LOCAL AND STABLE HOMOLOGICAL ALGEBRA IN GROTHENDIECK ABELIAN CATEGORIES ...

42 Pages·2009·0.39 MB·English

Homology, Homotopy and Applications, vol.11(1), 2009, pp.219–260
LOCAL AND STABLE HOMOLOGICAL ALGEBRA
IN GROTHENDIECK ABELIAN CATEGORIES
DENIS-CHARLES CISINSKI and FRE´DE´RIC DE´GLISE
(communicated by Brooke Shipley)
Abstract
Wedefinemodelcategorystructuresonthecategoryofchain
complexes over a Grothendieck abelian category depending on
the choice of a generating family, and we study their behaviour
with respect to tensor products and stabilization. This gives
convenient tools to construct and understand triangulated cat-
egories of motives.
Contents
1 Introduction 219
2 Model category structures on complexes 220
3 Derived tensor product 229
4 Localization of derived categories 234
5 Presentations of the derived category 240
6 How to get compact generators 243
7 Stabilization and symmetric spectra 245
1. Introduction
This paper is an attempt to apply the general techniques of abstract homotopical
algebra to describe derived categories of Grothendieck categories. Our motivation is
to describe model categories of complexes of sheaves “as locally as possible”, so that
we can easily get total left derived functors (like the tensor product). The formalism
we propose is made to stay close to something that looks like descent theory. Even
though the results proved here are just applications of a general theory (Bousfield
ThisworkispartiallysupportedbyANRgrantno.ANR-07-BLAN-0142.
ReceivedJanuary14,2008,revisedNovember21,2008;publishedonJune12,2009.
2000MathematicsSubjectClassification:18G55,18G35,18E15,14F42.
Keywordsandphrases:stablemodelcategories,Grothendieckabeliancategories,derivedfunctors,
motiviccomplexes.
Thisarticleisavailableathttp://intlpress.com/HHA/v11/n1/a11
Copyright(cid:176)c 2009,InternationalPress.Permissiontocopyforprivateusegranted.
220 DENIS-CHARLESCISINSKIandFRE´DE´RICDE´GLISE
localization, symmetric spectra), our contribution describes these model structures
withanemphasisonthespecificpropertiesinthesettingofGrothendieckcategories.
Asanexampleofpropertiesthatarenotknownforgeneralmodelcategories,wegive
explicit descriptions of fibrations after Bousfield localizations and prove that passing
to symmetric spectra can preserve the monoid axiom of Schwede and Shipley. We
also prove that any derived category of a Grothendieck category can be obtained as
a localization of the derived category of modules over an additive category. This is
usedtogiveasufficientconditionforthederivedcategoryofaGrothendieckcategory
to be compactly generated.
As an illustration, we give the construction of the categories of motivic complexes
and motivic spectra over a regular base scheme together with their natural struc-
tures: derived inverse and direct image functors, derived tensor product, and twisted
exceptionaldirectimagefunctors forsmoothmorphisms.Inthisconstruction,smooth
alwaysmeansabusivelysmoothoffinitetype.Thisconstructionwillbeamplifiedand
generalizedintheforthcomingpaper[6].Theresultsofthispaperarealsowidelyused
tounderstandthenotionofmixedWeilcohomology,whichisintroducedandstudied
in [7].
We wish to thank Marc Levine and the anonymous referee for useful comments
and suggestions.
2. Model category structures on complexes
2.1.
LetAbeanabeliancategory.IfEisanobjectofAandnaninteger,thenwedefine
SnE (resp. DnE) as the acyclic complex concentrated in degree n with (SnE)n =E
(resp.indegreesnandn+1with(DnE)n =(DnE)n+1 =E andasonlyanontrivial
differential the identity of E). We then have canonical inclusions Sn+1E −→DnE
induced by the identity of E. The complex DnE represents the functor
Comp(A)op −→Ab, C 7−→Hom (E,Cn),
A
and the complex SnE represents the functor
Comp(A)op −→Ab, C 7−→Hom (E,ZnC),
A
whereZnC =ker(d: Cn −→Cn+1)denotestheobjectofn-cocyclesofthecomplexC.
