A QUILLEN APPROACH TO DERIVED CATEGORIES AND
TENSOR PRODUCTS
JAMES GILLESPIE
Abstract.We put a monoidal model category structure (in the sense of
Quillen) on the category of chain complexes of quasi-coherent sheaves ov*
*er
a quasi-compact and semi-separated scheme X. The approach generalizes
and simplifies the method used by the author in [Gil04] and [Gil06] to b*
*uild
monoidal model structures on the category of chain complexes of modules *
*over
a ring and chain complexes of sheaves over a ringed space. Indeed, much *
*of
the paper is dedicated to showing that in any Grothendieck category G, a*
* nice
enough class of objects F (which we call a Kaplansky class) induces a mo*
*del
structure on the category Ch(G) of chain complexes. We also find simple *
*con-
ditions to put on F which will guarantee that our model structure in mon*
*oidal.
We see that the common model structures used in practice are all induced*
* by
such Kaplansky classes.
1.Introduction
The classical theory of derived functors was introduced in Cartan and Eilenbe*
*rg's
Homological Algebra. The idea is now very basic and is of fundamental importance
to many branches of mathematics. As we explain below, the author feels that the
best foundation for defining and treating derived functors is to use Quillen's *
*idea
of a model structure.
The definition of derived functor appearing in Cartan and Eilenberg's book de-
pended on the existence of projective and injective resolutions. Often however,
especially in algebraic geometry, one is usually dealing with a category in whi*
*ch
projective resolutions do not exist. For example, it is well-known that we do n*
*ot
have projective resolutions in the various types of sheaf categories. Therefor*
*e,
Grothendieck and his school generalized the definition of derived functor as to*
* make
no mention of projective or injective objects at all. This definition depends o*
*n first
defining the derived category of an abelian category A. This is the "category"
D(A) obtained by first forming the associated category Ch(A) of unbounded chain
complexes, and then formally inverting the homology isomorphisms. (The word
"category" is in quotes because it is only known in particular cases that the c*
*lass
of maps between two objects in D(A) actually form a set.) The derived category *
*of
an abelian category A is now widely accepted as the proper setting to define and
study derived functors of all sorts. However, in general D(A) is not at all eas*
*y to
understand. Even the morphism sets seem mysterious, without further analysis in
each case.
However, Quillen's notion of a model category, appearing in [Qui67], gave us a
language and theory designed to deal with categories exactly like derived categ*
*ories.
____________
Date: July 29, 2006.
Subject Classification: 55U35, 18G15, 18E30.
1
2 JAMES GILLESPIE
That is, categories which are obtained by localizing with respect to a class of*
* mor-
phisms. Indeed, with a model structure on Ch(A), the morphism set D(A)(X, Y )
for complexes X, Y 2 D(A) will be isomorphic to
Ch(A)(QX, RY )= ~
where QX and RY are some sort of resolutions of X and Y respectively (called
"cofibrant" and "fibrant" replacements, respectively) and ~ is the relation of *
*chain
homotopic maps. This alone gives us a better understanding of the derived cat-
egory, not to mention the versatility that different model structures correspond
to different cofibrant and fibrant replacements. As an illustration, take A to*
* be
the category of R-modules where R is a commutative ring with identity. There
is a "projective" model structure on Ch(R) in which cofibrant replacements corr*
*e-
spond to projective resolutions. There is also an "injective" model structure w*
*here
the fibrant replacements correspond to injective coresolutions. Both these model
structures are described in detail in [Hov99 ]. The important result of Spalten*
*stein
in [Spa88] which says that every unbounded complex has an injective resolution *
*is
automatic with the existence of the injective model structure.
Quillen's theory also includes an easy way to verify the existence of derived
functors and also to compute derived functors using the cofibrant and fibrant a*
*p-
proximations. If we look at the Ch(R) example again, both the projective model
structure and the injective model structure are suitable to prove the existence*
* and
compute the derived functors ExtnR(A, B). The first corresponds to the classic*
*al
computation of ExtnR(A, B) by taking a projective resolution of A and the sec-
ond corresponds to the classical definition of computing ExtnR(A, B) by taking a
coresolution of B.
However, not all model structures are suitable for studying all derived func-
tors. For example, the projective model structure on Ch (R) is suitable to stu*
*dy
TorRn(A, B) but the injective model structure is not. In general, if the catego*
*ry A
has a tensor product then in model category language we say that the model stru*
*c-
ture on Ch(A) is monoidal if it is compatible with the tensor product on Ch(A).
It is now abundantly clear: If G is a category without projective resolutions, *
*what
model structure allows for an easy treatment of the derived tensor product? In
particular, we would like an answer for this when G is the category Qco(X) of
quasi-coherent OX -modules on a scheme (X, OX ). Clearly, the literature sugges*
*ts
that there should exist a "flat model structure" on Ch(Qco (X)), which will all*
*ow
computing TorGn(A, B) via some sort of "flat resolution".
With this goal in mind, the author first constructed an analogous model struc*
*ture
on Ch(G) when G is the category of modules over a commutative ring R in [Gil04].
This was generalized to the case of when G is the category of sheaves of OX -
modules where (X, OX ) is any ringed space in [Gil06]. Through this experience *
*and
while working on the problem when G = Qco(X) the author developed a powerful
theorem for building such homological model structures. This method appears as
Theorem 4.12. The theorem basically says that a suitable class of objects F in a
Grothendieck category G gives rise to a model structure on Ch(G). The theorem
is very general and in fact we show in Section 7 that every (homological) model
structure the author has dealt with arises in this way.
In Theorem 5.1 we provide conditions which will guarantee that the class F
induces a monoidal model structure. In Section 6 we see that the class of flat
objects in Qco(X) induces a monoidal model structure on Ch(Qco (X)) when X is
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 3
a quasi-compact and semi-separated scheme. Results from [Tar97] and [EE05 ] are
essential and are used in the construction of the flat model structure.
We point out that with all of the current literature on model categories our *
*flat
model structure on Ch(Qco (X)) provides a strong foundation for doing homologic*
*al
algebra in Qco(X). For example, from [May3 ] it is now automatic that D(Qco (X))
is triangulated in a way that is strongly compatible with the derived tensor pr*
*oduct.
We also automatically have the additivity property of generalized trace maps as
defined in [May3 ].
We now summarize the layout of this paper. Our goal is to show that a nice
enough Kaplansky class F in a Grothendieck category G induces a (monoidal) model
structure. We do this by using Theorem 2.2 of [Hov02 ] relating cotorsion pairs*
* and
model structures. In Section 3 we develop some theory on small cotorsion pairs.
The word "small" was first used to describe a cotorsion pair in [Hov02 ]. An an*
*alogy
is that small is to cotorsion pair as cofibrantly generated is to model categor*
*y. The
material in this section was surely known to Hovey, but to the author's knowled*
*ge,
has not been written down in as much detail. The material in Subsection 3.2 is
new and is due to the author. In particular, Proposition 3.7 greatly simplifies
the problem of building the flat model structure and makes great chunks of the
work in [Gil04] and [Gil06] seem obsolete. In Section 4 we introduce the notion
of Kaplansky class in a Grothendieck category G. Our definition is a categoric*
*al
version of the definition given by Edgar Enochs in [ELR02 ]. Enochs has proved
the existence of flat covers and cotorsion envelopes in several algebraic categ*
*ories,
essentially by showing that the flat objects form a Kaplansky class. We see in
Section 4 that Kaplansky classes give rise to cotorsion pairs and the cotorsion
pairs give rise to homological model structures. In Section 5 we show that if t*
*he
Kaplansky class satisfies certain compatibilities with the tensor product, then*
* the
induced model structure is monoidal in the sense of [Hov99 ]. In Section 6 we p*
*rove
the existence of the flat model structure on Ch(Qco (X)) when X is a quasi-comp*
*act,
semi-separated scheme and in Section 7 we show that all of the usual homological
model structures come from Kaplansky classes.
The author would like to thank Mark Hovey for always answering questions
related to this work. A couple of lemmas which are entirely due to Hovey are
pointed out in the text. Thankyou also to Edgar Enochs, and to Sergio Estrada
and Leo Alonso Tarr'io for helping me understand a few things they already knew
about quasi-coherent sheaves.
2.Preliminaries
Prerequisites for this paper are a basic understanding of Grothendieck catego*
*ries,
chain complexes, model categories, cotorsion pairs, and (quasi-coherent) sheave*
*s.
The basic reference used here for Grothendieck categories is [Sten75]. The auth*
*or
has used the texts [Har77] and (to a lesser degree) [Lit82] as references for s*
*heaves,
schemes and quasi-coherent sheaves. Also see Appendix B of [TT90 ] for more
advanced topics such as the definition and basic facts concerning semi-separated
schemes. The author uses [Hov99 ] for referencing facts on model categories.
Our work in this paper heavily rests on the work in [Hov02 ] and [Gil04]. The
first paper laid out the basic interplay between cotorsion pairs and model cat-
egories, while the second focused exclusively on the interplay between cotorsion
pairs of chain complexes and (homological) model structures. In particular we w*
*ill
4 JAMES GILLESPIE
use definitions and basic results from Section 3 of [Gil04]. There is a warnin*
*g:
Definition 3.11 in [Gil04] only makes sense if our category has enough projecti*
*ves
and injectives. This mistake actually leaves a gap in the construction of the *
*flat
model structure on Ch(OX -Mod) which appeared in the sequel [Gil06]. The mis-
take is easily fixed. Indeed one should replace the "hereditary" hypothesis wit*
*h the
assumption appearing as condition (4) in Theorem 4.12 of this paper. In this way
we have avoided using the word "hereditary" at all in a category without enough
projectives. In any case, we show in Section 7 that the flat model structure of
[Gil06] is just another corollary of our Theorem 4.12.
We also use language associated with locally ~-presentable categories through*
*out
much of the paper. This language along with all the results we will need are su*
*m-
marized in Appendix A. The author used [AR94 ] , [Bor94] Chapter V of [Sten75]
to prepare this appendix.
We end this section by giving proofs to a few lemmas will be used again and
again throughout the paper. The first two are very basic and concern generators*
* in
Grothendieck categories. We will usually use Lemmas 2.1 and 2.2 without explicit
mention. The last lemma is a lifting property that the author learned from Mark
Hovey through personal communication.
Lemma 2.1. Let G be an object in an abelian category G. G is a generator for G
if and only if given any morphism d: C -!D, d is an epimorphism whenever d* is
an epimorphism. Here d*: G(G, C) -!G(G, D) is defined by d*(t) = dt.
Proof.Suppose G is a generator and d: C -! D is a morphism for which d* is an
epimorphism. To show d is an epimorphism, we will show that if hd = 0, then h =*
* 0.
By way of contradiction suppose h 6= 0. By definition of a generator, there exi*
*sts
a morphism s: G -!D such that hs 6= 0. Now since d* is an epimorphism, there
exists t for which dt = s. So now 0 = 0t = hdt = hs 6= 0 which is a contradicti*
*on.
So d must be an epimorphism.
Next, suppose d is an epimorphism whenever d* is an epimorphism and let
h: X -!Y be a nonzero map. To show G is a generator we need to find a morphism
s: G -! X such that hs 6= 0. Let k :Z -! X be the kernel of h and by way of
contradiction suppose hs = 0 for all s. By the universal property of kernel, ea*
*ch s
factors through k. I.e., k* is an epimorphism. By hypothesis, k is an epimorphi*
*sm.
But since k is a kernel it is also a monomorphism and therefore k is an isomorp*
*hism.
