc o s p ese e limi s. Bu
his is no a majo es ic ion: i is ul illed by all ou examples o descen ca ego ies so a .
2.1.3. Among he he edi a y esul s o descen ca ego ies, le us poin ou one we will be using
ime and again and whose p oo we lea e as an easy exe cise o he in e es ed eade :
Lemma 2.1.3 (T ans e Lemma).Le (D′,E′,s′, µ′, λ′)be a descen ca ego y. Gi en a unc o
ψ:D −→ D′, conside in D he weak equi alences E = ψ−1E′. Assume ha Dhas ini e
p oduc s and is equipped wi h a unc o s:∆D −→ D, oge he wi h compa ible na u al weak
equi alences µ:ss −→ sDand λ: idD−→ s(−×∆). Then, (D,E,s, µ, λ)is a descen ca ego y
p o ided he ollowing s a emen s hold:
(FD1)ψcommu es wi h ini e p oduc s up o equi alence. Tha is, he na u al map ψ(X×
Y)−→ ψ(X)×ψ(Y)is in E o all X, Y in D.

6 BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
(FD2)The e exis s a na u al weak equi alence θ:ψs−→ s′ψ illing he squa e
∆Dψ//
s

∆D′
s′

Dψ//
θ
⇒
D′
2.1.4. To end wi h, we desc ibe some examples o descen ca ego ies.
Example 2.1.4. Bounded complexes [Rod1, (3.4)].Le Abe an abelian ca ego y. Fo
a ixed in ege b∈Z, deno e by C≥b(A) he ca ego y o uni o mly bounded below cochain
complexes o A; ha is, An= 0 o all n < b and all A∗∈C≥b(A).
We will conside he ollowing descen s uc u e on C≥b(A). The weak equi alences E a e
he quasi-isomo phism (quis): hose maps inducing isomo phism in cohomology. The simple
unc o s:∆C≥b(A)−→ C≥b(A) a a gi en cosimplicial cochain complex Ais he (p oduc )
o al complex o he double complex induced by A:
s(A)n=Y
p+q=n
Apq .
Which, in his case, since Ahas ini e codiagonals, s(A)n=Lp+q=nApq.µZis jus he
Alexande -Whi ney map ssZ−→ sDZand λn
X:Xn−→ s(X×∆)nis he canonical inclusion.
Example 2.1.5. Unbounded complexes. The ca ego y C∗(A) o unbounded cochain com-
plexes o Ais also a descen ca ego y wi h weak equi alences, simple unc o , µand λde ined
as in he bounded case p o ided axiom (S4) holds. Fo ins ance, his is he case when A=R-
modules.
Example 2.1.6. Simplicial model ca ego ies [Rod1, Theo em 3.2].The subca ego y
o ib an objec s M o a model ca ego y Mis a descen ca ego y whe e E is he class o
weak equi alences o Mand he simple unc o is he Bous ield-Kan homo opy [BK] limi ,
holim
←− :∆M −→ M , as de ined in [Hi ]. I Mis a simplicial model ca ego y, he homo opy
limi o a cosimplicial objec Xis he end o he bi unc o XN(∆↓·):∆op ×∆−→ M ,
(n, m)7→ (Xm)N(∆↓n), ha is,
holim
←− X=Zn
(Xn)N(∆↓n).
Mo phisms µand λa e easily de ined using ha a unc o F:B −→ C induces a na u al map
holim
←− CX−→ holim
←− BF∗X.
Two pa icula ins ances o his example a e ele an when alking abou shea cohomology
heo ies. Fi s , he ca ego y sS o poin ed Kan complexes, wi h weak equi alences he weak
homo opy equi alences. Secondly, he ca ego y Sp o poin ed ib an spec a, as de ined in
[Th, 5.2]. The weak equi alences o he descen s uc u e a e hen he s able weak equi alences;
ha is, mo phisms o spec a inducing bijec ions in all homo opy g oups.
GODEMENT RESOLUTIONS 7
Example 2.1.7. Fil e ed complexes. Deno e by FC≥b(A) he ca ego y o il e ed complexes,
wi h objec s he pai s (A, F) whe e Ais in C≥b(A) and F is a dec easing il a ion o A.
Gi en ≥0, conside he class E o weak equi alences gi en by he E -quasi-isomo phisms o
FC≥b(A), ha is, mo phisms o il e ed complexes such ha he induced mo phism be ween
he E +1- e ms o he spec al sequences associa ed wi h he il a ions is an isomo phism.
I holds ha (FC≥b(A),E ) is a descen ca ego y wi h simple unc o (s, δ ) : ∆FC≥b(A)→
FC≥b(A) de ined as (s, δ )(A, F) = (s(A), δ (F)) whe e
δ (F)k(s(A)n) = M
i+j=n
Fk− iAi,j ,
and wi h na u al ans o ma ions λand µgi en a he le el o complexes by hose o C≥b(A).
I = 0, no e ha an E0-isomo phism is he same hing as a g aded quasi-isomo phism. Also,
(s, δ0)(A, F) is jus (s(A),s(F)). The ac ha his is a simple unc o o (FC≥b(A),E0) is
an easy consequence o he ans e lemma applied o he g aded unc o G : FC≥b(A)→
C≥b(A)Z.
To ea he gene al case, conside he decalage il a ion unc o Dec :FC≥b(A)→FC≥b(A),
(A, F) 7→ (A, DecF), whe e (DecF)kAn= ke {d: Fk+nAn→Fk+nAn+1/Fk+n+1An+1}. Since
Dec (s, δ +1) = (s, δ )Dec, by applying he ans e lemma induc i ely, we can conclude ha
(s, δ ) is a simple unc o o (FC≥b(A),E ), o each ≥0.
2.2. Ca an-Eilenbe g ca ego ies.
2.2.1. Ca an-Eilenbe g ca ego ies a e a new app oach o homo opical algeb a de eloped in
[GNPR1]. They use, we belie e, a minimum amoun o da a in o de o de i e unc o s, so i s
condi ions can be ul illed by a wide class o ca ego ies, as we a e going o show.