Weshallreferourselvesto[12]forgeneralresultsanddefinitionsonmodelcategories,
andto[11]forthedefinitionofcellularmodelcategories1 andforresultsonBousfield
localizations.
Theorem 2.1 (Beke). Let A be a Grothendieck abelian category. The category of
complexes Comp(A) is a proper cellular model category with quasi-isomorphisms as
weak equivalences and the monomorphisms as cofibrations.
1This notion is useful to ensure the existence of left Bousfield localizations. For such a purpose,
we can work instead with Jeff Smith’s notion of combinatorial model category [3]. The notion of
combinatorial model category might be seen as slightly simpler than the notion of cellular model
category, but it is very easy to check that the model structures considered in these notes are both
cellularandcombinatorial,soweinvitethereadertoconsiderhis/herfavouritenotion.
LOCALANDSTABLEHOMOLOGICALALGEBRA 221
Proof. See[3,Proposition3.13]fortheconstructionofthismodelcategorystructure.
The fact that this model category is cellular comes from the fact that any monomor-
phism in a Grothendieck category is effective, and that any object in a Grothendieck
category is compact (in fact accessible according to [1]). Remark that fibrations are,
in particular, epimorphisms. This is proved as follows: For any object E in A and
any integer n, the complex DnE is acyclic. Hence any fibration has the right lifting
property with respect to the inclusions 0−→DnE. This lifting property for a map
p: C −→D is equivalent to saying that for any object E of A and any integer n, the
map Hom (E,Cn)−→Hom (E,Dn) is surjective. This implies, in particular, that
A A
p is degreewise split, hence that p is an epimorphism. Properness now comes from
classical homological algebra as follows. Let
i (cid:47)(cid:47)
A B
p q
(cid:178)(cid:178) (cid:178)(cid:178)
(cid:47)(cid:47)
C D
j
beapushout(resp.apullback)squareofcomplexesinA.Supposethati(resp.q)isa
cofibration(resp.fibration).Asi(resp.q)isamonomorphism(resp.anepimorphism),
the same is true for j (resp. for p). Moreover, it is obvious that the cokernels of i
andj (resp.thekernelsofpandq)arecanonicallyisomorphic,whichimpliesthatwe
have a long exact sequence of shape:
Hn(A)−→Hn(B)⊕Hn(C)−→Hn(D)−→Hn+1(A).
We deduce then immediately that if p (resp. j) is a quasi-isomorphism, then so is q
(resp. i), which is what we wanted to prove.
2.2.
The above model category structure will be called the injective model structure.
The fibrations of this model structure will be called the injective fibrations, and the
fibrant objects the injectively fibrant objects.
The injective model structure gives a powerful tool to construct right derived
functors (well defined on unbounded complexes). To define left derived functors on
unbounded complexes, we have to consider other model structures depending on the
choice of a set of generators of A. We shall restrict ourselves to a special situation
that will occur in the cases we are interested in. This can be well formulated using
the intuition of descent theory.
Definition 2.2. Let A be a Grothendieck category.
If G is an essentially small set of objects of A, then we define a morphism in
Comp(A) to be a G-cofibration if it is contained in the smallest class of maps in
Comp(A) closed under pushouts, transfinite compositions and retracts, generated by
the inclusions Sn+1E −→DnE, for any integer n and any E in G. A chain complex
C isG-cofibrant ifthemap0−→C isaG-cofibration.Forexample,ifC isabounded
above complex such that for any integer n, Cn is a direct sum of elements of G, then
C is G-cofibrant.
222 DENIS-CHARLESCISINSKIandFRE´DE´RICDE´GLISE
A chain complex C in A is G-local, or simply local if no confusion can arise from
the context, if for any E in G and integer n, the canonical map
Hom (E[n],C)−→Hom (E[n],C)
K(A) D(A)
is an isomorphism.