This implies h = 0
If G is an abelian category and A is a class of objects, we say that G has en*
*ough
A-objects if for any object X 2 G we can find an epimorphism A -! X where
A 2 A.
Lemma 2.2. Let G be a Grothendieck category and A be a class of objects which is
closed under coproducts. Then A contains a generator if and only if G has enough
A-objects.
Proof.First assume A contains a generator G and that C is an arbitrary object in
C. It is well-known (for example, see [Bor94]) that the canonical map
M
G -! C
f2Hom(G,C)
is an epimorphism. Since we assume A is closed under coproducts we are done.
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 5
Next assume G has enough A-objects and let G be a generator for G. Then we
can find an epimorphism A -!G for which A 2 A. It is easy to check that A too
must be a generator for G.
Lemma 2.3. Let G be any abelian category. Suppose we have short exact sequences
A i-!B p-!C and K j-!L q-!M and a commutative diagram as shown below:
A ---f-! L
? ?
i?y ?yq
B ----!g M
If Ext1A(C, K) = 0 then there exists a lift h: B -!L so that hi = f and qh = g.
Proof.Consider the diagram below:
0 - ---! A _______A
?? ? ?
y ?y(if) ?yi
(01) (1 0)
L - ---! B L ----! B
? ? ?
q?y ?y(-g h) ?y
M _______ M ----! 0
Each column forms a chain complex and so the diagram is a short exact sequence *
*of
chain complexes. The associated long exact sequence in homology leads to a short
exact sequence K k-!T -r!C.
If we let Z denote the kernel of the map (-g q): B L -! M, then Z is the
pullback of the maps q and g as shown in the square below:
Z ---"g-! L
? ?
"q?y ?yq
B ---g-! M
The maps "gand "qare the projections B L -!L and B L -!B restricted to Z.
Some more notation is necessary. We set "':= if:A -! Z, and so T = cok"'.
We write "p:Z -!T for the quotient map and we set "k:= (0 j): K -!Z. Now one
can check that the diagram below commutes, the rows and columns are exact, and
the bottom right square is a pullback:
K _______K
? ?
"k?y ?yk
A --"'--!Z ---"p-!T
flfl ? ?
fl "q?y ?yr
A --i--! B ---p-! C
Now we can finally construct the lift. Since Ext1G(C, K) = 0, the sequence
K -k!T -r!C splits and so we have a map n: C -! T such that rn = 1C . By
6 JAMES GILLESPIE
the pullback property, there is a unique map "n:B -! Z with "q"n= 1B and "p"n=
np: B -! T . We claim that the map h := "g"n:B -! L is a lift. Indeed, we first
have qh = q"g"n= g"q"n= g. Next, one checks that "p("'- "ni) = 0 and "q("'- "ni*
*) = 0
and so the pullback property tell us that "ni = "':A -!Z. This gives us hi = "g*
*"'= f
as required.
3. Small cotorsion pairs
In the paper [Hov02 ] we learned a relationship between cotorsion pairs and m*
*odel
structures on an abelian category. In that paper Hovey defined small cotorsion
pairs, which are cotorsion pairs (A, B) along with a set of "generating monomor-
phisms" I. The basic analogy is that small is to cotorsion pairs as cofibrantly
generated is to model categories.
3.1. Properties of small cotorsion pairs. The results in this subsection are all
known and in fact much of it can be found in Section 6 of [Hov02 ]. The motivat*
*ion
for this subsection is proving Lemmas 3.4 and 3.5. Although these were certainly
known to Hovey (personal communication) they do not appear in [Hov02 ] and the
author could not find them in the literature. Lemma 3.4 will be used in the next
section while Lemma 3.5 will be used to prove that the flat model structure on
chain complexes of quasi-coherent sheaves is monoidal.
Definition 3.1. Let (A, B) be a cotorsion pair in a Grothendieck category G.
Suppose A contains a generator G for the category. The following conditions are
equivalent and we say that the cotorsion pair is small if it satisfies one of t*
*hese
conditions:
1) There is a set S which cogenerates (A, B) and for each S 2 S there is a
monomorphism iS, with cokiS = S, satisfying the following: For all X 2 G, if
G(iS, X) is surjective for all S 2 S, then X 2 B.
2) There is a set I of monomorphisms for which cokI = {coki : i 2 I} cogener-
ates (A, B) and which satisfies the following: For all X 2 G, if G(i, X) is sur*
*jective
for all i 2 I, then X 2 B.
3) There is a single monomorphism i for which {coki} cogenerates (A, B) and
such that for all X, G(i, X) surjective implies X 2 B.
We call a collection I, as in (2) above, together with the monomorphism 0 -!G,
a set of generating monomorphisms for (A, B).
The first definition is the original due to Hovey and can be found in [Hov02 *
*]. The
first definition clearly implies the secondLdefinition. The second definition i*
*mplies
the third by looking at the direct sum i2Ii. Finally the third condition clea*
*rly
implies the first.
The following lemma can be found implicitly by studying Section 6 of [Hov02 ].
See [Hov99 ] for the definition of I-inj, I-cof, and I-cell.
Lemma 3.2. Let (A, B) be a cotorsion pair in a Grothendieck category G. Also
suppose that A contains a generator G for the category and that the cotorsion
pair is small with generating monomorphisms I. Then I-inj is the class of all
epimorphisms p with kerp 2 B.
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 7
Proof.Say p: X -!Y is in I-inj. Since 0 -!G is in I, there is a lift in any dia*
*gram
of the form
0 ----! X
?? ?
y ?yp
G ----! Y
By Lemma 2.1, p must be an epimorphism.
Now kerp -!0 can be viewed as the pullback of p over the map 0 -!Y . Since
I-inj is closed under pullbacks we see that kerp -! 0 is also in I-inj. But th*
*is
is equivalent to saying G(i, kerp) is surjective for all i 2 I. So kerp 2 B by*
* the
definition of a set of generating monomorphisms.
On the other hand, let p be an epimorphism with kerp 2 B. We look for a lift
in a diagram of the form
M ----! X
? ?
i?y ?yp
N ----! Y
where i 2 I. Since Ext(coki, kerp) = 0, such a lift exists Lemma 2.3. This prov*
*es
the Lemma.
Lemma 3.3. Let (A, B) be a cotorsion pair in a Grothendieck category G. Also
suppose that A contains a generator G for the category and that the cotorsion
pair is small with generating monomorphisms I. Then I-cof is the class of all
monomorphisms f with cokf 2 A.
Proof.If f is a monomorphism with cokf 2 A, then f is in I-cof by combining
Lemma 3.2 and Lemma 2.3.
Conversely, let f :M -! N be in I-cof. Embed M ,! I in an injective. Then
(I -!0) 2 I-inj, so there is a lift in the diagram
M ----! I
? ?
f?y ?y
N ----! 0
and so f must be injective.
Now we wish to show that N=M 2 A. So let B 2 B be arbitrary and embed
B in an injective to get a short exact sequence 0 -! B -i!I -j!I=B -! 0. Since
Ext1(N=M, I) = 0, we will be done if we can show that any map h: N=M -! I=B
lifts to a map ~h:N=M -!I so that h = j~h. Notice we have a commutative diagram
M --0--! I _______ I
? ?
f?y ?yj
N ----!c N=M ----!h I=B
where c: N -! N=M is the canonical map to the cokernel. Since j 2 I-inj and
f 2 I-cof, there exists a lift _ :N -! I such that _f = 0 and j_ = hc. By the
universal property of cokernel, there exists a map ~h:N=M -!I such that _ = ~hc.
But now hc = j_ = j~hc, and since c is epi, it is right cancellable. Thus h = j*
*~h.
8 JAMES GILLESPIE
The next lemma basically says that a set I of monomorphisms (for which cokI =
S cogenerates (A, B)) is a set of generating monomorphisms if and only if I-inj*
* and
I-cof can be classified as in the last two lemmas.
Lemma 3.4. Let (A, B) be a cotorsion pair in a Grothendieck category G, cogen-
erated by a set S. Also suppose that A contains a generator G for the category.
Suppose for each S 2 S there is a monomorphism iS, with cokiS= S. Denote the
set of all iS together with 0 -!G by I. Then the following are equivalent:
1) (A, B) is a small cotorsion pair with I a set of generating monomorphisms.
2) I-inj is the class of all epimorphisms with kernel in B.
3) I-cof is the class of all monomorphisms with cokernel in A.
Proof.(1) implies (2) is Lemma 3.2 and (1) implies (3) is Lemma 3.3. We first
show (2) implies (1). So let X 2 G and suppose G(iS, X) is surjective for all S*
* 2 S.
This is equivalent to saying X -!0 is in I-inj. By hypothesis X 2 B as desired.
Next we show (3) implies (1). So let X 2 G and suppose G(iS, X) is surjective
for all S 2 S. Again, this is equivalent to saying X -! 0 is in I-inj. We need *
*to
conclude X 2 B, so we let A 2 A be arbitrary and argue that
0 -!X f-!Z g-!A -!0
must split. But by hypothesis, f 2 I-cof = I-inj-proj, so there is a lift in t*
*he
diagram
X _______X
? ?
f?y ?y
Z ----! 0
This lift is a splitting.
Lemma 3.5. Let (A, B) be a cotorsion pair in a Grothendieck category G. Also
suppose that A contains a generator G for the category and that the cotorsion p*
*air
is small with cogenerating set S and generating monomorphisms I. Then every
object A 2 A is a retract of a transfinite extension of objects in S. In partic*
*ular,
A is the smallest class containing S and closed under transfinite extensions and
retracts (summands).
Proof.Let A 2 A. Then 0 -! A is in I-cof by Lemma 3.3. By Corollary 2.1.15
of [Hov99 ] we see 0 -! A is a retract of a map 0 -! Y in I-cell, by a map which
fixes 0. That is, A is a retract of Y . But if 0 -! Y is in I-cell, then Y *
*is a
transfinite extension of the cokernels of maps in I. (This follows right from *
*the
definition of I-cell and the fact that cokernels are unchanged when pushing out
over a monomorphism.) So A is a retract of a transfinite extension of objects i*
*n S.
Say W is the smallest class containing S and is closed under transfinite exte*
*nsions
and retracts. Now A contains S and we know from Lemma 6.2 of [Hov02 ] that the
left side of a cotorsion pair is always closed under transfinite extensions and*
* retracts.
So A W. Conversely, from the last paragraph it is clear that A W.
3.2. An associated cotorsion pair of chain complexes is small. We continue
to let (A, B) represent a small cotorsion pair in a Grothendieck category G whi*
*ch
has a generator G 2 A. By Corollary 3.8 of [Gil04], we have induced cotorsion
pairs (dgAe, eB) and (Ae, dgBe) of chain complexes. We now show (dgAe, eB) is s*
*mall
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 9
whenever (A, B) is small. The author doesn't see a corresponding theorem for the
cotorsion pair (Ae, dgBe) without making further assumptions on the class A. Th*
*is
is the subject of Section 4.
Lemma 3.6. Let X be a chain complex in an abelian category G with generator G.
If any chain map f :Sn(G) -!X extends to Dn+1(G), then X is exact.
Proof.Let n be an arbitrary integer. By Lemma 2.1, showing exactness in degree
n requires showing that any morphism f :G -!ZnX lifts over d: Xn+1 -!ZnX.
But it is easy to see that this is the same as showing that the induced chain m*
*ap
^f:Sn(G) -!X extends to a morphism Dn+1(G) -!X.