De ini ion 2.2.1. Le (C,S,W) be a ca ego y wi h wo classes Sand Wo dis inguished
mo phisms, called espec i ely s ong and weak equi alences, and such ha S ⊂ W. An objec
Mo Cis called Ca an-Eilenbe g ib an ,CE- ib an o sho , i o each weak equi alence
w:Y−→ X∈ W and e e y mo phism ∈ C[S−1], he e is a unique mo phism g∈ C[S−1]
making he ollowing iangle commu a i e:
Yw//

X
g
~~
M
Rema k 2.2.2. He e Wdeno es he sa u a ion o W. Classes Sand Wo s ong and weak
equi alences conside ed la e in he s udy o shea es a e sa u a ed, i.e. S=Sand W=W.
In his case, Whi ehead’s heo em holds: a weak equi alence be ween CE- ib an objec s is a
s ong one.
2.2.2. A igh CE- ib an model o an objec Xo Cis a mo phism w:X−→ Mo C[S−1] ha
becomes an isomo phism in C[W−1], and such ha Mis CE- ib an . I Xadmi s a CE- ib an
model, i is unique up o unique isomo phism o C[S−1].
8 BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
De ini ion 2.2.3. A ca ego y wi h s ong and weak equi alences (C,S,W) is called a igh
Ca an-Eilenbe g ca ego y, o CE-ca ego y o sho , i each objec Xo Chas a CE- ib an
model. In his case, we will also say ha Chas enough CE- ib an models.
Example 2.2.4. I Cis a Quillen model ca ego y and S,Wa e he classes o i s igh homo opy
equi alences and weak equi alences, espec i ely, hen (Cc,S,W) is a igh Ca an-Eilenbe g
ca ego y. He e Ccis he ull subca ego y o Quillen co ib an objec s. In his case, e e y
Quillen ib an objec is CE- ib an , bu he con e se needs no be ue: by i s e y de ini ion,
CE- ib an objec s a e homo opically in a ian , while Quillen ib an objec s a e no .
Rema k 2.2.5. So, CE-ca ego ies na u ally include Quillen model ones and he inclusion is
“s ic ” in he sense ha , o ins ance, he class Smus no be any class o “homo opy equi -
alences”. This is pa icula ly impo an o us because, in he case o shea es, he global
equi alences canno indeed be he homo opy equi alences o any Quillen model s uc u e, as
shown in [GNPR2]. Since hese global equi alences a e such a na u al ing edien o shea es,
his seems o be signi ican . Global equi alences a e needed, o ins ance, o alk abou shea es
sa is ying Thomason descen , which a e p ecisely CE- ib an models, o close he ci cle.
2.2.3. In CE-ca ego ies, he de i abili y c i e ion o unc o s eads as ollows (see [GNPR1,
3.2.1]).
P oposi ion 2.2.6. Le (C,S,W)be a Ca an-Eilenbe g ca ego y and F:C −→ D a unc o
such ha F(s)is an isomo phism o e e y s ong equi alence s∈ S. Then Fhas a igh de i ed
unc o RF:C[W−1]−→ D whose alue on objec s may be compu ed as RF(X) = F(M), whe e
Mis a ib an model o X.
2.2.4. In he CE-ca ego ies conside ed la e on, he CE- ib an model o an objec Xwill
be unc o ial in he sense o [GNPR1, 2.5]: wha we call a esol en unc o . One o he
ad an ages o ha ing a esol en unc o is ha , i C ib deno es he ull subca ego y o Co
CE- ib an objec s, he e is an equi alence o ca ego ies ([GNPR1, P oposi ion 2.5.3(2)])
C ib[S−1]i
∼//C[W−1]
R
oo
3. Ca ego ies o shea es
We ecall some gene al de ini ions and esul s abou shea es o se s on a G o hendieck si e.. Ou
main objec i e is o poin ou o mulas (3.1.3) and (3.1.4) o s alks and skysc ape shea es,
espec i ely. Then we obse e ha hese o mulas s ill make sense o shea es wi h alues in
any ca ego y wi h il e ed colimi s and a bi a y p oduc s, and ha hey do indeed o m a pai
o adjoin unc o s. The associa ed iple gi es us he cosimplicial Godemen esolu ion.
We also show ha he ca ego y o shea es wi h alues in a descen ca ego y inhe i s a na u al
descen s uc u e, which will be used epea edly in he es o he pape .
3.1. Shea es o se s.
GODEMENT RESOLUTIONS 9
3.1.1. Le Xbe a ca ego y. Le b
X=P Sh(X,Se ) deno e he ca ego y o p eshea es on X
wi h alues in he ca ego y o se s Se . By he Yoneda embedding, e e y objec U∈ X can be
hough o as he ep esen able p eshea yU=X(−, U)∈b
X.
3.1.2. I Xis a G o hendieck si e,e
X=Sh(X,Se ) will deno e he ull subca ego y o
P Sh(X,Se ) whose objec s a e shea es.
Shea es may be cha ac e ized by he ollowing p ope y (see [McLM], page 122): a p eshea
F ∈ b
Xis a shea i and only i o e e y objec U∈ X and e e y co e S={Uα−→ U}o U,
he diag am
F(U)//QαF(Uα)////Qαβ F(Uαβ) (3.1.1)
is an equalize o se s. He e he second p oduc anges o e all composable pai s Uαβ −→ Uα,
Uα−→ Uwi h Uα−→ U∈S(hence also i s composi ion Uαβ −→ Ubelongs o S). I ollows
ha a unc o o p eshea es ha commu es wi h limi s will send shea es o shea es.
3.1.3. Le :X −→ Y be a mo phism o si es; ha is, a unc o be ween he unde lying
ca ego ies going in he opposi e di ec ion −1:Y −→ X which is con inuous. This means ha
he di ec image unc o ∗:b
X −→ b
Y,F 7→ F ◦ −1, es ic s o a unc o be ween shea es
∗:e
X −→ e
Y.
3.1.4. Recall ha a poin o a si e Xis by de ini ion a pai o adjoin unc o s x= (x∗, x∗)
e
Xx∗
//Se
x∗
oo,Se (x∗F, D) = e
X(F, x∗D)
such ha x∗commu es wi h ini e limi s. The igh adjoin x∗:Se −→ e
Xgi es o e e y se
D he so called skysc ape shea x∗Do Da he poin x. The le adjoin x∗:e
X −→ Se gi es
o e e y shea F he ib e o s alk x∗F=Fxo Fa x.