IfHisasmallfamilyofcomplexesinA,thenanobjectC ofComp(A)isH-flasque
(orsimplyflasque)ifforanyintegernandanyH inH,thegroupHom (H,C[n])
K(A)
vanishes.
A descent structure on a Grothendieck category A is a couple (G,H), where G
is an essentially small set of generators of A, and H is an essentially small set of
G-cofibrant acyclic complexes such that any H-flasque complex is G-local.
Example 2.3. LetS beanessentiallysmallGrothendiecksiteandRasheafofringson
S.ThenthecategoryR-ModofsheavesofR-modulesisaGrothendieckcategory.For
an object X of S, denote by R(X) the free sheaf of R-modules generated by X. The
family G of the R(X)’s form a generating set for R-Mod. If X is a hypercover of X,
S
then the simplicial R-module R(X) can be seen as a complex of R-modules with the
differentials given by the alternating sum of the face operators. For a hypercovering
X of an object X, denote by R(cid:101)(X) the cone of the structural map R(X)−→R(X).
The complexes R(X), R(X) and R(cid:101)(X) are then G -cofibrant. Let H be the family
S S
of all the complexes of the form R(cid:101)(X) for any hypercover X of an object X of S.
Then the couple (G ,H ) is a descent structure on the category R-Mod. One way to
S S
prove this comes from Verdier’s computation of hypercohomology; see [1, Expos´e V]
and [4]. If X is an object of S and C is a complex of R-modules, then we have the
formula
lim Hom (R(X),C)=Hom (R(X),C),
−→ K(R-Mod) D(R-Mod)
X∈HX
where H denotes the filtering category of hypercoverings of X up to simplicial
X
homotopy (and Hom (R(X),C) is nothing else but the hypercohomology of
D(R-Mod)
X withcoefficientsinC).WededucethatifC isH -flasque,aswehavedistinguished
S
triangles in K(R-Mod) of shape
R(X)−→R(X)−→R(cid:101)(X)−→R(X)[1]
for any X in H , then we have the isomorphisms
X
Hom (R(X),C)=Hom (R(X),C).
K(R-Mod) K(R-Mod)
As H is filtering, we deduce that the canonical map
X
Hom (R(X),C)−→ lim Hom (R(X),C)
K(R-Mod) −→ K(R-Mod)
X∈HX
is an isomorphism. Hence C is G -local.
S
This example generates new ones: let A be a Grothendieck abelian category en-
dowed with a functor
U: A−→R-Mod
which preserves colimits and limits, and which is conservative. Then the functor U
LOCALANDSTABLEHOMOLOGICALALGEBRA 223
has a left adjoint
F: R-Mod−→A,
and we can see that (F(G ),F(H )) is a descent structure on A.
S S
A natural example of this shape is the following: Let S =X be the small
Zar
Zariski site of a quasi-separated and quasi-compact scheme, and let R=O be
X
the structural sheaf of rings on X. We can consider the Grothendieck abelian cate-
gory A=QCoh(X) of quasi-coherent O -modules and see that the forgetful functor
X
U: QCoh(X)−→O -Mod preserves colimits and limits, and is conservative.
X
Another example which fits in this kind of presentation comes from the theory of
algebraic cycles.
Example 2.4. Let S be a noetherian regular scheme, and Ntr be the category of
S
Nisnevich sheaves with transfers over S defined in [8, 4.2.3]. Let G be the category
S
of sheaves with transfers of type L (X) for any smooth S-scheme X. Then it follows
S
from Proposition 4.2.8 of loc. cit. that A is a Grothendieck abelian category and
S
that G is a set of generators. We denote by H the set of complexes obtained as the
S S
conesofthemapsL (X)−→L (X),whereX isanysmoothS-schemeandX isany
S S
Nisnevich hypercovering of X. From the example above and from Propositions 4.2.5
and 4.2.9 of loc. cit., it follows that (G ,H ) is a descent structure on Ntr.
S S S
Theorem 2.5. Let A be a Grothendieck category endowed with a descent structure
(G,H). Then the category Comp(A) is a proper cellular model category with quasi-
isomorphisms as weak equivalences, and G-cofibrations as cofibrations. Furthermore,
for a complex C in A, the following conditions are equivalent:
(i) C is fibrant;
(ii) C is H-flasque;
(iii) C is G-local.