Proposition 3.7. Let (A, B) be a cotorsion pair in a Grothendieck category G
which has a generator G 2 A. If (A, B) is cogenerated by a set {Ai}i2I, then the
induced cotorsion pair (dgAe, eB) is cogenerated by the set
S = { Sn(G) | n 2 Z } [ { Sn(Ai) | n 2 Z , i 2 I }.
Furthermore, suppose (A, B) is small with generating monomorphisms the map
0 -!G together with monomorphisms ki as below (one for each i 2 I):
0 -!Yi-ki!Zi-! Ai-! 0.
Then (dgAe, eB) is small with generating monomorphisms the set
n(ki)
I = { 0 -!Dn(G) } [ { Sn-1(G) -!Dn(G) } [ { Sn(Yi) S----!Sn(Zi) }.
Proof.Clearly S dgAe, so we have S? (dgAe)? = eB. Conversely if X 2 S? , th*
*en
0 = Ext1Ch(G)(Sn(Ai), X) for all i 2 I. But Ext1Ch(G)(Sn(Ai), X) ~=Ext1G(Ai, Zn*
*X)
(Lemma 3.1 in [Gil04]). So Ext1G(Ai, ZnX) = 0 which implies ZnX 2 B since the
set {Ai} cogenerates the cotorsion theory.
Next we want to show that X is exact. Consider the short exact sequence
0 -!Sn-1(G) -!Dn(G) -!Sn(G) -!0.
It induces an exact sequence of abelian groups
Hom Ch(G)(Dn(G), X) -!Hom Ch(G)(Sn-1(G), X) -!ExtCh(G)(Sn(G), X).
But again Lemma 3.1 of [Gil04] gives us ExtCh(G)(Sn(G), X) ~=ExtG(G, ZnX) and
this last group equals 0 by the last paragraph. Therefore Lemma 3.6 tells us X *
*is
exact. Since X is exact and has cycles in B, we see that X 2 eB. So S? = eB. Th*
*is
shows that S cogenerates the cotorsion pair (dgAe, eB).
Next we prove the statement about smallness. First note that since G generates
G, the complexes Dn(G) generate Ch(G). Also Dn(G) 2 dgAe, and so dgAecontains
the generators {Dn(G)}. Now let X be any chain complex. We wish to show that
"extending through monomorphisms in I" implies X 2 eB. But again, if morphisms
Sn-1(G) -!X can extend over Dn(G), then X must be exact by Lemma 3.6. Next
let ZnX be a cycle. Any map Yi -!ZnX determines a morphism Sn(Yi) -! X,
which we assume extends over Sn(ki) to a map Sn(Zi) -! X. Thus any map
Yi -! ZnX extends over ki to a map Zi -! ZnX. By hypothesis this implies
ZnX 2 B.
10 JAMES GILLESPIE
As mentioned at the beginning of this subsection, we know from Corollary 3.8
of [Gil04] that if A contains a generator for G and (A, B) is a cotorsion pair,*
* then
we have two induced cotorsion pairs on Ch(A). We denote them by (dgAe, eB)
and (Ae, dgBe) as in [Gil04]. We will see in the proof of Theorem 4.12 that a c*
*rucial
step in building a model category structure on Ch(A) is showing that these indu*
*ced
cotorsion pairs are compatible. This means dgAe\E = eAand dgBe\E = eB. Condition
(4) of Corollary 3.8 below will be used at that time to guarantee that the indu*
*ced
cotorsion pairs are compatible.
Corollary 3.8. Let (A, B) be a small cotorsion pair in a Grothendieck category *
*G.
Assume A contains a generator for G. Then the following are equivalent:
(1) The induced cotorsion pairs (dgAe, eB) and (Ae, dgBe) are compatible.
(2) A is resolving and B is coresolving. That is, A is closed under taking ke*
*rnels
of epimorphisms and B is closed under taking cokernels of monomorphisms.
(3) ExtnA(A, B) = 0 for any n > 0 and any A 2 A and B 2 B.
(4) Ae= dgAe\ E
Proof.First we show (1) implies (2). Say we are given a short exact sequence
0 -! X -! A0 -!A -! 0 where A0 and A are in the class A. Since A contains a
generator we can complete the short exact sequence to obtain an "A-resolution"
. .-.!A2 -!A1 -!A0 -!A -!0. Since this resolution is a bounded below complex
with objects in A, it is in dgAe by Lemma 3.4 of [Gil04]. But it is also in E. *
*Since
dgAe\ E = eA, we see that the resolution is in eA. Therefore X 2 A by the defin*
*ition
of the class Ae. This shows A is resolving. The dual shows that B is coresolvin*
*g.
Next we show (2) implies (3). We will prove by induction that Extn(A, B) = 0
for all n > 0 , A 2 A and B 2 B. Obviously Ext1(A, B) = 0 for any A 2 A and B.
Now suppose k > 0 and that Extk(A, B) = 0 for any A 2 A and B 2 B. We now let
A 2 A and B 2 B be arbitrary but fixed and wish to argue Extk+1(A, B) = 0. We
start by embedding B inside an injective, 0 -!B -!I -!B0-! 0. Note that I 2 B
and so by hypothesis B02 B. We now apply the functor Hom (A, -) to get the long
exact sequence . .-.!Extk(A, B0) -!Extk+1(A, B) -!Extk+1(A, I) -!. ...By the
induction hypothesis Extk(A, B0) = 0 and since I is injective Extk+1(A, I) = 0.*
* It
follows that Extk+1(A, B) = 0. Therefore (2) implies (3).
It is easy to see that (3) implies (2), so we have that (2) and (3) are equiv*
*alent.
Now we show (2) implies (4). First use Lemma 3.10 of [Gil04] to see that Ae
dgAe\E. Next, using only the coresolving hypothesis in (2), we can perform the *
*(dual
of the) argument in the proof of Theorem 3.12 of [Gil04] to conclude eA dgAe\ *
*E.
So Ae= dgAe\ E.
It is left to show that (4) implies (1) and this amounts to showing eB= dgBe\*
* E.
But by Proposition 3.7 we see that (dgAe, eB) is a small cotorsion pair. By Cor*
*ol-
lary 6.6 of [Hov02 ] we get that (dgAe, eB) has enough injectives. Finally Lemm*
*a 3.14
of [Gil04] tells us eB= dgBe\ E.
4. Locally cogenerated classes and Kaplansky classes
Throughout this section we again assume that G is a Grothendieck category.
In each of the categories R-Mod, OX -Mod and Qco(X), the class of flat objects
satisfies an important property. In a loose sense, every flat object is "built"*
* from
"smaller" flat objects. "Built" in this case means "is a transfinite extension*
* of"
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 11
and by "smaller" we essentially mean "of smaller cardinality". We axiomatize th*
*is
to get the concept of a locally cogenerated class. We will prove that a locally
cogenerated class F which is closed under transfinite extensions and retracts g*
*ives
rise to a small cotorsion pair (F, C). However, in light of Section 3.2, to ge*
*t the
flat model structure (on either R-Mod, OX -Mod or Qco(X)) we really need the
induced cotorsion pair (Fe, dgCe) to be small. We will do this by showing Feis*
* a
locally cogenerated class. Doing so will lead us to the notion of a Kaplansky c*
*lass,
which is a slight strengthening of a locally cogenerated class. The author got *
*the
term "Kaplansky class" from Edgar Enochs whose work (with several coauthors)
on proving the existence of flat covers in categories such as R-Mod, OX -Mod and
Qco(X) is closely related to the idea of a Kaplansky class. See [BBE01 ] , [ELR*
*02 ] ,
[EE05 ] , [EEGO ] , and [EO01 ]. In each category mentioned above, Enochs and
coauthors have essentially shown that the class of flat objects form a Kaplansky
class. So as a result of Theorem 4.12 we then have an induced (flat) model stru*
*cture
on the associated chain complex category. We explain in section 6 using results
of [EE05 ] why the class of flat objects in Qco(X) is a Kaplansky class.
The reader may want to skim through Appendix A at this point since many
definitions and theorems we use in this section can be found there.
4.1. Locally cogenerated classes.
Definition 4.1. Let ~ be a regular cardinal. Given a class F of objects in G, we
say F is locally ~-cogenerated if for every 0 6= F 2 F, there exists 0 6= S F
with S 2 Gen ~ F and F=S 2 F. We say F is locally cogenerated if it is locally
~-cogenerated for some regular cardinal ~.
Fact 4.2. Let ~0and ~ be regular cardinals with ~0 ~. It follows from Fact A.2
that if F is locally ~-cogenerated, then F is locally ~0-cogenerated.
Lemma 4.3. Suppose the class F is locally ~-cogenerated. Then given a monomor-
phism f :A ,! B with B=A 2 F, we may write B as a transfinite extension of A
by objects in Gen ~ F. In particular, every object of F is a transfinite extens*
*ion of
Gen ~ F
Proof.Set X0 = A. Now use the definition of locally ~-cogenerated to find 0 6=
X1=X0 B=X0 with X1=X0 2SGen ~ F and B=X1 2 F. Continue, by transfinite
induction, setting Xfl= ff ! is a regular cardinal, then X is ~-generated if and only if Xn is ~-ge*
*nerated
for each n. X is !-generated (finitely generated) if and only if X is bounded, *
*above
and below, and each Xn is !-generated (finitely generated).
Proof.SayP~ is any regular cardinal, and let n be an arbitraryPinteger. Let Xn =
i2IXi where Xi Xn. By Fact A.5 we wish to show Xn = i2JXi where
J I and |J| < ~. Each Xi gives rise to a subcomplex C(Xi) of X by defining
C(Xi) = . .-.!d-1n+2(d-1n+1(Xi)) -!d-1n+1(Xi) -!Xi-! Xn-1 -!Xn-2 -!. . .
P
and furthermorePX = i2IC(Xi). Since X is ~-generated,PFact A.5 tells us
X = i2JC(Xi) where J I and |J| < ~. Thus Xn = i2JXi. This shows
that if X is ~-generated, then Xn is ~-generated in G. For the special case of *
*when
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 15
P
~ = !, we still need to argue that X is bounded. For this, write X = n=ZSn
where
Sn = . .-.!0 -!Xn -!Xn-1 -!Xn-2 -!. .-.!X-n -! B-n-1 -!0 -!. . .
Now Fact A.5 tells us X = Sn for some n. So X is bounded.
P For the converse assume that each Xn is ~-generated whereP~ > !. Let X =
i2ISi where each Si is a subcomplex of X. ThenPXn = i2I(Si)n and soSthere
exists a setPJn I such that |Jn| < ~ and Xn = i2Jn(Si)n. Let J = n2ZJn.
Then X = i2JSi and |J| < ~ as desired. For the case of when ~ = ! it is clear
that we need the complex X to be bounded so that the collection J is finite.
Proposition 4.11. Let F be a ~-Kaplansky class. If G is locally ~-generated, th*
*en
the class eFin Ch(G) is locally ~-cogenerated, as long as ~ > !. (If ~ = !, the*
*n eF
is still ~0-cogenerated for any regular ~0> !.)
Proof.Fe is the class of all exact chain complexes F with ZnF 2 F for all n.
Suppose 0 6= F 2 F is given. We wish to construct a nonzero exact complex S F
in such a way that each ZnS, ZnF=ZnS 2 F and each Sn is ~-generated. It follows
that Fe is locally ~-cogenerated if ~ > !. If ~ = ! then S may not be locally
!-presentable because it need not be bounded. However in this case we still have
S ~0-generated for any regular ~0> ~. So eFis locally ~0-cogenerated.
Since F 6= 0 there must be some integer n for which ZnF 6= 0. We start by
finding 0 6= S0n ZnF with S0n~-presentable and S0n2 F, ZnF=S0n2 F. We set
Si = 0 for all i < n and set Sn = S0n. We now want to inductively build Si for
i > n.