The ollowing “compu a ional” o mulas o x∗and x∗a e o us o u mos impo ance, since
hey allow us o ex end hem o ou ca ego ies o coe icien s D. Fi s , we ha e a canonical
and unc o ial isomo phism
x∗F=Fx= colim
−→ (U,u)F(U),(3.1.2)
whe e (U, u) uns o e he opposi e ca ego y o neighbou hoods o x([SGA4], expos´e IV, 6.8).
This colimi is a il e ed one.
Fo a se D∈Se , he shea x∗Dalso admi s he ollowing desc ip ion: o U∈ X ,
(x∗D)(U) = Y
u∈x∗(yU)
Du,(3.1.3)
whe e Du=D o all u∈x∗(yU).
16 BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
Hence, applying he simple unc o we deduce ha s◦(θ′◦
G•(F))s◦(η◦
s•G•(F)) = s◦s•(η◦
G•(F)). As-
sume i p o ed ha s◦s•(η◦
G•(F))∈ S. In his case,
φ=s◦s•(η◦
G•(F))λs•G•(F)=s◦(θ′◦
G•(F))s◦(η◦
s•G•(F))λs•G•(F)=s◦(θ′◦
G•(F))ρHX(F)
is an isomo phism o Sh(X,D)[S−1], so σF=φ−1s◦(θ′◦
G•(F)) is a sec ion o ρHX(F). To inish, i
emains o be shown ha s◦s•(η◦
G•(F))∈ S. This happens i and only i s•s◦(η◦
G•(F))∈ S. Fo
a ixed n≥0, he coaugmen a ion η◦
Gn(F)=η◦
Tn+1(F):c◦Tn+1(F)−→ G◦Tn+1(F) has an ex a
degene acy. Hence we in e ha s◦(η◦
Gn(F)) is in S o each n≥0. Bu hen i ollows om
axiom (S4) ha s(n→s◦(η◦
Gn(F))) = s•s◦(η◦
G•(F))∈ S as equi ed. 
4.2.4. The class Wo local equi alences is by de ini ion equal o (p∗)−1E. Below we p o e ha
W=T−1S=H−1
XSas well.
P oposi ion 4.2.6. Assume ha Xand (D,E) sa is y he hypo heses (4.1.1). Then, o a
mo phism :F −→ G o shea es, he ollowing condi ions a e equi alen :
(1) is a local equi alence.
(2) T( ) : T(F)−→ T(G)is a global equi alence.
(3) HX( ) : HX(F)−→ HX(G)is a global equi alence.
P oo . (1) implies (2) since T(W)⊂ S. Con e sely, i T( ) is a global equi alence, i is in
pa icula a local one, so p∗T( )∈E. On he o he hand, i ollows om he iangle iden i ies
o he adjoin pai (p∗, p∗) ha p∗( ) is a e ac o p∗T( ) = p∗p∗p∗( ). Bu E being sa u a ed,
i is closed unde e ac s, and we deduce ha p∗( )∈E as well. Bu his is he same as saying
ha ∈ S. The e o e, (1) and (2) a e equi alen .
Le us see ha (2) implies (3). Assume ha T( )∈ S. Since T(S)⊂T(W)⊂ S, hen
Gn( ) = Tn+1( )∈ S o all n≥0, and i ollows om (S4) ha HX( ) = sG•( )∈ S as
equi ed. Finally, i HX( )∈ S hen also THX( )∈ S. By Lemma 4.2.4 T( ) is a e ac o
THX( ), so T( )∈ S and (2) and (3) a e equi alen as well. 
4.2.5. As announced, we deduce ha he hype cohomology shea is always CE- ib an .
P oposi ion 4.2.7. Assume ha Xand (D,E) sa is y hypo heses (4.1.1). Then, o any shea
F,HX(F)is a CE- ib an shea .
P oo . Hypo hesis 4.1.1 gua an ee ha T(S)⊂ S, hence HX(S)⊂ S. By Lemma 4.2.4, i is
equipped wi h na u al ans o ma ions ρ: id −→ HXand σ:H2
X−→ HXsuch ha σ ρ = id.
As a i s consequence, a mo phism g:G −→ HX(F) o Sh(X,D)[S−1] is uniquely de e mined
by HX(g). Indeed, om he commu a i e diag am
Gg//
ρG

HX(F)1//
ρHX(F)

HX(F)
HX(G)HX(g)//HX(F)2
σF
88
q
q
q
q
q
q
q
q
q
q

GODEMENT RESOLUTIONS 17
we deduce ha g=σFHX(g)ρGas claimed. Conside now a li ing p oblem
Gw//

G′
HX(F)
whe e is a mo phism o Sh(X,D)[S−1] and wis a mo phism o Sh(X,D) ha is a local
equi alence. Since HX(W)⊂ S, gi en wo solu ions g, g′:G′−→ HX(F) o his li ing p oblem,
we would ha e HX(g) = HX( ) (HX(w))−1=HX(g′). Hence g=g′, and we need only see ha
he e is a leas one li ing o he abo e diag am. Bu g=σFHX( ) (HX(w))−1ρHX(F)is easily
seen o sa is y g w = , so we a e done. 
4.3. Cha ac e iza ion. In iew o he las p oposi ion, we conclude ha i o any shea
ηF:F −→ HX(F) we e in W, hen (Sh(X,D),S,W) would be a Ca an-Eilenbe g ca ego y
wi h (HX, ρ) as a esol en unc o . Below we show ha his ac is indeed equi alen o wo
o he condi ions: one o hem is Thomason’s descen p ope y o hype cohomology shea es,
while he o he one consis s o a weak commu a ion be ween he simple unc o and s alks.
4.3.1. Le us s a e p ecisely wha we mean by he la e condi ion.
De ini ion 4.3.1. Le Xbe a G o hendieck si e and (D,E) a descen ca ego y. We say ha
he simple unc o commu es weakly wi h s alks i o each shea F he map θG•F:p∗HX(F) =
p∗sG•(F)−→ sp∗G•(F) in (4.2.1) belongs o E.
Equi alen ly, scommu es weakly wi h s alks i o each poin x∈X he canonical map θG•F(x) :
(sG•F)x−→ s(G•F)xis a weak equi alence.
4.3.2. We can now s a e and p o e ou i s main esul .
Theo em 4.3.2. Le Xbe a G o hendieck si e and (D,E) a descen ca ego y sa is ying he
hypo heses (4.1.1). Then, he ollowing s a emen s a e equi alen :
(1) (Sh(X,D),S,W)is a igh Ca an-Eilenbe g ca ego y and o e e y shea F ∈ Sh(X,D),
ρF:F −→ HX(F)is a CE- ib an model.
