cohomology of quasi-coherent sheaves …cohomology of quasi-coherent sheaves 3 remark 1.5. the...

COHOMOLOGY OF QUASI-COHERENT SHEAVES VIA

MODEL CATEGORIES AND APPROXIMATION THEORY

JAN STOVICEK

Abstract. Short lecture notes for the doctoral course Cohomology ofquasi-coherent sheaves via model categories and approximation theorygiven at the University of Padova on March 26–29, 2012.

Contents

Foreword 11. Grothendieck categories 12. Quasi-coherent modules over diagrams of rings 43. Weak factorization systems 74. Cotorsion pairs in a Grothendieck category 135. The derived category of an abelian category 216. Homological algebra for complexes 217. Model categories 21Appendix A. Quasi-coherent modules versus sheaves 21References 21

Foreword

The course was intended as an overview of methods available to studycategories of quasi-coherent sheaves over schemes and the related homolog-ical algebra. The topic being rather broad, we had to make choices whatto include. Several proofs are either skipped or only outlined, with a ref-erence to literature when possible. Most remarkably, however, we do notinclude some standard definitions like those of schemes or quasi-coherentsheaves of modules over a scheme. On the other hand, this does not preventus from describing in detail what the categories of quasi-coherent sheaveslook like. The readers interested in the algebraic geometric background arekindly referred to very nice books, such as [5] or [12].

1. Grothendieck categories

As already mentioned, rather than giving the definition of what a quasi-coherent sheaf is, we rather focus on the important abstract properties whichsuch categories have. We start with the classical concepts of a abelian anda Grothendieck category. An example to keep in mind while reading thissection is the category Mod-R of all rightR-modules, whereR is a associativebut not necessarily commutative ring with unit.

Date: March 30, 2012.

1

2 JAN STOVICEK

Definition 1.1. A category A is called abelian provided that

(1) A is additive, that is A(X,Y ) is an abelian group for every pair ofobjects X,Y ∈ A, the composition of morphisms is bilinear, andfinite coproducts exist in A;

(2) every morphism f : X → Y has a kernel k : K → X and a cokernelc : X → C;

(3) for every f : X → Y , the cokernel of the kernel of f is canonicallyisomorphic to the kernel of the cokernel of f . In other words, weconsider the kernel k : K → X of f and the cokernel c : X → C ofk, and dually the cokernel c : X → C of f and the kernel k : K → Yof c. Then there is a unique morphism g : C → K, making thefollowing diagram commutative:

Xf //

c ��???

????

? Yc

��???

????

?

K

k??~~~~~~~~

C g// K

k

??~~~~~~~C

We require that g be an isomorphism

Remarks 1.2.

(1) Note that under the assumption (1) of Definition 1.1, A has alsofinite products. Moreover, for a finite collection (X1, . . . , Xn) ofobjects of A we have

n∏i=1

Xi∼=

n∐i=1

Xi.

(2) Condition (3) of Definition 1.1 allows us to identify C and K. That

is, we can without loss of generality assume that C = K, and thisobject is usually called the image of f (check what this means inMod-R!)

(3) Since we have kernels and images at our disposal, it makes sense tospeak of exact sequences in abelian categories. As a particular case,a diagram in A of the form

0 −→ Xi−→ Y

p−→ Z −→ 0

is called a short exact sequence whenever i is the kernel of p and atthe same time p is the cokernel of i.

There is a close connection among module categories and abelian cate-gories in terms of Mitchell’s full embedding theorem [11, Theorem 4.4]:

Theorem 1.3. Every small abelian category A admits a full and exact em-bedding F : A → Mod-R for some ring R.

Remark 1.4. If the abelian category is not small, we still can do the following:For any set of objects S, we can take the smallest full subcategory A′ ⊆ Awhich contains S and is closed under taking kernels and cokernels in A.That is, A′ is a skeletally small abelian subcategory of A and as such it isembeddable as a full and exact subcategory also to Mod-R for some ring R.

COHOMOLOGY OF QUASI-COHERENT SHEAVES 3

Remark 1.5. The theorem together with the latter observation is very im-portant since it implies that properties of module categories involving onlyfinite diagrams and existence of finitely many homomorphisms are satisfiedalso in an arbitrary abelian category. For example, the Snake Lemma, theFive Lemma or the 3× 3 Lemma all hold in all abelian categories.

Warning 1.6. The functor F from Theorem 1.3 may not (and typicallydoes not) preserve infinite limits or colimits. Thus, although a product ofinfinitely many short exact sequences is a short exact sequence in any modulecategory, one cannot use Theorem 1.6 to prove an analogous statement forgeneral abelian categories. There is a good reason for that: there are abeliancategories in which a product of infinitely many epimorphisms may not bean epimorphism, e.g. the ones in Examples 2.11 and 2.12.