This model structure on the category Comp(A) will be called the G-model structure.
Remark 2.6. It is possible to give a nice description of the fibrations for this model
category structure; see Corollary 5.5.
2.3.
The proof of this theorem will be completed by the results below. We first have to
fix some notation.
WedefineI tobethesetofinclusionsoftheformSn+1E −→DnE foranyinteger
n and any E in G.
If C is a complex, then Cyl(C) is the complex defined by
Cyl(C)n =Cn⊕Cn+1⊕Cn
with the differential
d(x,y,z)=(dx−y,−dy,y+dz).
Theprojectionσ: Cyl(C)−→C,definedbytheformulaσ(x,y,z)=x+z,isaquasi-
isomorphism. We also have a canonical inclusion (i ,i ): C⊕C −→Cyl(C) defined
0 1
byi (x)=(x,0,0)andi (z)=(0,0,z),sothatthecompositeσ(i ,i )isequaltothe
0 1 0 1
224 DENIS-CHARLESCISINSKIandFRE´DE´RICDE´GLISE
codiagonal from C⊕C to C. It is an easy exercise to check that to define a chain
homotopy between two maps of complexes u and u from C to C0 is equivalent to
0 1
defining a map h from Cyl(C) to C0 such that hi =u for e=0,1.
e e
We define at last J =J0∪J00, where J0 is the set of maps 0−→DnE for any
integer n, any E in G, and J00 is the set of maps H ⊕H[n]−→Cyl(H)[n] for any H
in H and any integer n.
To prove Theorem 2.5, we are going to verify the conditions of [12, Theorem
2.1.19] for I and J. We adopt the notation of [12]: I-cof (resp. J-cof) is the class
of G-cofibrations (resp. the smallest class of maps closed under pushouts, transfinite
compositions and retracts which contains J), and I-inj (resp. J-inj) is the class of
maps which have the right lifting property with respect to I (resp. J). We will prove
that I-cof, J-cof, I-inj and J-inj are respectively the class of cofibrations, the class
of trivial cofibrations, the class of trivial fibrations and the class of fibrations for this
model structure.
Lemma 2.7. For any E in G and any integer n, the complexes SnE and DnE are
G-cofibrant. For any G-cofibrant complex C, the inclusion C⊕C −→Cyl(C) is a G-
cofibration. Furthermore, if C is acyclic, then this map is also a quasi-isomorphism.
Proof. The pushout square
SnE (cid:47)(cid:47)0
(cid:178)(cid:178) (cid:178)(cid:178)
Dn−1E (cid:47)(cid:47)Sn+1E
implies the first assertion.
Remember that the cone Cone(p) of a morphism of complexes p: X −→Y is
definedbyCone(p)n =Yn⊕Xn+1,withthedifferentialsgivenbytheformulad(y,x)
=(dy+px,−dx). We then have an obvious short exact sequence
0 (cid:47)(cid:47)Y u (cid:47)(cid:47)Cone(p) v (cid:47)(cid:47)X[1] (cid:47)(cid:47)0,
where u(y)=(y,0), and v(y,x)=x.