Since G is locally ~-generated, we can use Lemma 4.4 to find a ~-generated
subobject Xn+1 Fn+1 for which d|Xn+1:Xn+1 -!S0nis an epimorphism. Since
S0nis ~-presentable and Xn+1 is ~-generated and G is locally ~-generated, it fo*
*llows
from Lemma A.12 that kerd|Xn+1is ~-generated as well. Now using the definition
of Kaplansky class, we can find a ~-presentable S0n+1such that kerd|Xn+1 S0n+1
Zn+1F and S0n+1, Zn+1F=S0n+12 F. Now
d|Xn+1+S0n+1:Xn+1 + S0n+1-!S0n
is an epimorphism whose kernel is S0n+1. So we set Sn+1 = Xn+1 + S0n+1. In this
way we continue inductively to construct an exact subcomplex 0 6= S F with
ZnS, ZnF=ZnS 2 F and ZnS ~-presentable (and therefore Sn is ~-generated by
Facts A.3 and A.4).
Theorem 4.12. Let G be a locally ~-presentable Grothendieck category. Suppose
F is a class of objects which satisfies the following:
(1) F is a ~-Kaplansky class.
(2) F contains a ~-presentable generator G for G.
(3) F is closed under transfinite extensions and retracts.
(4) dgFe\ E = eF.
Then we have an induced model category structure on Ch (G) where the weak
equivalences are the homology isomorphisms. The cofibrations (resp. trivial c*
*ofi-
brations) are the monomorphisms whose cokernels are in dgFe (resp. eF). The
fibrations (resp. trivial fibrations) are the epimorphisms whose kernels are in*
* dgCe
16 JAMES GILLESPIE
(resp. eC), where C = F? . Furthermore this model structure is cofibrantly gene*
*rated.
The generating cofibrations form the set
I = { 0 -!Dn(G) } [ { Sn-1(G) -!Dn(G) } [ { Sn(A) -!Sn(B) }
where A -!B ranges over all possible monomorphisms with B a ~-generated object
for which B=A 2 F. The generating trivial cofibrations are
J = { 0 -!Dn(G) } [ { X -!Y }
where X -! Y ranges over all monomorphisms where Y is a ~-generated complex
and Y=X 2 eF
Proof.All of the work has been done. Since F is a locally cogenerated class, cl*
*osed
under transfinite extensions and retracts, and containing a generator for G, Pr*
*opo-
sition 4.8 tells us (F, C) is a small cotorsion pair. Now by Corollary 3.8 of [*
*Gil04] we
have the induced cotorsion pairs (Fe, dgCe) and (dgFe, eC) of chain complexes. *
*These
cotorsion pairs are compatible by Corollary 3.8 above.
Having the cotorsion pairs of complexes (dgFe\E, dgCe) and (dgFe, dgCe\E) at *
*hand
we are in position to use Hovey's Theorem 2.2 from [Hov02 ]. By Proposition 3.7*
* we
know that (dgFe, dgCe\E) is small, so this cotorsion pair is complete by Coroll*
*ary 6.6
of [Hov02 ]. Next dgFe\E is locally cogenerated by Proposition 4.11, so (dgFe\E*
*, eC)
is small too and hence complete. Theorem 2.2 and Lemma 6.7 of [Hov02 ] tell us
we have an induced model structure on Ch(G) which is cofibrantly generated. One
can see that the weak equivalences, fibrations and cofibrations are as stated i*
*n the
theorem by looking at Section 5 of [Hov02 ].
5. Monoidal model structures
Suppose our ground category G has a tensor product making it a closed sym-
metric monoidal Grothendieck category. We would like to have a condition on our
Kaplansky class F which will guarantee that the model structure induced from
Theorem 4.12 will be monoidal. We refer the reader to Chapter 4 of [Hov99 ] for*
* a
detailed discussion of monoidal model structures. There is also a nice discussi*
*on of
the essential ideas in Section 7 of [Hov02 ]. In fact our proof below will rely*
* on basic
results from Section 7 of [Hov02 ]. Also, the crucial argument of the proof bel*
*ow,
part (ii), is due to Hovey.
Theorem 5.1. Suppose G is a Grothendieck category with a closed symmetric
monoidal structure - G - and let F be a class of objects such that G and F sat-
isfy the hypotheses of Theorem 4.12. Then the induced model structure on Ch(G) *
*is
monoidal with respect to the usual tensor product of chain complexes if the fol*
*lowing
conditions hold:
(1) Each object in F is flat. I.e., F G - is exact for each F 2 F.
(2) If F and G both belong to F, then F G G also belongs to F.
(3) U 2 F, where U is the unit for the monoidal structure on G.
Proof.Suppose the three stated conditions hold. We wish to check the four con-
ditions of Theorem 7.2 from [Hov02 ]. In this case the four conditions translat*
*e to
the following:
(i) Every cofibration is a pure injection in each degree.
(ii) If X and Y are in dgFe, then X Y is in dgFe.
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 17
(iii) If X is in eFand Y is in dgFe, then X Y is in eF.
(iv) The unit for the monoidal structure on Ch(G) is in dgFe.
Before we check each condition we make a few observations. First, as we can s*
*ee
from Proposition 3.7, the hypotheses guarantee that (dgFe, eC) is a small cotor*
*sion
pair with cogenerating set consisting of spheres on objects from F. That is, t*
*he
cogenerating set consists of complexes of the form Sn(F ) where F 2 F. For the
rest of the proof we will call any such object Sn(F ) an "F-sphere". Second, dg*
*Fe
contains a set of generators for Ch(G) since if G 2 F generates G, then the set
of complexes Dn(G) generates Ch(G). (Each Dn(F ) is in dgFe by Lemma 3.4
of [Gil04].) It now follows from Lemma 3.5 that each complex in dgFe is a direct
summand of a transfinite extension of F-spheres. Lastly, we note that if X 2 dg*
*Fe,
then X - is exact. Indeed one can easily check that a complex X is tensor exa*
*ct
if and only if Xn is flat for each n. We now check the four conditions above.
(i) This follows from a basic property of flat objects. Cofibrations are mono*
*mor-
phisms with cokernels in dgFe. In particular, the cokernel is flat in each deg*
*ree.
Therefore each cofibration is a pure injection in each degree. (See for example*
*, the
proof of Lemma XVI.3.1 in [Lan97].)
(ii) Step 1: We first show that if X is a transfinite extension of F-spheres *
*and
if Sn(F ) is an F-sphere, then X Ch(G)Sn(F ) is in dgFe. To see this first not*
*ice
that the tensor product of two F-spheres is again an F-sphere and in particular*
* is
in dgFe. Now say X is a transfinite extension of a sequence such as
X0 ,! X1 ,! . .,.! Xff,! Xff+1,! . . .
where X0 and each Xff+1=Xffare F-spheres. Since the functor - Ch(G)Sn(F ) is
exact and also preserves direct limits, applying it to the above sequence will *
*display
X Ch(G)Sn(F ) as a transfinite extension of objects in dgFe. Therefore it too *
*is in
dgFe by Lemma 6.2 of [Hov02 ].
Step 2: Now we see that if X 2 dgFeand if Sn(F ) is an F-sphere, then X Ch(G)
Sn(F ) is in dgFe. The reason is that such an X is a direct summand of a transf*
*inite
extension of F-spheres. Since the functor - Ch(G)Sn(F ) commutes with direct
sums, the result follows from the fact that dgFe is closed under direct summand*
*s.
(The left side of any cotorsion pair is closed under direct summands.)
Step 3: Now we argue the same way to see that X Ch(G)Y is in dgFe whenever
X and Y are in dgFe. For example if Y is in dgFethen it is a retract of a trans*
*finite
extension of F-spheres. As above, since X Ch(G)- commutes with direct sums and
dgFeis closed under taking direct summands we just need to show that X Ch(G)Y
is in dgFe when X is in dgFe and Y is a transfinite extension of F-spheres. But*
* in
this case Y can be viewed as the transfinite extension of a sequence such as
Y0 ,! Y1 ,! . .,.! Yff,! Yff+1,! . . .
where Y0 and each Yff+1=Yffare F-spheres. Then we apply the functor X Ch(G)-
and use the fact from Step 2 to argue that X Ch(G)Y is a transfinite extension*
* of
complexes in dgFe. So X Ch(G)Y is in dgFe too.
(iii) The approach is similar to our proof of (ii). Note that if E is the cla*
*ss of
exact complexes in Ch(G), then E is closed under transfinite extensions and dir*
*ect
18 JAMES GILLESPIE
summands. One reason this is true is that E is the left side of the cotorsion p*
*air
(E, dgeI) where dgeIis the class of dg-injective complexes.
Step 1: First we will prove that if E is an exact complex in Ch(G) and Y 2 dg*
*Fe,
then E Y is exact. We leave it to the reader to see that, as in our proof of *
*(ii),
we can assume that Y is a transfinite extension of F-spheres. So suppose Y is t*
*he
direct limit of a sequence such as
Y0 ,! Y1 ,! . .,.! Yff,! Yff+1,! . . .
where Y0 and each Yff+1=Yffare F-spheres. Since each short exact sequence
0 -!Yff-!Yff+1-!Yff+1=Yff-!0
is pure in each degree, the sequence below is also exact:
0 -!E Yff-!E Yff+1-!E Yff+1=Yff-!0
Now by applying the functor E - to the entire sequence
Y0 ,! Y1 ,! . .,.! Yff,! Yff+1,! . . .
we can argue that E Y is a transfinite extension of exact complexes and so is
itself exact.
Step 2: Now say X 2 eFand Y 2 dgFe. Then X 2 dgFe by our assumption that
eF= dgFe\ E. So by part (ii) we have that X Y is in dgFe. But by Step 1 of th*
*is
proof we also have that X Y is in E. Therefore X Y is in eF.
(iv) Since U 2 F, we have from Lemma 3.4 of [Gil04] that S(U) = S0(U) is in
the class dgFe. Since S(U) is the the unit for the tensor product on Ch(G) we a*
*re
done.
6.Chain complexes of quasi-coherent sheaves
For the rest of the paper we turn to applications of Theorem 4.12 and The-
orem 5.1. In this section we prove the existence of the flat model structure on
Ch(Qco (X)) when X is a quasi-compact and semi-separated scheme.
So let (X, OX ) be a scheme on a topological space X. We denote the category *
*of
all sheaves of OX -modules by OX -Mod. We denote by Qco(X), the full subcategory
of OX -Mod consisting of all quasi-coherent OX -modules. The class F of all fl*
*at
quasi-coherent sheaves will be the Kaplansky class inducing the model structure.
As far as the author can tell we need some assumptions on the scheme X in order
to satisfy the hypotheses of Theorem 4.12. From a study of the current literatu*
*re
it seems as though assuming X is quasi-compact and semi-separated is the best
we can do. In this case we see from [Tar97] or [Mur ] that every quasi-coherent
OX -module is the quotient of a flat quasi-coherent OX -modules. We start this
section by recalling some facts about the category Qco(X) as well as defining s*
*emi-
separated. The author has used [Har77], [Lit82] and Appendix B of [TT90 ] as ba*
*sic
references on this material. We will also use recent results from [EE05 ].