Now we define Grothendieck categories:

Definition 1.7. An abelian category G is called a Grothendieck category if

(1) G has all set-indexed coproducts (equivalently: G is a cocompletecategory).

(2) G has exact limits. That is, given a direct system

(0 −→ Xjij−→ Yj

pj−→ Zj −→ 0)j∈I

of short exact sequence, then the colimit diagram

0 −→ lim−→j∈I Xj −→ lim−→j∈I Yj −→ lim−→j∈I Zj −→ 0

is again a short exact sequence in G. This is sometimes called theAB5 condition following an equivalent requirement in [6, p. 129].

(3) G has a generator. That is, there is an object G ∈ G such that every

X ∈ G admits an epimorphism G(I) → X → 0. Here, G(I) standsfor a coproduct

∐j∈I Gj of copies Gj of G.

An important property of a Grothendieck category is that it always hasenough injective objects, which is very good from the point of view of ho-mological algebra. This is in fact a reason to look at infinitely generatedmodules or sheaves of infinitely generated modules: injective objects are of-ten always infinitely generated in any reasonable sense. We summarize thecomment in a theorem:

Theorem 1.8. Let G be a Grothendieck category. Then each X ∈ G admitsan injective envelope X → E(X). Moreover, G admits all set-indexed prod-ucts (equivalently: it is complete) and has an injective cogenerator C. Thatis, C is injective in G and each X ∈ G admits a monomorphism of the form0→ X → CI .

Proof. The fact that G every object X ∈ G admits an monomorphism 0 →X → E with E injective was showed already in [6, Theoreme 1.10.1]. Theexistence of injective envelopes and an injective cogenerator is proved in [11,Theorem 2.9] and [11, Corollary 2.11], respectively. The fact that G hasproducts and many other properties of G are clear from the Popescu-Gabrieltheorem, see e.g. [14, Theorem X.4.1]. �

4 JAN STOVICEK

2. Quasi-coherent modules over diagrams of rings

The simplest examples of Grothendieck categories are module categoriesG = Mod-R. In this section, we construct more complicated examples,involving diagrams of rings and diagrams of modules over these rings. Infact, for suitable choices we obtain a category equivalent to the categoryof quasi-coherent sheaves over any given scheme. The presentation is asomewhat adjusted version of [2, §2]. Since the discussion in [2, §2] is ratherbrief and many details are omitted, we also discuss the translation betweenquasi-coherent sheaves and the Grothendieck categories we describe belowin Appendix A.

We start the discussion of our examples with a definition.

Definition 2.1. Let (I,≤) be a partially ordered set. Then a representationR of the poset I in the category of rings is given by the following data:

(1) for every i ∈ I, we have a ring R(i),(2) for every i < j, we have a ring homomorphism rij : R(i)→ R(j), and

(3) we require that for every triple i < j < k, the morphism Rik : R(i)→R(k) coincides with the composition Rjk ◦R

ij .

Remark 2.2. If we view I as a thin category, whose objects are elements ofI and where there is exactly one morphism i→ j if i ≤ j and no morphismsotherwise, then R is none other than a functor

R : I −→ Rings.

Remark 2.3. Although all of our examples and the geometrically mindedmotivation will involve only representations of posets in the category ofcommutative rings, non-commutative rings can be potentially useful too.For instance, one can consider sheaves of algebras of differential operatorsand ring representations coming from them. In any case, the commutativityis not necessary for the very basic properties which we discuss in this section,so we do not include it in our definitions either.

Having defined representations of I in the category of rings, we can definemodules over such representations in a straightforward manner.

Definition 2.4. Let R be a representation of a poset I in the category ofrings. A right R-module is

(1) a collection (M(i))i∈I , where M(i) ∈ Mod-R(i) for each i ∈ I(2) together with a morphisms of the additive groups M i

j : M(i)→M(j)for each i < j

(3) satisfying the compatibility condition M ik = M j

k ◦Mij for every triple

i < j < k, and such that(4) the ring actions are respected in the following way: Given x ∈ R(i)

and m ∈M(i) for i ∈ I, then for any j ≥ i we have the equality

M ij(m · x) = M i

j(m) ·Rij(x).


Remark 2.5. Another way to express condition (3) is that for each i < j,the following diagram commutes

M(i)⊗R(i)mult.−−−−→ M(i)

M ij⊗Rij

y yM ij

M(j)⊗R(j)mult.−−−−→ M(j)

In general, the tensor product is over the ring of integers, but if there isanother base ring k (i.e. k is commutative, each R(i) is a k-algebra and allRij and M i

j are k-linear), we can use the tensor product over k as well.

All our modules in the section are going to be right modules, so we willomit the adjective “right”. In order to obtain a category, it only remains totell what a morphism of R-modules is.