(2) Fo e e y shea F ∈ Sh(X,D),ρF:F −→ HX(F)is in W.
(3) The simple unc o commu es weakly wi h s alks.
(4) Fo e e y shea F ∈ Sh(X,D),HX(F)sa is ies Thomason’s descen ; ha is, ρHX(F):
HX(F)−→ H2
X(F)is in S.
De ini ion 4.3.3. We say ha a descen ca ego y (D,E) is compa ible wi h he si e Xi he
equi alen condi ions o his heo em a e sa is ied.
Rema k 4.3.4. As we will see in he examples, his is no necessa ily he case o gene al Xand
(D,E). Fu he mo e, i may happen ha (Sh(X,D),S,W) is indeed a Ca an-Eilenbe g ca e-
go y, bu he CE- ib an model o a shea Fdoes no ag ee wi h HX(F) in gene al. Howe e ,
his does no pose much o a p oblem, and hese d awbacks only occu when Xis “coho-
mologically big”: a sui able ini e cohomological dimension hypo hesis on Xensu es ha he
hype cohomology shea HX(F) is always a (CE- ib an ) model o F.
18 BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
P oo o Theo em 4.3.2. By P oposi ion 4.2.7 we know ha HX(F) is CE- ib an o any
shea F. Hence, he equi alence be ween (1) and (2) is clea . Le us see ha (2) and (3) a e
equi alen . On he one hand, by de ini ion, (2) holds i and only i p∗(ρF) is in EX o any shea
F. On he o he hand, he cosimplicial Godemen esolu ion is such ha he coaugmen a ion
p∗ηF:cp∗(F)−→ p∗G•(F) has an ex a degene acy. I hen ollows om P oposi ion 3.2.3 ha
sp∗(ηF) belongs o E. Since λG:G −→ sc(G) is also in E o any shea G, we ha e he ollowing
commu a i e diag am in which he a ows deco a ed wi h ∼a e in E:
p∗(F)p∗(λF)
∼//
λp∗(F)
∼
%%
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑
p∗sc(F)p∗s(ηF)//
θc(F)

p∗sG•(F) = p∗HX(F)
θG•(F)

sp∗c(F)∼
sp∗(ηF)//sp∗G•(F)
No e ha he composi ion o he mo phisms in he op ow is p ecisely p∗(ρF) : p∗(F)−→
p∗HX(F). By he 2-ou -o -3 p ope y, we conclude ha p∗(ρF) is in E i and only i θG•(F)is in
E. In o he wo ds, (2) and (3) a e equi alen .
To inish wi h, we now show ha (4) and (2) a e equi alen . Because o P oposi ion 4.2.6,
W=H−1
XS. Hence, ρF:F −→ HX(F) is in Wi and only i HX(ρF) is in S. I is hen enough
o check ha ρHX(F)is in Si and only i HX(ρF) is. As in he p oo o Lemma 4.2.4, he
i e a ion o θ′gi es a canonical mo phism o cosimplicial objec s θ′◦
F•:G◦s•(F•)−→ s•G◦(F•)
ha makes he ollowing diag ams commu e
s◦s•G◦c•(F)∼
s◦s•G◦(η•
F)//s◦s•G◦G•(F)s◦s•c◦G•(F)
∼
s◦s•(η◦
G•(F))
oo
s◦(η◦
s•G•(F))
ww♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥
s◦G◦(F)
HX(ρF)
88
∼
s◦G◦(λF)//
∼
s◦(λG◦(F))
88
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
s◦G◦s•c•(F)s◦G◦s•(η•
F)//
∼
s◦(θ′◦
c•(F))
OO
s◦G◦s•G•(F)
s◦(θ′◦
G•(F))
OO
s•G•(F)
∼
λs•G•(F)
OO
ρHX(F)
oo
No e ha all he a ows deco a ed wi h ∼a e global equi alences: o hose a ows in ol ing λ
his is clea (in pa icula his is so o s◦(θ′◦
c•(F))). We al eady p o ed ha s◦s•(η◦
G•(F))∈ S, and
again using an ex a degene acy a gumen i eadily ollows ha s◦s•G◦(η•
F)∈ S. Consequen ly,
ρHX(F)∈ S i and only i HX(ρF)∈ S.
4.3.3. As a oy example, le ’s check wha ou main heo em says o he case o a opological
space wi h jus one poin .
Le X={x}be a opological space wi h jus one poin and wi h i s unique possible opology;
namely, i s open se s a e Open(X) = {∅,{x}}. So, e e y shea F ∈ Sh(X, D) is de e mined
by i s alue on x:F(x)∈ D. The co espondence φ:Sh(X, D)−→ D,F 7−→ F(x) de ines
an isomo phism o ca ego ies whose in e se is ψ:D −→ Sh(X, D), D7−→ D, whe e Dis he
shea de ined by D(x) = D.
GODEMENT RESOLUTIONS 19
Nex , in a sobe space such as X, he poin s o he si e Open(X) a e in a bijec i e co -
espondence wi h he poin s o Xas a plain opological space. So, we ha e exac ly one
G o hendieck poin ; ha is, a couple o adjoin unc o s x∗:Sh(X, D)⇄D:x∗, de ined
by x∗(F) = Fx=F(x) and (x∗D)(x) = D. In o he wo ds, x∗=φand x∗=ψ. Hence, i we
iden i y Sh(X, D) wi h Dusing φand ψ,x∗and x∗become he iden i y unc o o C. Hence, he
Godemen cons uc ion G•:D −→ ∆Dis simply he cons an cosimplicial unc o . Applying
he simple unc o , we ge HX(D) = sG•(D) = scD ≃D, because o axiom (S3) o a descen
ca ego y.
This en ails ha e e y objec Dshould be ib an wi h he CE-s uc u e gi en on Dby ou
main heo em. The eade can easily check ha i is so: unde he iden i ica ions φand ψ,
classes o local and global equi alences a e jus E: W=S= E and, wi h hese local and global
equi alences, e e y descen ca ego y is a CE-ca ego y in which e e y objec is ib an .