The inclusion functor Qco(X) -! OX -Mod is exact and Qco(X) is an abelian
subcategory of OX -Mod. Therefore finite limits and colimits in Qco(X) are taken
as in OX -Mod. As stated in the beginning of Appendix B of [TT90 ], Qco(X) is
cocomplete with all small colimits taken as in OX -Mod. Therefore direct limits
are exact in Qco(X). In fact Qco(X) also has a set of generators making it a
Grothendieck category. This last fact was stated as Lemma 2.1.7 in [Con00 ] and
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 19
apparently is due to Gabber. A proof is not given in [Con00 ] but an independent
proof can be found in [EE05 ].
Since we know Qco(X) has a generator and the inclusion functor Qco(X) -!
OX -Mod preserves small colimits the inclusion has a right adjoint Q: OX -Mod
-!Qco (X) by the special adjoint functor theorem. Thus Qco(X) is a coreflective
subcategory of OX -Mod. In particular, Q satisfies a universal property dual in
nature to the sheafification of a presheaf. One can now easily check that all s*
*mall
limits exist in Qco(X) and are taken by applying Q to the usual limit taken in *
*OX -
Mod. Lastly, we recall that Qco(X) is closed under extensions and tensor produc*
*ts
taken in OX -Mod. The tensor product makes Qco(X) into a closed symmetric
monoidal category. The closed structure is given by applying the functor Q after
the usual "sheafhom" functor.
A quasi-coherent OX -module F is flat if the tensor functor F OX - is exact.
Clearly the class of flat modules in Qco(X) is merely the class of flat modules*
* in
OX -Mod intersected with the class of all quasi-coherent OX -modules. We denote
the class of all flat quasi-coherent OX -modules by F. We point out that since
flatness is defined in terms of the tensor product which is a left adjoint, F i*
*s closed
under direct limits and retracts (summands). Furthermore, if
0 -!F 0-!F -! F 00-!0
is a short exact sequence of quasi-coherent sheaves with F 002 F, then F 02 F if
and only if F 2 F. These assertions are all true by using categorical arguments
with the tensor product, similar to those found in [Lan97] for modules over a
ring. Alternatively, they can be shown using the characterization of flatness a*
*s a
"stalkwise" property along with the fact that the corresponding property holds *
*for
modules over a ring. In any case, we point out that F is closed under retracts *
*and
transfinite extensions and also that F is resolving, meaning it is closed under*
* taking
kernels of epimorphisms between objects in F.
We say a scheme X is semi-separated if there exists an affine basis V = {Vff}
for sp(X) which is closed under finite intersections. The basis V is called a s*
*emi-
separating affine basis of X. As pointed out in B.7 of [TT90 ] a semi-separated
scheme is quasi-separated. Furthermore, a separated scheme is semi-separated wi*
*th
the set of all affine subsets serving as a semi-separating affine basis.
It is well-known that both OX -Mod and Qco(X) do not, in general, have enough
projectives. While the category OX -Mod has a set of flat generators, the author
does not know whether or not for a general scheme X, Qco(X) has a set of flat
generators, or equivalently whether each quasi-coherent sheaf can be written as*
* the
quotient of a flat quasi-coherent sheaf. However, Proposition 1.1 of [Tar97] im*
*plies
that this is indeed the case if we assume X is a quasi-compact, separated schem*
*e.
In fact as pointed out by Daniel Murfet their proof works for any quasi-compact
semi-separated scheme X. Murfet's proof can be found as Proposition 16 in [Mur *
*].
Since the quasi-compact semi-separated hypothesis is the best one the author has
found for the existence of enough flats in Qco(X) we will build the flat model
structure on Ch(Qco (X)) under these assumptions on X.
The following subsections will contain lemmas breaking down the proof of The-
orem 6.7 which of course amounts to checking the four hypotheses of Theorem 4.12
for the class F of flat modules in Qco(X). We start with the Kaplansky class co*
*ndi-
tion. As we will see, it readily follows from recent work of Enochs and Estrada*
* that
F is indeed a Kaplansky class for any scheme X [EE05 ]. Since the term Kaplansky
20 JAMES GILLESPIE
class was not used in [EE05 ] we now check carefully that their work proves F i*
*s a
Kaplansky class in the sense of Definition 4.9.
6.1. Quasi-coherent modules on a representation of a quiver. By a quiver
we mean a directed graph Q = (V, E) where V is the set of vertices and E is the
set of edges. By a representation of Q, we will mean a functor R: Q -! CRng
where Q is thought of as a category in the obvious way and CRng is the category
of commutative rings with identity. Let R be a representation of Q. By an R-
module M we will mean an R(v)-module, M(v), for each v 2 V , and an R(v)-linear
map M(e): M(v) -! M(w) for each edge e: v -! w in E. We refer the reader
to Section 2 of [EE05 ] for the definition of a flat representation and the def*
*inition
of a quasi-coherent R-module. It is also pointed out there that the category C *
*of
quasi-coherent R-modules is a Grothendieck category when R is flat.
We define the cardinality of an R-module as
a
|M| = | M(v)|.
v2V
Lemma 6.1. Let R be a flat representation of a quiver Q = (V, E) and M be a
quasi-coherent R-module. Let ~ be a regular cardinal for which ~ > |R(v)| for a*
*ll v
and ~ > max{|E|, |V |}. Then the following are equivalent.
(1) |M| < ~.
(2) M is ~-generated.
(3) M is ~-presentable.
P
Proof.(1) ) (2). We use Fact A.5. Suppose |M| < ~ and say M = i2IMi is a
~-filtered union of quasi-coherentPsubmodules. For each x 2 M, there corresponds
some i 2 I such that x 2 Mi. But i2IMi is ~-filtered and |M| < ~, so there
exists i 2 I such that M = Mi. `
(2) ) (1). Let S be the collection of all subsets S v2VM(v) such that
|S| < ~. For each S 2 S, let MS represent the quasi-coherent submodule generated
by S. Then |MS| < ~. (One way to see this is to use Proposition 3.3 of [EE05 ].
All signs in that Proposition can be changed toP< signs.) Note that (S, ) is
~-filtered and in fact M is the ~-filtered union S2SMS. By Fact A.5, M = MS
for some S 2 S.
(3) ) (2) is automatic. We now prove (2) ) (3), using that (1) iff (2). First
we point out that the category of quasi-coherent R-modules isPlocally ~-generat*
*ed.
Indeed each M can be expressed as the ~-filtered union M = S2SMS where each
MS is ~-generated as in the last paragraph. Therefore, we may use the charac-
terization of ~-presentable objects in Fact A.12. Suppose M is ~-generated and
N -! M is an epimorphism with N a ~-generated R-module. Then |N| < ~. So
of course | ker(N -!M) | < ~ (kernels are computed componentwise), which means
ker(N -!M) is ~-generated. This proves M is ~-presentable.
An R-module M is called flat if each M(v) is a flat R(v)-module.
Lemma 6.2. Let R be a flat representation of a quiver Q = (V, E). Let ~ be a
regular cardinal for which ~ > |R(v)| for all v and ~ > max{|E|, |V |}. Then the
class of all flat quasi-coherent R-modules constitute a ~-Kaplansky class.
Proof.This follows immediately from Lemma 6.1 along with the Proposition 3.3
of [EE05 ]. Again, all signs can be changed to < signs in the statement and p*
*roof
of Proposition 3.3 of [EE05 ].
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 21
6.2. A category equivalent to Qco(X) . If X is a scheme, then the collection
of all affine open subsets determines a directed graph, AX . The opposite graph
(reverse all arrows) is also a quiver which we will denote QX . We get a (flat)*
* rep-
resentation R by letting R(U) = OX (U) for each U 2 AX and using the restriction
maps to get R(V ) -! R(U) whenever U V . In the same way, a quasi-coherent
sheaf S on X gives rise to a quasi-coherent R-module M by letting M(U) = S(U)
for U 2 AX . In fact, as explained in [EE05 ], the category C of quasi-coherent
R-modules is categorically equivalent to Qco(X). Furthermore, the equivalence
restricts to an equivalence between the flat objects in C and the flat objects *
*of
Qco(X).
Lemma 6.3. Let A0 A and B0 B where A and B are abelian categories. Let
G: A -! B be an additive functor which is an equivalence. If G restricts to an
equivalence between A0 and B0, then A0 is a ~-Kaplansky class if and only if B0*
* is
a ~-Kaplansky class.
Proof.Since G: A -!B is an equivalence, there exists H :B -!A such that HG ~=
1A and GH ~=1B. Furthermore, G is both a left and right adjoint of H (and so
each preserves colimits). We start by showing that G preserves ~-generated and *
*~-
presentable objects. Indeed if X 2 A is ~-presentable then the functor A(X, H(-*
*))
preserves ~-filtered colimits. But A(X, H(-)) ~=B(G(X), -), so G(X) must be
~-presentable. On the other hand, if G(X) is ~-presentable, then B(G(X), G(-))
preserves ~-filtered colimits. Therefore, B(G(X), G(-)) ~=A(HG(X), -) preserves
~-filtered colimits and HG(X) ~=X must be ~-presentable. The same proof (but
with ~-filtered colimits of monomorphisms) shows that an equivalence preserves
~-generated objects.
Now we suppose G restricts to an equivalence between A0 and B0 and that B0
is a ~-Kaplansky class of B. We will show that A0 is a ~-Kaplansky class of A.
So let X F 6= 0 where F 2 A0 and X is a ~-generated object in A. Then
G(X) G(F ) 6= 0 and G(X) is ~-generated and G(F ) 2 B0. Since B0 is a ~-
Kaplansky class, there exists a nonzero ~-presentable object S 2 B0 for which
G(X) S G(F ) and G(F )=S 2 B0. Therefore H(S) 2 A0 is a nonzero ~-
presentable object for which X = HG(X) H(S) HG(F ) = F and F=H(S) =
HG(F )=H(S) = H(GF=S) 2 A0.
Proposition 6.4. Let (X, OX ) be quasi-compact and semi-separated scheme. Let
F be the class of flat quasi-coherent OX -modules and define C = F? . Then F is*
* a
Kaplansky class and (F, C) is a small cotorsion pair.
Proof.It follows from the lemmas above that F is a Kaplansky class. In particul*
*ar
F is locally cogenerated. Furthermore as explained in the introduction to this *
*sec-
tion, F is closed under transfinite extensions and retracts and contains a gene*
*rator.
Therefore (F, C) is a small cotorsion pair by Proposition 4.8.
6.3. Exact dg-flat complexes are flat. Let (X, OX ) be a quasi-compact semi-
separated scheme. For what follows we fix a semi-separating affine basis V and *
*again
let F be the class of all flat modules in Qco(X). We will prove that eF= dgFe\ E
where E is the class of all exact chain complexes.
Recall that if f :Y -! Z is any morphism of schemes then by Proposition II.5.8
of [Har77] the inverse image functor f* preserves quasi-coherence. In particula*
*r, if
Vff2 V and j :Vff-!X is the inclusion, then we have the functor j*: Qco(X) -!
22 JAMES GILLESPIE
Qco(X|Vff). In this case the functor is merely restriction of a quasi-coherent *
*OX -
module to the affine subset Vffand it is clear that j* is an exact functor.
On the other hand, recall that the direct image functor f* does not always
preserve quasi-coherence. However it is easy to see that j* does preserve quas*
*i-
coherence. Indeed j is separated since it is an open immersion (see Theorem 1.17
of [Lit82]). Also since affine subsets are quasi-compact we see by Proposition *
*1.51
of [Lit82] (with our semi-separating basis V used as the affine open cover requ*
*ired
in the Proposition), that j* preserves quasi-coherence.
Lemma 6.5. Let (X, OX ) be a semi-separated scheme with semi-separating affine
basis V = {Vff}. Let Vff2 V and j :Vff-!X be the inclusion. Then j*: Qco(X) -!