Consider an element E of G and an integer n. One checks Cone(1 )=DnE
Sn+1E
and the inclusion of Sn+1E in Cone(1 ) is the canonical inclusion of Sn+1E in
Sn+1E
DnE. The complex Cone(1 ) is the split acyclic complex
DnE
0−→E −→E⊕E −→E −→0,
where E⊕E is placed in degree n+1. One deduces that the canonical map
DnEq Cone(1 )−→Cone(1 )
Sn+1E Sn+1E DnE
is a G-cofibration: it corresponds to the obvious inclusion of the complex
0−→0−→E⊕E −→E −→0
inCone(1 ).Inparticular,themapCone(1 )−→Cone(1 )isaG-cofibra-
DnE Sn+1E DnE
tion. Let C be the class of maps A−→B such that the map
Bq Cone(1 )−→Cone(1 )
A A B
LOCALANDSTABLEHOMOLOGICALALGEBRA 225
induced by the commutative square
A (cid:47)(cid:47)Cone(1A)
(cid:178)(cid:178) (cid:178)(cid:178)
B (cid:47)(cid:47)Cone(1B)
is a G-cofibration. This class of maps is stable by pushouts, transfinite compositions
and retracts (the exercise is left to the reader). As C contains I, it has to contain
all of I-cof. Hence, if C is G-cofibrant, as the map 0−→C is in C, then the map
C −→Cone(1 ) is a G-cofibration. We deduce, for any G-cofibrant complex C, that
C
thecanonicalinclusionC −→Cone(1 )isaG-cofibration.Indeed,wehaveapushout
C
square of the following form:
C (cid:47)(cid:47)C⊕C
(cid:178)(cid:178) (cid:178)(cid:178)
Cone(1 ) (cid:47)(cid:47)Cyl(C)
C
and the G-cofibrations are stable by pushout.
If moreover C is acyclic, then so is C⊕C. As the map Cyl(C)−→C is always a
quasi-isomorphism, the complex Cyl(C) is acyclic as well. Hence the inclusion C⊕
C −→Cyl(C) has to be a quasi-isomorphism.
Lemma 2.8. Any element of J-cof is both a G-cofibration and a quasi-isomorphism.
Proof. AnyG-cofibrationisamonomorphismandtheclassofmapswhicharemono-
morphisms and quasi-isomorphisms is stable by pushouts, transfinite compositions
and retracts. Hence the class of G-cofibrations, which are quasi-isomorphisms is also
stable by any of these operations. It is then sufficient to check that any map in J is
a G-cofibration and a quasi-isomorphism. This follows from Lemma 2.7.
Lemma 2.9. Let C be a complex in A. Then the map C −→0 is in J-inj if and only
if C is H-flasque.
Proof. Let H be in H, and u a map from H to C[n]. The data of a chain homotopy
betweenuandthezeromapisequivalenttothedataofaliftingkinthecommutative
square below.
H ⊕H (u,0) (cid:47)(cid:47)C[n]
(cid:58)(cid:58)
k
(cid:178)(cid:178) (cid:178)(cid:178)
Cyl(H) (cid:47)(cid:47)0.
As the right lifting property of C −→0 with respect to maps of shape 0−→A is
always verified, this proves the assertion.
Lemma 2.10. A H-flasque complex C is acyclic if and only if the map C −→0 has
the right lifting property with respect to I.
226 DENIS-CHARLESCISINSKIandFRE´DE´RICDE´GLISE
Proof. For any complex C in A, as the cohomology group Hn(Hom (E,C)) is the
A
cokernel of the map
Hom (Dn−1E,C)−→Hom (SnE,C),
Comp(A) Comp(A)
itisclearthatC −→0isinI-inj ifandonlyifthecomplexesHom (E,C)areacyclic
A
for any E in G. If C is H-flasque, then for any E in G and integer n, we have the
canonical isomorphisms
Hn(Hom (E,C))=Hom (E,C[n])=Hom (E,C[n]).
A K(A) D(A)
To finish the proof, it is then sufficient to say that a complex C in A is acyclic if and
only if for any E in G and integer n, the group Hom (E,C[n]) vanishes.
D(A)
Lemma 2.11. A map in J-inj is a quasi-isomorphism if and only if it is in I-inj.
Proof. Let p: X −→Y be map in J-inj. We can see that it is an epimorphism: for
any element E of G and any integer n, the map
Hom (E,Xn)=Hom (DnE,X)−→Hom (DnE,Y)=Hom (E,Yn)
A Comp(A) Comp(A) A
issurjective.AsG isasetofgeneratorsofA,thisimpliesthatpisanepimorphism.In
particular,kerpisacyclicifandonlyifpisaquasi-isomorphism.IfpisinI-inj,then
themapkerp−→0isalsoinI-inj.ButJ ⊂I-cof (see2.8),whichby,byLemma2.9,
implies that kerp is H-flasque. Hence by Lemma 2.10, the complex kerp is acyclic,
and p is a quasi-isomorphism.