So, condi ion (1) o ou main heo em is indeed ul illed. The eade can check, o ins ance,
ha condi ion (3), he commu a ion be ween s alks and simple unc o , is also i ially ul illed
oo.
4.3.4. The i s consequence o ou main heo em is he ollowing cha ac e iza ion o Thoma-
son’s descen p ope y o shea es o spec a.
Co olla y 4.3.5. I (D,E) is compa ible wi h he si e X, hen a shea F ∈ Sh(X,D)sa is ies
Thomason’s descen i and only i i is a CE- ib an shea .
4.3.5. The exis ence o an associa ed shea unc o , o shea i ica ion, (−)a:P Sh(X,D)−→
Sh(X,D) gua an ees ha he homo opy heo y o p eshea es is he same as he homo opy
heo y o shea es, because he adjoin pai (−)a:P Sh(X,D)⇄Sh(X,D) : i, whe e iis he
inclusion unc o , induces an equi alence o ca ego ies P Sh(X,D)[W−1]≃Sh(X,D)[W−1].
Al hough an associa ed shea unc o may no exis o D, when (D,E) is compa ible wi h he
si e X he hype cohomology shea may be hough o as a ‘homo opical’ shea i ica ion unc o .
Mo e p ecisely, he adjoin pai (p∗, p∗) is also an adjoin pai
P Sh(X,D)
p∗
//DX
p∗
oo
and he induced iple on P Sh(X,D) allows an analogous de ini ion HX(F) = sG•(F) o a
p eshea F, which enjoys he same p ope ies as in he shea case. In addi ion, T(F) = p∗p∗(F)
is a shea , and so is HX(F).
Co olla y 4.3.6. Le (D,E) be a descen ca ego y compa ible wi h he si e X. Then
P Sh(X,D)[W−1]
HX//Sh(X,D)[W−1]
i
oo
a e in e se equi alences o ca ego ies.
P oo . By hypo hesis, ρF:F −→ HXi(F) is in W, so i is an isomo phism o Sh(X,D)[W−1]
o any shea F. I emains o be shown ha i Fis now a p eshea hen ρF:F −→ iHX(F) is in
W. Since HX(F) is a shea , ρHX(F)∈ W. Bu ρHX(F)is a mo phism be ween CE- ib an shea es
20 BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
and hence belongs o S. By he same p oo as in Theo em 4.3.2, we in e ha HX(ρF)∈ S as
well. Again, his means ha ρFis a local equi alence as equi ed. 
4.3.6. We ha e seen ha a descen s uc u e on (D,E) always induces one on (Sh(X,D),S)
de ined objec wise. We ha e ano he descen s uc u e, hough.
P oposi ion 4.3.7. Assume ha a descen ca ego y (D,E) is compa ible wi h he si e Xand
ha il e ed colimi s commu e wi h ini e p oduc s in D. Then, (Sh(X,D),W)is a descen
ca ego y wi h simple unc o
s′=sHX:∆Sh(X,D)−→ Sh(X,D).
P oo . The commu a ion o ini e p oduc s wi h il e ed colimi s gua an ees ha WQW ⊂ W.
The ac ha s′is a simple unc o o (Sh(X,D),W) may be p o ed using ha HX(W)⊂ S
and ha sis a simple unc o o (Sh(X,D),S). 
I ollows om he esul s in [Rod1] ha pa h and loop unc o s may be cons uc ed o
(Sh(X,D),W) in a na u al way. They gi e ise o well beha ed ibe sequences, sa is ying
he usual p ope ies in Sh(X,D)[W−1]. In pa icula , Sh(X,D)[W−1] is a iangula ed ca e-
go y p o ided ha he loop unc o is an equi alence o ca ego ies.
4.4. De i ed unc o s o shea es.
4.4.1. The second consequence o ou cha ac e iza ion o CE- ib an shea es, he exis ence o
he igh de i ed di ec image unc o , ollows immedia ely (c . [B , h.6 ]).
Co olla y 4.4.1. Le :X −→ Y be a con inuous unc o o G o hendieck si es and (D,E) a
descen ca ego y compa ible wi h he si e X. Then, ∗:Sh(X,D)−→ Sh(Y,D)admi s a igh
de i ed unc o R ∗:Sh(X,D)[W−1]−→ Sh(Y,D)[W−1]gi en by
R ∗(F) = ∗HX(F).
P oo . In iew o Theo em 4.3.2 and P oposi ion 2.2.6, we only need o show ha ∗sends
global equi alences o local equi alences. Bu his is ob ious: i ϕ:F −→ G ∈ S, hen, o
e e y objec V∈ Y, we ha e ∗(ϕ)(V) = ϕ( −1(V)) : F( −1(V)) −→ G( −1(V)) ∈E. So ∗(ϕ)
is also a global equi alence and hence, a o io i, a local one. 
I Uis an objec o X, he same p oo wo ks o he U-sec ions unc o Γ(U, −) : Sh(X,D)−→ D
because, by de ini ion, Γ(U, F) = F(U) sends global equi alences in Sh(X,D) o equi alences
in D. Hence,
Co olla y 4.4.2. Le (D,E) be a descen ca ego y compa ible wi h he si e X. Then Γ(U, −) :
Sh(X,D)−→ D admi s a igh de i ed unc o RΓ(U, −) : Sh(X,D)[W−1]−→ D[E−1]gi en
by
RΓ(U, F) = Γ(U, HX(F)) .
GODEMENT RESOLUTIONS 21
4.4.2. When Xhas a e minal objec X, e.g. in case Xis he si e associa ed wi h a opological
space X,shea cohomology is by de ini ion he igh de i ed unc o o he global sec ions unc o
Γ(X, −) : Sh(X,D)−→ D. So, unde he abo e assump ions, shea cohomology is well de ined
and ag ees wi h Γ(X, HX(F)).
Following [SGA4, 4.3.6.1], i he coe icien ca ego y Dhas limi s, he no ion o global sec ions
unc o Γ(X,−) : Sh(X,D)−→ D gene alizes o a gene al si e X, possibly wi hou a e minal
objec , as:
Γ(X,F) = lim
←− U∈X F(U).