Qco(X|Vff) is left adjoint to j*: Qco(X|Vff) -! Qco(X). Furthermore j* is exact
and preserves cotorsion objects.
Proof.It is a standard fact that inverse image functors are left adjoint to dir*
*ect
image functors. For example, see Section II.5 of [Har77]. In particular, j* is *
*left ad-
joint to j*. Since j* is a right adjoint it is left exact. We will show that j**
* preserves
surjections. So suppose we have a surjection F -! G of quasi-coherent sheaves in
Qco(X|Vff). Then given any affine subset Vfl Vff, we know F(Vfl) -! G(Vfl) is a
surjection. It follows that for any affine Vfi2 V, [j*(F)](Vfi) -![j*(G)](Vfi) *
*is sur-
jective. Indeed the definition of [j*(F)](Vfi) -![j*(G)](Vfi) is just F(Vfl) -!*
*G(Vfl)
where Vfl= Vff\ Vfi2 V. It follows that j* is exact since V is a basis for the
topology on X.
We now show that for any quasi-coherent sheaf F 2 Qco(X) and any quasi-
coherent sheaf C 2 Qco(X|Vff) we have an isomorphism
Extn(j*F, C) ~=Extn(F, j*C).
First note that j* being the right adjoint of an exact functor preserves inject*
*ive
objects by Proposition 2.3.10 of [Wei94]. Since j* itself is also exact it pres*
*erves
injective resolutions. The result now follows by taking an injective resolution*
* of C,
applying j* to the resolution, and then applying the adjoint relationship betwe*
*en
j* and j*.
Now suppose C 2 Qco(X|Vff) is cotorsion and F is a flat quasi-coherent OX -
module. We wish to show that j*C cotorsion. But j*F is clearly flat since j* is*
* just
restriction, and so Ext1(F, j*C) ~=Ext1(j*F, C) = 0. Therefore j*C is cotorsion.
Proposition 6.6. Let (X, OX ) be a semi-separated scheme with semi-separating
affine basis V = {Vff} and let F be the class of flat quasi-coherent OX -module*
*s.
Then eF= dgFe\ E where E is the class of exact complexes.
Proof.We saw in Proposition 6.4 that (F, C) is a cotorsion pair. Now Lemma 3.10
of [Gil04] says that eF dgFe\ E. So we wish to prove eF dgFe\ E. We let Y
be an exact dg-flat complex of quasi-coherent OX -modules. We want to show that
it is a flat complex. I.e. that is has flat cycles in each degree. One way t*
*o do
this is to show that the complex j*(Y ) is a flat complex of O|Vff-modules for *
*each
inclusion j :Vff-!X where Vff2 V. So let j :Vff-!X be such an inclusion. Then
by the usual equivalence between the category of Aff-modules and the category of
quasi-coherent sheaves on A"ffwhere Affis a ring in which A"ff~=O|Vff, the chain
complex j*(Y ) = Y |Vffcorresponds to a chain complex of Aff-modules. Since the
equivalence preserves flat objects by Proposition III.9.2 of [Har77] it also pr*
*eserves
cotorsion objects. Therefore, the complex of Aff-modules corresponding to j*(Y *
*) is
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 23
flat (respectively dg-flat) if and only if j*(Y ) is flat (respectively dg-flat*
*). Finally,
since it is already known from [Gil04] that exact, dg-flat complexes of Aff-mod*
*ules
are flat complexes we will be able to conclude that j*(Y ) is flat by just prov*
*ing it
is exact and dg-flat.
It is clear that j*(Y ) is exact so we will show that j*(Y ) is a dg-flat com*
*plex.
So let j*(Y ) -!C be a morphism where C is a cotorsion complex of quasi-coherent
OVff-modules. (This means that C is exact and each cycle is cotorsion.) Using
the definition of dg-flat we wish show this map is null homotopic. But using the
adjoint property of j* and j* gives us a morphism Y -! j*(C) of complexes. Since
j*(-) is exact and preserves cotorsion modules by Lemma 6.5 we see that j*(C)
is a cotorsion complex and by definition of dg-flat Y -! j*(C) is null homotopi*
*c.
From this we can deduce that j*(Y ) -!C is also null homotopic.
6.4. The main theorem. We are now ready to prove the main theorem. By
applying Definition 3.3 of [Gil04] to the cotorsion pair (F, C) we get four cla*
*sses
of chain complexes in Ch (Qco (X)). We call the complexes in dgFe the dg-flat
complexes, and the complexes in eFthe flat complexes. We call the complexes in
dgCethe dg-cotorsion complexes, and the complexes in eCthe cotorsion complexes.
Theorem 6.7. Let X be a quasi-compact semi-separated scheme and let F
Qco(X) be the class of flat quasi-coherent sheaves. Then we have a cofibrantly *
*gen-
erated model category structure on Ch(Qco (X)) which is described as follows: T*
*he
weak equivalences are the homology isomorphisms. The cofibrations (resp. trivial
cofibrations) are the monomorphisms whose cokernels are dg-flat complexes (resp.
flat complexes). The fibrations (resp. trivial fibrations) are the epimorphisms*
* whose
kernels are dg-cotorsion complexes (resp. cotorsion complexes). Furthermore, th*
*is
model structure is monoidal with respect to the usual tensor product of chain c*
*om-
plexes.
Proof.From B.3 of [TT90 ] we see that if X is quasi-compact and quasi-separated,
then Qco(X) is locally finitely presentable [TT90 ]. That is, Qco(X) is locally*
* ~-
presentable (and hence ~-generated) for any regular cardinal ~. Since Qco(X) is*
* a
Grothendieck category it has a generator. By writing this generator as the quot*
*ient
of a flat quasi-coherent OX -module we obtain a flat generator. By Lemmas 6.2
and 6.3 we can find arbitrarily large regular cardinals ~ for which F is a ~-Ka*
*plansky
class. In light of Fact A.2 we can find a single regular cardinal ~ such that *
*(i)
Qco(X) is locally ~-presentable, (ii) F is a ~-Kaplansky class, and (iii) F con*
*tains
a ~-presentable generator. This verifies properties (1) and (2) of Theorem 4.1*
*2.
Property (3) holds as explained in the introduction to this section. Property (*
*4) is
Proposition 6.6. This finishes the proof of the existence of the flat model str*
*ucture
on Ch(Qco (X)).
We now check that the model structure is monoidal. Condition (1) of Theo-
rem 5.1 is trivial. Condition (2) is true since the tensor product of two flat *
*quasi-
coherent sheaves is again a flat quasi-coherent sheaf. The unit for the tensor *
*product
is the structure sheaf OX . It is both quasi-coherent and flat and so condition*
* (3)
holds.
Many questions come to mind that the author has not been able to figure out
or has not had time to consider. First, is there an easier proof that Qco(X) has
enough flat objects when X is quasi-compact and semi-separated? Can we construct
24 JAMES GILLESPIE
some actual flat generators for Qco(X) that are strongly dualizable in D(Qco (X*
*))?
Is D(Qco (X)) a stable homotopy category in the sense of [HPS97 ]? What are
the implications of May's additivity theorem in [May3 ]? Can one easily treat t*
*he
derived inverse and direct image functors from the model category viewpoint?
7.Other applications
In this section we point out other applications of Theorem 4.12. In particula*
*r we
will show that all model structures the author has previously constructed in [G*
*il04]
and [Gil06] as well as the more common "injective" and "projective" model struc-
tures on Ch(R) can each be construed as corollaries to Theorem 4.12. Note that
condition (1) (the Kaplansky class condition) in Theorem 4.12 is defined in ter*
*ms
of ~-generated and ~-presentable objects. In practice it may be cumbersome to
work with the definitions of ~-generated or ~-presentable. In each corollary be*
*low
we find it convenient to pick ~ large enough so that the ~-generated objects and
~-presentable objects coincide. In the case when G is a concrete category we go
further and relate the notions of ~-generated and ~-presentable to the cardinal*
*ity
of the underlying set. As mentioned after the definition of Kaplansky class in *
*Sec-
tion 4.2, ~-generated and ~-presentable become good categorical replacements for
the notion of cardinality.
Corollary 7.1. Let G be any Grothendieck category and let A be the class of all
objects in G. Then A is a Kaplansky class and in induces the injective model
structure on Ch(G). Furthermore this model structure is cofibrantly generated.
Proof.G is locally ~-presentable for some regular cardinal ~ by Proposition 3.10
of [Bek00]. Note A? = I is the class of injectives and (A, I) is the "injecti*
*ve
cotorsion pair". In order to use Theorem 4.12 we would like to say that A is a
~-Kaplansky class, but the author suspects that this is just not true. (If it w*
*ere,
then clearly every locally ~-generated Grothendieck category would be locally ~-
presentable. See the paragraph before Fact A.7 in Appendix A.) In any case there
is a trick: Pick ~0 to be a regular cardinal large enough so that (i) G is loc*
*ally
~0-presentable, (ii) the class of ~0-generated objects coincides with the class*
* of ~0-
presentable objects, and (iii) there is a ~0-presentable generator for G. Condi*
*tion (i)
is possible by FactA.8, condition (ii) is possible by Appendix B, and condition*
* (iii)
is possible by Fact A.2. Now using ~0and A we can easily check the conditions of
Theorem 4.12. Conditions (1) and (3) hold trivially and condition (2) holds by *
*our
choice of ~0. Finally if I 2 eI, it is a well-known fact that every chain map i*
*nto I is
null homotopic. Therefore dgAeis simply the class of all chain complexes in Ch(*
*G).
It is also clear that Ae= E, so condition (4) from Theorem 4.12 holds too. The
conclusion of Theorem 4.12 translates to the usual injective model structure wh*
*ere
the cofibrations are the monomorphisms and the fibrations are the epimorphisms
with dg-injective kernels.
It was in [ELR02 ] that the term "Kaplansky class" first appeared. The reason*
*ing
is that the class of projective modules (over a ring R) form a Kaplansky class *
*due to
Theorem 1 of [Kap58 ]. The details in interpreting Kaplansky's Theorem in terms
of our Definition 4.9 are in the proof of the next corollary.
Corollary 7.2. Let G = R-Mod where R is a commutative ring with 1 and let
P be the class of projective modules. Then P is a Kaplansky class and it induces
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 25
the usual projective model structure on Ch(G). Furthermore this model structure*
* is
cofibrantly generated and monoidal with respect to the usual tensor product of *
*chain
complexes.
Proof.First we note that G is locally finitely presentable. For a proof of this*
* fact,
see the footnote to Theorem 4.34 of [Lam99 ]. So G is locally ~-presentable for*
* every
regular cardinal ~ by Fact A.8.
We now show P is a Kaplansky class. Since projective modules are direct sum-
mands of free modules, it follows immediately from Theorem 1 of [Kap58 ] that e*
*ach
projective module is a direct sum of countably generated modules. We let ~ be a
regular cardinal with ~ > max { |R|, ! } and argue that P is a ~-Kaplansky clas*
*s.
So let X P 6= 0 where X is ~-generated and P is projective. Using Lemmas B.1
and B.2 we are done if we can find a directLsummand S of P which contains X and
for which |S| < ~. We start by writing P = i2ISi where each Si is a countably
generated (hence ~-generated) summand.LLet J = { j 2 I : Sj \ X 6= 0 }. Clearly
|J| < ~ since |X| < ~. Therefore S = j2JSj is the desired summand since
|S| < ~.
With the choice of ~ as above we will now check parts (1) - (4) of Theorem 4.*
*12.