Consider now a quasi-isomorphism p in J-inj. As p is then an epimorphism, kerp
is acyclic, and as right lifting properties are stable by pullback, the map kerp−→0
is in J-inj. One deduces from 2.9 and 2.10 that the map kerp−→0 is in I-inj. Let
Sn+1E x (cid:47)(cid:47)X
p
(cid:178)(cid:178) (cid:178)(cid:178)
DnE (cid:47)(cid:47)Y
y
be a commutative square in Comp(A) where E is in G, and n is an integer. It cor-
responds to the maps x: E −→Xn+1 and y: E −→Yn such that px=dy. To show
that the square above admits a lift is equivalent to showing that there is a map
ξ: E −→Xn such that dξ =x and pξ =y. As p has the right lifting property with
respecttoJ,Hom (E,pn)issurjective,andsothereexistsamapx0: E −→Xn such
A
that px0 =y. We know that pdx0 =dpx0 =dy =px, so that x−dx0 factors through
kerp. As the map kerp−→0 has the right lifting property with respect to I, this
implies that there is a map x00: E −→Xn such that px00 =0 and dx00 =x−dx0. We
can now set ξ =x0+x00. We have that pξ =px0+px00 =y and dξ =dx0+dx00 =x.
Hence the desired lift.
Proof of Theorem 2.5. One deduces from 2.8 and 2.11 that all the conditions of [12,
Theorem 2.1.19] are satisfied. In particular, we obtain that we have defined a cofi-
brantlygeneratedmodelcategorywithquasi-isomorphismsasweakequivalencesand
withI (resp.J)asasetofgeneratorsforthecofibrations(resp.forthetrivialcofibra-
tions).Toseethatthisstructureiscellular,itissufficienttosaythatanyG-cofibration
LOCALANDSTABLEHOMOLOGICALALGEBRA 227
isamonomorphismandthatanymonomorphisminaGrothendieckcategoryiseffec-
tive.Propernesscomesfromthefactthatcofibrations(resp.fibrations)aremonomor-
phisms(resp.epimorphisms)andstandardresultsofhomologicalalgebra(seetheend
of the proof of 2.1).
It remains to prove the equivalence of conditions (i)–(iii). Using the above defini-
tionsand2.9,itissufficienttoshowthat(iii)implies(ii).LetC beaG-localcomplex.
One can prove that the full subcategory of K(A) made of G-cofibrant complexes is
containedinthelocalizingsubcategoryofK(A)generatedbyG (inthesenseof[19]).
Then for any G-cofibrant complex A, the map below is an isomorphism.
Hom (A,C)−→Hom (A,C)
K(A) D(A)
In particular, for any H in H and integer n, we have Hom (H,C[n])=0. Hence
K(A)
C is H-flasque.
Remark 2.12. ThemodelstructureofTheorem2.5dependsonlyonG andnotonthe
wholedescentstructure(G,H).ThedataHisonlyatooltogetanicecharacterization
of the fibrant objects in terms of flasque properties (as we will see below, that will
be useful to construct total left derivedfunctors). One can provedirectly by abstract
nonsense that for any Grothendieck category A with a choice of an essentially small
set of generators G, there is a Quillen model category structure on Comp(A) with
quasi-isomorphismsasweakequivalencesandG-cofibrationsascofibrations.Toprove
this,weremarkthattheclassofquasi-isomorphismsisaccessible(seetheproofof[3,
Proposition 3.13]), that any map with the right lifting property with respect to G-
cofibrationsisaquasi-isomorphism(byvirtueof[13,Proposition1.4]),andapplyJeff
Smith’s theorem (see [3, Theorem 1.7 and Proposition 1.15]). In fact one can show
that any small set G of generators of A can be completed in a descent structure for
A, in such a way that the description of this model category structure given by 2.5
is always available (but in practice, we essentially always have a canonical choice of
descent structure). One can proceed as follows: Choose a set J of generating trivial
0
cofibration (with respect to the model structure above). Define H to be the set of
the cokernels of the maps in J . Then (G,H) is a descent structure. Note that the
0
model structure of Theorem 2.5 coincides with the model structure of Christensen
and Hovey [5] in the particular case where all the elements of G are projective. In
general, the model structure of Theorem 2.5 is a left Bousfield localization of the
model structure associated to the projective class generated by G.