No e ha in his case Γ(X,−) does no necessa ily send a global equi alence o a weak equi -
alence o D. Bu , being (D,E) a descen ca ego y in which a bi a y p oduc s a e E-exac , he
igh de i ed unc o o lim
←− X:DX−→ D exis s, and is gi en by he composi ion o he simple
unc o wi h he cosimplicial eplacemen DX−→ ∆D(see [Rod2]). The esul ing unc o
holim
←− X:Sh(X,D)−→ D sends global equi alences o weak ones; hence, i admi s a igh
de i ed unc o Sh(X,D)[W−1]−→ D[E−1] ha may be seen o ag ee wi h he igh de i ed
unc o o Γ(X,−). Tha is, RΓ(X,−) : Sh(X,D)[W−1]−→ D[E−1] exis s and is gi en by
RΓ(X,F) = holim
←− U∈X HX(F)(U).
4.4.3. Recall ha when he e is a shea i ica ion unc o hen Sh(X,D) is comple e ( esp.
cocomple e) when Dis. A homo opical e sion o his ac is ha when HXis a ‘homo opical’
shea i ica ion unc o ( ha is, when (D,E) is compa ible wi h he si e X) hen (Sh(X,D),W)
is homo opically comple e, and homo opically cocomple e p o ided (D,E) is.
The key poin s o seeing his a e ha he esol en unc o (HX, ρ) is also a esol en unc o o
p eshea es, and ha i may be li ed o diag am ca ego ies: o each small ca ego y I, (HX, ρ)
induces objec wise a esol en unc o on (P Sh(X,D)I=P Sh(X,DI),S,W). This in u n
implies ha he e is an adjunc ion na u al in I
P Sh(X,D)I[S−1]id //P Sh(X,D)I[W−1]≃Sh(X,D)I[W−1]
HX
oo
whe e he igh adjoin HXis ully ai h ul. This na u al adjunc ion hen ans e s homo opy
limi s and colimi s exis ing o (P Sh(X,D),S) = (DX,EX) o Sh(X,D)[W−1]. In pa icula
(Sh(X,D),W) is homo opically comple e and
holim
←−
(Sh(X,D),W)
I= holim
←−
(Sh(X,D),S)
IHX.
5. Examples
In his sec ion we show how he abo e esul s apply o classic and no so classic examples o
ca ego ies o shea es. Mo e conc e ely, we will p o e ha a ini e cohomological dimension
assump ion on he si e Xgua an ees i s compa ibili y wi h he na u al descen s uc u es seen
on ca ego ies o coe icien s Dsuch as complexes, simplicial se s and spec a. Consequen ly,
om he esul s o he p e ious sec ion we conclude ha o such Xand Dwe ha e:

22 BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
•Fo e e y shea F, he na u al a ow ρF:F −→ HX(F) is a ib an model o F. O ,
wha amoun s o he same, (Sh(X,D),S,W) is a CE-ca ego y wi h esol en unc o
(HX, ρ).
•The localized ca ego y Sh(X,D)[W−1] is na u ally equi alen o Sh(X,D) ib[S−1].
•The CE- ib an objec s o Sh(X,D) a e p ecisely hose shea es sa is ying Thomason’s
descen .
•De i ed sec ions RΓ(U, −) and de i ed di ec image unc o R ∗may be compu ed by
p ecomposing wi h HX.
•The hype cohomology shea HXis a ‘homo opical’ shea i ica ion unc o ha gi es an
equi alence Sh(X,D)[W−1]≃P Sh(X,D)[W−1].
5.1. Bounded complexes o shea es.
5.1.1. Conside he descen ca ego y s uc u e on he ca ego y o uni o mly bounded cochain
complexes C≥b(A) desc ibed in example 2.1.4.
In his case he simple unc o is s=To Q=To ⊕:∆C≥b(A)−→ C≥b(A) by he boundedness
assump ion. The ca ego y o shea es o uni o mly bounded cochain complexes Sh(X,C≥b(A))
is a descen ca ego y whe e he weak equi alences a e he global equi alences and he simple
unc o is he o al-sum unc o applied objec wise: (To F)(U) = To (F(U)).
I ollows ha scommu es in his case wi h all colimi s, since i is de ined deg ee-wise h ough a
ini e di ec sum. Hence, scommu es i ially wi h s alks. The e o e, we deduce om Theo em
4.3.2
Theo em 5.1.1. Assume ha Ais an abelian ca ego y sa is ying (AB4)∗and (AB5)( ha is,
a bi a y p oduc s and il e ed colimi s exis and a e exac ). Then, he descen ca ego y C≥b(A)
is compa ible wi h any si e X. In pa icula , p ope ies 5hold o (Sh(X,C≥b(A)),S,W).
In his case a local equi alence ∈ W is jus a quasi-isomo phism o Sh(X,C≥b(A)) =
C≥b(Sh(X,A)). On he o he hand, a global equi alence ∈ S is a mo phism :F −→ G o
complexes o shea es such ha (U) is a quasi-isomo phism o C≥b(A) o each objec U∈ X.
Consequen ly, a unc o F : Sh(X,C≥b(A)) −→ C sending global equi alences o isomo phisms
admi s a igh de i ed unc o RF : D≥b(Sh(X,A)) = Sh(X,C≥b(A))[W−1]−→ C gi en by
RF(F) = F(HX(F)). No e ha his de i abili y c i e ion does no assume he exis ence o
enough injec i es in A. Pa icula ly, o he case A=R−modules, we eco e he classic
cons uc ion o abelian shea hype cohomology and de i ed di ec image o shea es cons uc ed
h ough canonical Godemen esolu ions by lasque shea es.
5.2. Unbounded complexes o shea es.
5.2.1. When he boundedness assump ion on complexes o shea es is d opped, Theo em 5.1.1
is no longe ue o a gene al si e X, e en in he case A=R−modules.
Conside he ca ego y C∗(R) o unbounded cochain complexes o R-modules wi h he descen
s uc u e o example 2.1.5. In his case, he simple unc o s=To Q:∆C∗(R)−→ C∗(R)
GODEMENT RESOLUTIONS 23
is an in ini e p oduc deg ee-wise, and consequen ly i does no commu e (e en weakly) wi h
il e ed colimi s. This in u n means ha he hype cohomology shea HX(F) associa ed wi h
an unbounded complex Fo shea es o R-modules does no necessa ily p oduce a CE- ib an
model o Fin (Sh(X,C∗(R)),S,W), o a gene al si e X.