We just verified property (1) and property (2) is clearly true since R 2 P is ~-
generated. Property (3) is true by Lemma 6.2 of [Hov02 ] since P is the left si*
*de of
a cotorsion pair. For (4) we use Corollary 3.13 of [Gil04]. Basic facts of proj*
*ective
modules show that P is a resolving class, that is, P is closed under taking ker*
*nels
of epimorphisms. Referring to Definition 3.11 of [Gil04] this says that the cot*
*orsion
pair (P, A) (where A is the class of all modules) is hereditary. Corollary 3.13
of [Gil04] now tells us that eP= dgPe\ E. Thus we have a cofibrantly generated
model structure on Ch (R) where the cofibrations are the monomorphisms with
dg-projective cokernels and the fibrations are the epimorphisms.
Now using Theorem 5.1 it is simple to check that the model structure is monoi*
*dal.
It is a standard fact that projective modules are flat. The tensor product of t*
*wo
projectives is again projective since Hom R(P1 R P2, -) ~=Hom R(P1, Hom R(P2-)
is an exact functor if Hom R(P1, -) and Hom R(P2, -) are exact. Finally, the un*
*it
for the tensor product is R which is projective.
Another monoidal model structure on Ch(R) which is induced by a Kaplansky
class is the flat model structure which first appeared in [Gil04].
Corollary 7.3. Let G = R-Mod where R is a commutative ring with 1 and let
F be the class of flat modules. Then F is a Kaplansky class and it induces a
model structure on Ch(G) called the flat model structure. The cofibrations are *
*the
monomorphisms with dg-flat cokernels and the fibrations are the epimorphisms wi*
*th
dg-cotorsion kernels. Furthermore this model structure is cofibrantly generated*
* and
monoidal with respect to the usual tensor product of chain complexes.
Proof.Again we note that G is locally finitely presentable. So it is locally ~-
presentable for every regular cardinal ~. We let ~ be a regular cardinal with ~*
* > |R|
and argue that F is a ~-Kaplansky class.
So let X F 6= 0 where X is ~-generated and F is flat. By Lemma B.1, |X| < ~.
For each x 2 X we use Lemma 2 of [BBE01 ] to findSa flat submodule Fx F with
x 2 Fx, |Fx| < ~ and F=Fx 2 F. The direct union x2X Fx contains X and is flat.
26 JAMES GILLESPIE
Furthermore [
F=( Fx) ~=limF=Fx
x2XS x2X
is flat. Finally, since | x2X Fx| < ~, Lemmas B.1 and B.2 allow us to conclude
that F is a ~-Kaplansky class.
This proves property (1) of Theorem 4.12 and property (2) is clearly true sin*
*ce
R is a flat R-module. Proposition XVI.3.1 of [Lan97] tells us that F is closed
under retracts (direct summands). It is also a standard fact about flat modules
that F is closed under extensions and direct limits. It follows that F is clos*
*ed
under transfinite extensions. So it is left to check property (4) of Theorem 4.*
*12. By
Proposition XVI.3.4 of [Lan97] F is closed under taking kernels of epimorphisms.
Therefore we can argue as in the proof of Corollary 7.2 that eF= dgFe\ E.
Thus we have a cofibrantly generated model structure on Ch(R) where the cofi-
brations are the monomorphisms with cokernels in dgFe(dg-flat complexes) and the
fibrations are the epimorphisms with kernels in dgCe(dg-cotorsion complexes).
Again it is easy to see that the model structure is monoidal using Theorem 5.*
*1.
The tensor product of two flat modules is clearly flat and the unit R is flat.
As described in [Gil06] the (monoidal) flat model structure of Corollary 7.3
generalizes to the category Ch(OX -Mod) of complexes of OX -modules where OX
is a ringed space. It too is obtained from a Kaplansky class as we will see nex*
*t in
Corollary 7.9. The first lemma below concerns cotorsion modules and skyscraper
sheaves. We now recall the concept of a skyscraper sheaf.
Let OX be a ringed space, p 2 X be a point, and M be an Op-module. The
skyscraper sheaf Sp(M) is the sheaf on X defined by U 7! M if p 2 U and U 7! 0
if p =2U. It is in fact an OX -module by viewing M as an O(U)-module via_the ri*
*ng
homomorphism O(U) -!Op. One can_check_that [Sp(M)]q = M for each q 2 { p }
and [Sp(M)]q = 0 for each q =2{ p }. It is also standard that Sp(-) is an exact
functor from the category Op-Mod to the category OX -Mod and is right adjoint to
the (also exact) "stalk functor" which sends a OX -module F to the Op-module Fp.
Therefore Sp(-) preserves injective objects by Proposition 2.3.10 of [Wei94].
Lemma 7.4. Let OX be a ringed space and p 2 X be a point. The skyscraper
functor Sp(-) preserves cotorsion objects.
Proof.First notice that for any OX -module F and any Op-module C we have an
isomorphism Extn(Fp, C) ~=Extn(F, Sp(C)). (This can be proved using the adjoint
relationship discussed above along with the fact that Sp(-) is exact and preser*
*ves
injective objects, and therefore preserves injective resolutions.)
Now suppose C is a cotorsion Op-module and F is a flat OX -module. We wish
to show that Sp(C) is a cotorsion OX -module. But Fp is a flat Op-module, so
Ext1(F, Sp(C)) ~=Ext1(Fp, C) = 0. Therefore Sp(C) is cotorsion.
In order to verify the first hypothesis of Theorem 4.12, we would like to find
a large enough cardinal ~ so that the ~-generated OX -modules coincide with the
~-presentable OX -modules and so that these notions may be used in place of car*
*di-
nality. Lemma 7.8 will allow us to do this. Our next lemma below however states
that the category of OX -modules has a set of flat generators. This is a standa*
*rd
fact but we document it now since it will be used in the proof of Lemma 7.8 and*
* is
needed for Corollary 7.9.
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 27
Let OX be a ringed space. For each open U X, extend O|U by 0 outside of U
to get a presheaf, denoted OU . Now sheafify to get an OX -module, which we will
denote j!(OU ).
Lemma 7.5. Let OX be a ringed space. Then Hom (j!(OU ), G) ~=G(U) for any
OX -module G and each open set U X. Furthermore, { j!(OU ) : U X } is a set
of flat generators for OX -Mod.
Proof.One can prove without much difficulty that Hom (OU , G) ~=G(U). So by the
universal property of sheafification we get Hom (j!(OU ), G) ~=G(U). It follows*
* at
once that the set forms a generating set since the modules j!(OU ) "pick out po*
*ints".
Also, each j!(OU ) is flat since [j!(OU )]p ~=(OU )p, which equals Op if p 2 U *
*and 0
if p 2 X\U.
Definition`7.6. We define the cardinality of a presheaf (or sheaf), F , to be |*
*F | =
| U X F (U)| where U X ranges over all the open sets in X.
In Lemmas 7.7 and 7.8 we let OX be our ringed space and let U represent the
set of all open sets U X. We note now that if fi is an infinite cardinal in w*
*hich
fi > |OX |, then automatically fi > |U|, since O(U) is nonempty for each U X.
Lemma 7.7. Let fi be an infinite cardinal such that fi > max { |X|, |OX | }. Now
let ~ = 2fi. If S is a presheaf of OX -modules and |S| < ~, then |S+ | < ~ wher*
*e S+
is the sheafification.
Proof.Clearly |Sp| < ~ for each point p 2 X. So by the sheafification construct*
*ion
(see the proof of Proposition-Definition II.1.2 of [Har77]) one can easily see *
*that for
each open U X we have |S+ (U)| < ~fi.`However, ~fi= (2fi)fi= 2(fi2)= 2fi= ~,
so |S+ (U)| < ~. Therefore |S+ | = | U X S+ (U)| < ~.
Next suppose we have a presheaf of OX -modules, S, and suppose we have a
subset a
W S(U)
U X
where again U X ranges over all the open sets in X. We set WU = W \ S(U).
The set W generates a presheaf S0 and it is given by
X X
S0(U) = rV,U([O(V )]w).
V Uw2WV
That is, S0(U) consists of all finite sums of the form
rV1,U(ae1w1) + rV2,U(ae2w2) + . .+.rVn,U(aenwn)
where
Vi U, wi2 WVi, aei2 O(Vi)
and rVi,U:Vi -!U are the restriction maps. It is straightforward to see that S0
is a presheaf and upon reflection it is clearly the smallest subpresheaf contai*
*ning
W . Furthermore, if ~ > |OX | is an infinite cardinal, and ~ > |W |, then we w*
*ill
have ~ > |S0|. If S was a sheaf of OX -modules to start with, then (S0)+ is the
OX -submodule generated by W which we will denote SW . In this case note that
if ~ = 2fiwhere fi > max { |X|, |OX | }, then by Lemma 7.7, ~ > |SW | whenever
~ > |W |.
28 JAMES GILLESPIE
Lemma 7.8. Let fi be an infinite cardinal such that fi > max{ |X|, |OX | }. Now*
* let
~ = 2fi. Also assume that ~ is large enough that each j!(OU ) is ~-generated. T*
*hen
the following are equivalent for an OX -module S.
(1) |S| < ~.
(2) S is ~-generated.
(3) S is ~-presentable.
P
Proof.(1) ) (2). We use Fact A.5. Suppose |S| < ~ and say S = i2ISi is a
~-directed union of OX -submodules. Note that we will be done if we can show th*
*at
for each open U X and each x 2 S(U), there exists an i 2 I such that x 2 Si(U*
*).
Indeed if this were true, then by the large choice of ~ and the fact that the u*
*nion
is ~-filtered, we would take the union of all such Si(U) to display S = Si for *
*some
particular i 2 I. (Note also that the assertion we wish to prove is not obvious*
* since
we must sheafify when taking the direct union.)
Now let x 2 S(U). From our choice of ~ we know that each j!(OU ) is ~-
generated and so the canonical map colimi2IHom(j!(OU ), Si) -! Hom (j!(OU ), S)
is an isomorphism. Now using Lemma 7.5 this translates to an isomorphism
colimi2ISi(U) ~= S(U). Through this isomorphism we see that x 2 Si(U) for
some i 2 I. `
(2) ) (1). Let W be the collection of all subsets W U X S(U) (where U
ranges over all open subsets of X) which satisfy |W | < ~. For each W 2 W, let
SW represent the OX -submodule generated by W . Then |SW | < ~ by LemmaP7.7.
Note that (W, ) is ~-filtered and in fact S is the ~-filtered union W2W SW *
*. By
Fact A.5, S = SW for some W 2 W. So |S| < ~.
(3) ) (2) is automatic. We now prove (2) ) (3), using that (1) iff (2). First
we point out that the category of OX -modules isPlocally ~-generated. Indeed ea*
*ch
S can be expressed as the ~-filtered union S = W2W SW where each SW is ~-
generated as in the last paragraph. Therefore, we may use the characterization
of ~-presentable objects in Fact A.12. Suppose S is ~-generated and T -! S is
an epimorphism with T a ~-generated OX -module. Then |T | < ~. So of course
| ker(T -! S)| < ~, which means ker(T -! S)is ~-generated. This proves S is
~-presentable.
Finally we prove that the flat model structure on Ch(OX -Mod) comes from the
Kaplansky class of flat OX -modules. The proof of Corollary 7.9 will again rely*
* on
a result of Enochs, et.al. which can be found in [EO01 ].
If S F is a subpresheaf (or subsheaf) we call S presheaf pure if S(U) is a *
*pure
O(U)-submodule of F (U) for each open U. We say S F is stalkwise pure if Sp is
a pure Op-submodule of Fp for each p 2 X.