Example 2.13. Consider the notation of 2.4.
Let C be a complex of sheaves with transfers over S.
For any smooth S-scheme X, we denote by HnC(X) (resp. Hn(X,C)) the n-th
cohomologyofthecomplexofabeliangroupsmadebythesectionsofC overX (resp.
the n-th Nisnevich hypercohomology of X with coefficients in the complex C where
we forget transfers).
We will say that C is Nisnevich fibrant if it is fibrant with respect to the descent
structure (G ,H ). According to the previous theorem and [8, 4.2.9], this means one
S S
of the following equivalent conditions are true:
228 DENIS-CHARLESCISINSKIandFRE´DE´RICDE´GLISE
(i) For any smooth S-scheme X and any integer n, the canonical map
Hn(C(X))→Hn (X;C)
Nis
is an isomorphism.
(ii) For any smooth S-scheme X, any Nisnevich hypercovering X of X, and any
integer n, the canonical map Hn(C(X))→Hn(TotC(X)) is an isomorphism.
2.4.
LetAandA0 betwoGrothendieckcategories.Supposethat(G,H)(resp.(G0,H0))
isadescentstructureonA(resp.A0).Afunctorf∗: A0 −→Asatisfies descent (with
respect to the above descent structures) if it satisfies the following conditions:
(i) The functor f∗ preserves colimits.
(ii) For any E0 in G0, f∗(E0) is a direct sum of elements of G.
(iii) For any H0 in H0, f∗(H0) is in H.
ItisastandardfactthatafunctorbetweenGrothendieckcategoriespreservescolimits
ifandonlyifithasarightadjoint.Notealsothatanyfunctorwhichpreservescolimits
preserves,inparticular,finitedirectsums,sothatanyfunctorwhichsatisfiesdescent
is additive.
Theorem 2.14. Under the assumptions above, let f∗: A0 −→A be a functor that
preserves colimits, and f : A−→A0 be its right adjoint. If f∗ satisfies descent, then
∗
the pair of adjoint functors
f∗: Comp(A0)−→Comp(A) and f : Comp(A)−→Comp(A0)
∗
is a Quillen pair with respect to the model structures associated to G and G0 by The-
orem 2.5. In particular, the functors f∗ and f have the functors
∗
Lf∗: D(A0)−→D(A) and Rf : D(A)−→D(A0)
∗
as left and right derived functors respectively, and Lf∗ is left adjoint to Rf .
∗
Proof. If f∗ satisfies descent, then one checks easily that it preserves cofibrations;
see [12, Lemma 2.1.20]. It is then sufficient to show that it sends trivial cofibrations
toquasi-isomorphisms.Asf∗ preservescolimitsandcofibrations,byvirtueofloc. cit,
it is sufficient to prove that any element of f∗(J0) is a quasi-isomorphism, where J0
is the set of trivial cofibrations defined as in 2.3. This comes directly from the fact
that f∗(H0) is contained in H and that f∗ commutes to the formation of cones and
cylinders (just because it is additive).
Remark 2.15. Consider three Grothendieck categories endowed with descent struc-
turesA,A0,andA00.Letf0∗: A00 −→A0 andf∗: A0 −→Abetwofunctorscommut-
ing to colimits and satisfying descent, and denote by f0 and f their corresponding
∗ ∗
right adjoints. Then it follows easily from general abstract nonsense about Quillen
adjunctions and the preceding theorem that we havecanonical isomorphisms of total
derived functors
Lf∗◦Lf0∗ ’L(f∗◦f0∗) and R(f0 ◦f )’Rf0 ◦Rf .
∗ ∗ ∗ ∗