Example 5.2.1. To illus a e his ac , conside a amily {F−k}ko abelian shea es o which
(Qk>0Hk(−,F−k))x6= 0 ( o ins ance hose desc ibed in [We, A.5] o [MV, 1.30]). Then
cons uc he complex o shea es Fwi h ze o di e en ial ha is 0 in posi i e deg ees and
equal o F−kin nega i e deg ees. I is no ha d o e i y ha ρF:F−→ HX(F) is no a
quasi-isomo phism in his case, so i does no p o ide a CE- ib an model o F.
We ema k howe e ha (Sh(X,C∗(R)),S,W) is s ill a Ca an-Eilenbe g ca ego y o any si e
X: K-injec i e complexes o shea es a e easily seen o be CE- ib an , and by [Sp] each complex
o shea es is locally equi alen o some K-injec i e one (see also [We], appendix). Hence he
CE- ib an model o an unbounded complex Fo shea es does no ag ee in gene al wi h i s
hype cohomology shea HX(F), unless some ex a assump ion is imposed on si e X.
5.2.2. We a e going o show ha ini e cohomological dimension is a su icien condi ion o
he si e Xin o de ha he hype cohomology shea HXp oduces a esol en unc o o he
Ca an-Eilenbe g ca ego y (Sh(X,C∗(R)),S,W).
Recall ha a sys em o neighbou hoods o a poin x∈ X is, by de ini ion, a ull co inal
subca ego y o he ca ego y o neighbou hoods o xin X([SGA4] 6.8.2).
De ini ion 5.2.2 ([GS]).A si e Xis said o ha e ini e cohomological dimension i o any
poin x∈ X he e exis s d≥0 and a sys em Λ o neighbou hoods o xsuch ha o any shea
o abelian g oups F ∈ Sh(X,Ab) and any neighbou hood U∈Λ i holds ha Hn(U;F) = 0
whene e n > d.
Fo ins ance, he ollowing si es ha e ini e cohomological dimension:
(1) The small Za iski si e o a noe he ian opological space o ini e K ull dimension; e.g.,
he Za iski si e o a noe he ian scheme o ini e K ull dimension. This ollows om
G o hendieck’s anishing Theo em ([Ha ] III, Theo em 2.7).
(2) The big Za iski si e o a noe he ian scheme Xo ini e K ull dimension consis ing o all
schemes o ini e ype o e X, o all noe he ian schemes o bounded K ull dimension
([GS], page 6).
(3) The small si e o a opological mani old o ini e dimension. This ollows om he
anishing Theo em o [KS].
Theo em 5.2.3. The descen ca ego y C∗(R)is compa ible wi h any ini e cohomological di-
mension si e X. In his case, p ope ies 5hold o (Sh(X,C∗(R)),S,W).
The p oo is based on a spec al sequence a gumen , he Colimi Lemma, o which we need
some p elimina ies. The same spec al sequence a gumen will also be used in he examples o
simplicial se s and spec a.
24 BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
5.2.3. Le Cbe a ca ego y wi h il e ed colimi s and Ia il e ed indexing se . Fo us “spec al
sequence” means a unc o ial igh hal -plane cohomological spec al sequence E∗o abelian
g oups, commu ing wi h il e ed colimi s: E∗(colim
−→ iXi) = colim
−→ iE∗(Xi).
Fo an objec X∈ C, we say ha he spec al sequence E∗(X) is bounded on he igh i
he e exis s dsuch ha Ep∗
2(X) = 0 o p > d. No e ha , o condi ionally con e gen spec al
sequences, his implies s ong con e gence ([Boa], Theo em 7.4). Gi en a il e ed sys em {Xi}i∈I
o objec s o C, we say ha he amily o spec al sequences {E∗(X)}i∈Iis uni o mly bounded
on he igh i he e is a ixed d ha wo ks o all i∈I.
P oposi ion 5.2.4 (Colimi Lemma).Assume as gi en he ollowing da a:
(1) An objec X∈ C and a il e ed sys em X•={Xi}i∈Io objec s Xi∈ C.
(2) A cone { i:Xi−→ X}i∈I om he base X• o he e ex Xand, hence, an induced map
: colim
−→ iXi−→ X.
Mo eo e , assume also ha :
(1) The spec al sequences {E∗Xi}i∈Iand E∗Xcon e ge condi ionally o {Hi}i∈Iand H,
espec i ely.
(2) The spec al sequences {E∗Xi}i∈Ia e uni o mly bounded on he igh .
(3) The map E ( ) : colim
−→ iE (Xi)−→ E (X)is an isomo phism o some ≥0.
Then he map H( ) : colim
−→ iHi−→ His an isomo phism oo.
P oo . See [Mi ], P oposi ion 3.3. 
P oo o Theo em 5.2.3. The i s il a ion o a double complex K∈C∗∗(R),
Fp(To QK)n=Qs≥pKs,n−sgi es us a condi ionally con e gen spec al sequence
Epq
2(To QK) = Hp
hHq
(K) =⇒Hp+q(To QK), p ≥0.
By Theo em 4.3.2, o p o e ha o any shea F ∈ Sh(X,C∗(R)) i holds ha ρF:F −→
HX(F) is a CE- ib an model we may equi alen ly show ha he canonical mo phism
θF(x) : colim
−→ (U,u)∈Nbh(x)To Q(G∗F)(U)−→ To Qcolim
−→ (U,u)∈Nbh(x)(G∗F)
is a quis o C∗(R) o any shea Fand any poin xin he se o enough poin s X. These
colimi s may be compu ed using he neighbou hoods (U, u) in he sys em o neighbou hoods Λ
ha exis s by assump ion.
The e o e, we ha e an objec To Qx∗(G∗F)∈C∗(R), a il e ed sys em nTo Q(G∗F)(U)o(U,u),
whe e (U, u) uns o e all neighbou hoods o xin Λ and he induced map θF(x).
Le us e i y he hypo heses o he Colimi Lemma: he spec al sequences
Epq
2(U) = Hp
Hq
h((G∗F)(U)) =⇒Hp+q(To Q(G∗F)(U)) , p ≥0
GODEMENT RESOLUTIONS 25
and
Epq
2(x) = Hp
Hq
h((G∗F)x) =⇒Hp+q(To Q((G∗F)x)) , p ≥0
con e ge condi ionally.