Corollary 7.9. Let G = OX -Mod where OX is a sheaf of rings on a topological
space X and let F be the class of flat OX -modules. Then F is a Kaplansky class
and it induces a model structure on Ch(G) we call the flat model structure. The
cofibrations are the monomorphisms with dg-flat cokernels and the fibrations are
the epimorphisms with dg-cotorsion kernels. Furthermore this model structure is
cofibrantly generated and monoidal with respect to the usual tensor product of *
*chain
complexes.
Proof.Let U represent the set of all open sets U X. Let fi be an infinite car*
*dinal
such that fi > max{ |X|, |U|, |OX | }. Now let ~ = 2fi. Using Fact A.2 we can a*
*lso
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 29
assume that ~ is large enough that each j!(OU ) is ~-generated. We let F be the
class of flat sheaves and claim that F is a ~-Kaplansky class.
So let S0 F 6= 0 where S0is ~-generated and F is flat. By Lemma 7.8, |S0| < *
*~.
With minor adjustments to the proof of Proposition 2.4 of [EO01 ] we can find an
OX -submodule S F which is presheaf pure and such that S0 S and |S| < ~. It
follows that S S+ and S+ F is stalkwise pure. (This is not hard, but one co*
*uld
also see a proof in Lemma 4.7 of [Gil06].) It follows immediately that S+ and F*
*=S+
are flat. Furthermore, by Lemma 7.7 we have |S+ | < ~ and so by Lemma 7.8 we
have that S+ is ~-presentable. This completes the proof that F is a ~-Kaplansky
class.
Note that OX -Mod is locally ~-presentable by the same argument given in the
last paragraph of the proof of Lemma 6.8. So we are done verifying properties (*
*1)
and (2) of Theorem 4.12.
It is a standard fact that F is closed under extensions and direct limits. It*
* follows
that F is closed under transfinite extensions. It is also standard that F is cl*
*osed
under retracts. So we focus on proving property (4). We let Y be an exact dg-fl*
*at
complex of sheaves. We want to show that it is a flat complex. I.e. that is *
*has
flat cycles in each degree. Note that we will be done if we can show that the"s*
*talk
complex" Yp is a flat complex of Op-modules for each p 2 X. We know from the
proof of Corollary 7.3 that an exact dg-flat complex of Op-modules is a flat co*
*mplex.
So we will simply show that Yp is a dg-flat complex. So let Yp -!C be a morphism
where C is a cotorsion complex of Op-modules. (This means that C is exact and
each cycle is cotorsion.) We want to show this map is null homotopic. But using
the skyscraper functor and its adjoint property gives us a morphism Y -! Sp(C)
of complexes. Since Sp(-) is exact and preserves cotorsion modules by lemma 7.4
we see that Sp(C) is a cotorsion complex and by definition of dg-flat Y -! Sp(C*
*) is
null homotopic. From this we can deduce that Yp -!C is also null homotopic.
Thus we have a cofibrantly generated model structure on Ch(G) where the cofi-
brations are the monomorphisms with cokernels in dgFe(dg-flat complexes) and the
fibrations are the epimorphisms with kernels in dgCe(dg-cotorsion complexes).
We check now that the model structure is monoidal using Theorem 5.1. Again
this is easy. The tensor product of two flat sheaves is also flat. The unit i*
*s the
structure sheaf OX which is flat too.
Appendix A. locally presentable categories
Tibor Beke showed (Proposition 3.10 of [Bek00]) that Grothendieck categories
are locally presentable. Here we gather some properties of locally presentable *
*cate-
gories from the literature which will be needed for this paper. Virtually every*
*thing
here can be found scattered throughout [AR94 ] and [Sten75].
Throughout this section, we will assume all cardinals are regular. These are
infinite cardinalsPwhich are not the sum of a smaller number of smaller cardina*
*ls.
For example, @! = n |R|. Then M is ~-generated iff |M| < ~.
Proof.We use the characterization of ~-generated provided by Example A.
First, if |M| < ~, then M is clearly ~-generated since we can take the genera*
*ting
set to be M itself. Conversely,Psuppose M is ~-generated soPthat there exists a
setLS M for which M = x2SRx and |S| < ~. Then |M| = | x2SRx|
| x2SRx| < ~.
Lemma B.2. Let R be a ring and M an R-module. Let ~ be a regular cardinal
such that ~ > |R|. Then M is ~-generated iff M is ~-presentable.
Proof.Of course ~-presentable objects are always ~-generated, so it remains to
show that under the given hypothesis, ~-generated objects are ~-presentable. But
this is easy using Lemma B.1 and the fact that M is ~-presentable iff M is ~-
generated and every epimorphism N -!M with N ~-generated, has a ~-generated
kernel. Indeed suppose M is ~-generated and N -! M is an epimorphism with N
~-generated. Then |N| < ~. So of course | ker(N -!M) | < ~.
Next we can prove that the notion of ~-generated coincides with ~-presentable
in any Grothendieck category if we take ~ to be large enough. In the proof we u*
*se
the fact that an equivalence of categories preserves ~-generated and ~-presenta*
*ble
objects. Indeed if F :A -! B is an equivalence, then there exists G: B -! A
such that GF ~=1A and F G ~=1B. Furthermore, F is then both a left and right
adjoint of G (and so each preserves colimits). So if X 2 A is ~-presentable the*
*n the
functor A(X, G(-)) preserves ~-filtered colimits. But A(X, G(-)) ~=B(F (X), -),
so F (X) must be ~-presentable. On the other hand, if F (X) is ~-presentable,
then B(F (X), F (-)) preserves ~-filtered colimits. Therefore, B(F (X), F (-))*
* ~=
A(GF (X), -) preserves ~-filtered colimits and GF (X) ~=X must be ~-presentable.
The same argument (but with ~-filtered colimits of monomorphisms) shows that
an equivalence preserves ~-generated objects.
Proposition B.3. Let G be a Grothendieck category. Then there exists a regular
cardinal ~ for which the ~-generated objects coincide with the ~-presentable ob*
*jects.
Proof.By the Gabriel-Popescu Theorem ([Sten75]), G is equivalent to a subcatego*
*ry
of R-Mod for some ring R. (The Gabriel-Popescu Theorem says more than this, but
we don't need the whole statement.) Since an equivalence of categories preserve*
*s ~-
generated and ~-presentable objects, the result follows from Lemma B.2 by choos*
*ing
~ to be a regular cardinal ~ > |R|.
References
[AR94]J. Ad'amek and J. Rosick'y, Locally presentable and accessible categories*
*, Number 189 in
London Mathematical Society Lecture Note Series, Cambridge University Press, *
*1994.
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS 33
[Bek00]Tibor Beke, Sheafifiable homotopy model categories, Math. Proc. Camb. Ph*
*il. Soc.
vol.129, no.3, 2000, pp.447-475.
[BBE01]L. Bican, R. El Bashir and E. Enochs, All modules have flat covers, Bull*
*. London Math
Soc., vol. 33, no. 4, 2001, pp. 385-390.
[Bor94]Francis Borceux, Handbook of Categorical Algebra. 1-3, Encyclopedia of M*
*athematics and
its Applications 50-52. Cambridge University Press, Cambridge, 1994.
[Con00]Brian Conrad, Grothendieck dualilty and base change, Lecture Notes in Ma*
*thematics,
Vol. 1750, Springer-Verlag, 2000.
[EEGO]E. Enochs, S. Estrada, J.R. Garc'ia Rozas and L. Oyonarte, Flat covers in*
* the category
of quasi-coherent sheaves over the projective line, Comm. Algebra 32, 2004, p*
*p.1497-1508.
[EE05]E. Enochs, S. Estrada, Relative homological algebra in the category of qu*
*asi-coherent
sheaves, Advances in Mathematics vol. 194, no. 2, 2005, pp.284-295.
[ELR02]E. Enochs, J.A. L'opez-Ramos, Kaplansky classes, Rend. Sem. Mat. Univ. P*
*adova,
Vol. 107, 2002.
[EO01]E. Enochs, L Oyonarte, Flat covers and cotorsion envelopes of sheaves, Pr*
*oceedings of
the American Mathematical Society vol. 130, no. 5, 2001, pp.1285-1292.
[Gil04]James Gillespie, The flat model structure on Ch(R), Trans. Amer. Math. S*
*oc. vol. 356,
no. 8, 2004, pp. 3369-3390.
[Gil06]James Gillespie, The flat model structure on complexes of sheaves, Trans*
*. Amer. Math.
Soc. vol. 358, no. 7, 2006, pp. 2855-2874.
[Har77]Robin Hartshorne, Algebraic Geometry, Grauate Texts in Mathematics vol. *
*52, Springer-
Verlag, New York, 1977.
[Hov02]Mark Hovey, Cotorsion theories, model category structures, and represent*
*ation theory,
Mathematische Zeitschrift, vol. 241, 553-592, 2002.
[Hov99]Mark Hovey, Model categories, Mathematical Surveys and Monographs vol. 6*
*3, American
Mathematical Society, 1999.
[HPS97]M. Hovey, J. Palmieri, and N. Strickland, Axiomatic stable homotopy theo*
*ry, Mem.
Amer. Math. Soc., Vol. 128, no. 610, 1997.
[Kap58]I. Kaplansky, Projective modules, Annals of Mathematics, vol. 68, no. 2,*
* pp.372 - 377 ,
September 1958.
[Lam99]T.Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics v*
*ol. 189,
Springer-Verlag, New York, 1999.
[Lan97]S. Lang, Algebra, Addison-Wesley, third edition, 1997.
[Lip]J. Lipman, Notes on derived categories and derived functors, pr*
*eprint,
www.math.purdue.edu/"lipman.
[Lit82]Shigeru Litaka, Algebraic Geometry, Graduate Texts in Mathemamtics vol. *
*76, Springer-
Verlag, New York 1982.
[May1]J.P. May, Picard groups, Grothendieck rings, and Burnside rings of a cate*
*gory, Advances
in Mathemamtics, vol. 163, no. 1, 2001 pp.1-16.
[May3]J.P. May, The additivity of traces in triangulated categories, Advances i*
*n Mathemamtics,
vol. 163, no. 1, 2001 pp.34-73.
[Mur]D. Murfet, Derived categories of quasi-coherent sheaves, online at www.the*
*risingsea.org.
[Spa88]N. Spaltenstein, Resolutions of unbounded complexes, Compos. Math., vol.*
* 65, no. 2,
1988, pp. 121-154.
[Sten75]Bo Stenstr"om, Rings of Quotients, Die Grundlehren der mathematischen W*
*issenschaften
in Einzeldarstellungen Band 217, Springer-Verlag, New York 1975.
[Tar97]L.A. Tarr'io, A.J. L'opez, J. Lipman, Local homology and cohomology of s*
*chemes, Ann.
Sci. 'Ecole Norm. Sup. (4), vol. 30, no. 1, 1997, pp.1-39.
[TT90]R.W. Thomason, Thomas Trobaugh, Appendix B of Higher algebraic K-theory o*
*f schemes
and of derived categories, The Grothendieck Festschrift, Vol. III, pp.409-417*
*, Progr. Math.,
88, 1990
[Qui67]Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics no.*
* 43, Springer-
Verlag, 1967.
[Wei94]Charles A. Weibel, An Introduction to Homological Algebra, Cambridge Stu*
*dies in Ad-
vanced Mathematics vol. 38, Cambridge University Press, 1994.
4000 University Drive, Penn State McKeesport, McKeesport, PA 15132-7698
E-mail address, Jim Gillespie: jrg21@psu.edu