To compu e Epq
2(U) we use ha T:Sh(X,C∗(R)) −→ Sh(X,C∗(R)) commu es wi h cohomol-
ogy in Sh(X,C∗(R)). A he p eshea le el, clea ly H∗(T(F)) = T(H∗(F)) o any p eshea F,
because cohomology in C∗(R) commu es wi h p oduc s and il e ed colimi s. Since he s alks o
a p eshea Ga e isomo phic o he ones o i s associa ed shea Ga, hen T(G) = T(Ga). Hence,
i Fis a shea
T(H∗F) = T((H∗F)a) = T(H∗F) = H∗(TF).
In pa icula H∗(TF) = T(H∗F) is a shea , so i ag ees wi h i s associa ed shea . The e o e
H∗(TF) = H∗(TF) = T(H∗F), and H∗(G•F) = G•(H∗F). We hen ha e, o all p > d,
Epq
2(U) = Hp
Hq
h(G∗F)(U) = Hp(Γ(U, HqG∗F)) = Hp(Γ(U, G∗HqF)) = Hp(U, HqF) = 0
because o he ini e cohomological dimension assump ion. Finally, al eady o = 0, we ha e
an isomo phism
colim
−→ Epq
0(U) = colim
−→ GpFq(U) = (GpFq)x=Epq
0(x).
Hence he Colimi Lemma ells us ha
Hn(To Q(G∗F))x−→ HnTo Q((G∗F)x)
is an isomo phism o all n.
5.3. Shea es o ib an simplicial se s.
5.3.1. Le D=sS wi h he descen s uc u e o 2.1.6. As in he case o unbounded complexes,
he simple unc o may no commu e weakly wi h s alks. Again, o his o hold we mus ei he
es ic o simplicial se s wi h anishing highe homo opy g oups, o impose some ini eness
assump ion on he si e X. He e we s udy he second al e na i e, showing ha ρF:F −→
HX(F) is a CE- ib an model o each Fin (Sh(X, sS ),W,S) i and only i Xis a si e o ini e
ype in he sense o [MV].
5.3.2. By a heo em o Joyal, he ca ego y Sh(X, sS) possesses a simplicial model ca ego y
s uc u e in which all objec s a e co ib an and he weak equi alences a e he local equi alences
[Ja]. The ib an objec s in his model s uc u e a e hen de ined h ough a li ing p ope y,
and hey a e objec wise ib an simplicial se s. The e o e, he e is a ib an eplacemen unc o
Ex ha akes a simplicial shea o a ib an one, in pa icula Ex(F)∈Sh(X, sS ).
Gi en a simplicial shea F ∈ Sh(X, sS) and n≥0, le e
P(n)Fbe he simplicial shea associa ed
o he p eshea U7→ P(n)F(U) = Im{F(U)−→ cosknF(U)}. I is equipped wi h na u al maps
F −→ e
P(n)Fand e
P(n+1)F −→ e
P(n)F.
I he s alks o Fa e ib an simplicial se s, he owe {x∗e
P(n)F=P(n)x∗F} is p ecisely he
Moo e-Pos niko owe o x∗F. In his case he na u al map x∗F ≃ lim
←− n≥0x∗e
P(n)F −→
holim
←− n≥0x∗e
P(n)Fis a weak equi alence.
32 BEATRIZ RODR´
IGUEZ GONZ´
ALEZ AND AGUST´
I ROIG
Co olla y 6.2.2. Unde he same hypo heses o he p e ious p oposi ion, o any objec U∈ X
and any shea F ∈ Sh(X,D), we ha e
RΓ(U, φF) = φRΓ(U, F).
Example 6.2.3. As we ha e seen in he p e ious sec ion o examples, he descen ca ego ies
(C≥b(A),E) and (FC≥b(A),E ) o ( il e ed) complexes o Examples 2.1.4, 2.1.7 a e compa ible
wi h any si e, p o ided Ais (AB4)∗and (AB5). Then he abo e esul applies o bo h he
o ge ul unc o U: (FC≥b(A),E0)−→ (C≥b(A),E) and he decalage il a ion unc o Dec :
(FC≥b(A),E +1)→(FC≥b(A),E ). This in pa icula eco e s he classic esul ha il e ed
shea hype cohomology and il e ed highe di ec images ag ee wi h he usual abelian ones when
we o ge he il a ions.
Consequen ly, we can now ex end 2.1.3 o ca ego ies o shea es, ob aining a ans e lemma o
CE-s uc u es be ween hem.
P oposi ion 6.2.4. Assume ha Dis closed unde p oduc s and il e ed colimi s, and ha
(D′,E′)is a descen ca ego y compa ible wi h he si e X. I ψ:D −→ D′sa is ies he hypo heses
o he ans e lemma 2.1.3 and (1) and (2) o he p e ious p oposi ion, hen
(I) (D,E = ψ−1E′)is a descen ca ego y compa ible wi h X.
(II) ψ:Sh(X,D)−→ Sh(X,D′)is a mo phism o CE-ca ego ies.
P oo . By he ans e lemma 2.1.3 and he p e ious p oposi ion, he only s a emen emaining
o be p o ed is he ac ha he esul ing descen ca ego y (D,E = ψ−1E′) is compa ible wi h
X. Fi s no e ha by de ini ion (D,E) sa is ies hypo heses (4.1.1). Then, using Theo em 4.3.2,
i su ices o show ha o any F ∈ Sh(X,D), ρHXFis in S. By de ini ion o E, his holds i
and only i ψ(ρHXF) is in S′. Bu a guing as in he p e ious p oo , his happens i and only i
ρHXψ(F)is in S′, which holds because (D′,E′) is compa ible wi h X.
Examples 6.2.5. In ac , he unc o s in he p e ious example also sa is y hese s onge
hypo heses and hen may be used o ans e compa ibili y wi h he si e.
To close he pape , le us b ie ly desc ibe a classical si ua ion also co e ed by his ans e
lemma. I ’s a well-known ac ha , i (X, OX) is a inged space, hen he de i ed unc o
RΓ(X, F) na u ally inhe i s a module s uc u e o any shea Fo OX-modules. Bu , i we
o ge his module s uc u e h ough he o ge ul unc o ψ:Mod −→ Ab, his de i ed
unc o ag ees wi h he usual cohomology as an abelian shea : RΓ(X, ψF) = ψRΓ(X, F). In a
o hcoming a icle, we will show ha an analogous esul holds o shea es o ope ad algeb as.
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a ica Aplicada I, Uni e si a Poli `
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