Definition 2.6. Let R be a representation of a poset I in the category ofrings and M,N be R-modules. A morphism f : M → N is a collection of(f(i) : M(i)→ N(i))i∈I , where f(i) is a morphism of R(i)-modules for everyi ∈ I and the square

M(i)f(i)−−−−→ N(i)

M ij

y yN ij

M(j)f(i)−−−−→ N(j)

commuted for every i < j.

Let us denote the category of all R-modules by Mod-R. As we quicklyobserve:

Proposition 2.7. Let (I,≤) be a poset and R a representation of I inthe category of rings. Then Mod-R is a Grothendieck category. Moreoverlimits and colimits of diagrams of modules are computed component wise—we compute the corresponding (co)limit in Mod-R(i) for each i ∈ I andconnect these by the (co)limit morphisms.

Proof. Everything is very easy to check except for the existence of a gener-ator in Mod-R. In fact, there is a generating set {Pi | i ∈ I} of projectivemodules described as follows:

Pi(j) =

{R(j) if j ≥ i,0 otherwise

and the homomorphism Pi(j) → Pi(k) for j < k either coincides withR(j)→ R(k) if i ≤ j < k or vanishes otherwise.

One directly checks that a morphism f : Pi → M is determined by thevalue of f(i)(1R(i)) ∈ M(i) on one hand and every choice of the valuef(i)(1R(i)) yields a morphism f : Pi → M on the other hand. The argu-ment is similar to the one for the Yoneda lemma. Hence,

HomR(Pi,M) ∼= M(i) for each i ∈ I and M ∈ Mod-R

6 JAN STOVICEK

and the canonical homomorphism∐i∈I

P(M(i))i −→M

is surjective for every M ∈ Mod-R. Thus, G =∐i∈I Pi is a projective

generator. �

Although being valid Grothendieck categories, the categories Mod-R asabove are not the categories of our interest yet. In order to get a descriptionof categories of quasi-coherent sheaves as promised, we must consider cer-tain full subcategories instead. In order for this to work, we need an extracondition on R:

Definition 2.8. Let R be a representation of a poset I in rings. We callR a flat representation if for each pair i < j in I, the ring homomorphismRij : R(i)→ R(j) gives R(i) the structure if a flat left R(i)-module. That is,

−⊗R (i)R(j) : Mod-R(i) −→ Mod-R(j)

is an exact functor.

As discussed in Appendix A, the representation coming from structuresheaves of schemes always satisfy this condition. For such an R, we cansingle out the modules we are interested in:

Definition 2.9. Let R be a flat representation of I in rings. A moduleM ∈ Mod-R is called quasi-coherent, if for every i < j, the R(j)-modulehomomorphism

M(i)⊗R(i) R(j) −→M(j)

m⊗ x 7−→ m · xis an isomorphism.

Denote the full subcategory of Mod-R formed by quasi-coherentR-modulesby Qco(R).

Again, we obtain a Grothendieck category.

Theorem 2.10. Let (I,≤) be a poset and R a flat representation of I inthe category of rings. Then Qco(R) is a Grothendieck category. Moreovercolimits of diagrams and limits of finite diagrams are computed componentwise—that is, for each i ∈ I separately.

Proof. Again, the main task is to prove that Qco(R) has a generator andthe rest is rather easy, since taking colimits and kernels (hence also finitelimits) commutes with the tensor products −⊗R(i) R(j), where i, j ∈ I andi ≤ j. The fact that a generator exists follows from [2, Corollary 3.5], whoseproof is rather technical. �

Now we exhibit particular examples of flat representations of posets ofgeometric origin and quasi-coherent modules over them.

Example 2.11. Consider the three element poset be given by the Hasse dia-gram

• −−−−→ • ←−−−− •


and a representation in the category of rings of the form

R : k[x]⊆−−−−→ k[x, x−1]

⊇←−−−− k[x−1],

where k is an arbitrary commutative ring. Clearly R is a flat representationsince the inclusions are localization morphisms.

For each n ∈ Z, we have a quasi-coherent R-module

O(n) : k[x]⊆−−−−→ k[x, x−1]

xn·−←−−−− k[x−1].

One can easily check that O(m) 6∼= O(n) whenever m 6= n, since by directcomputation HomR(O(m),O(n)) = 0 for m > n.

In fact, the category Qco(R) is equivalent to the category of quasi-coherentsheaves over P1

k, the projective line over k.

Example 2.12. Given a commutative ring k, let us now show a flat represen-tation of a poset corresponding to the scheme P2

k, the projective plane overk. The Hasse diagram of the poset has the following shape:

•

��@@@

@@@@

''OOOOOOOOOOOOOO •

��~~~~

~~~

��@@@

@@@@

•

��~~~~

~~~

wwoooooooooooooo

•

��@@@

@@@@

•

��

•

��~~~~

~~~

•To describe the representation R in the category of rings corresponding

to P2k, it is enough to define the ring homomorphisms corresponding to

arrows in the Hasse diagram. Such a description is given in the followingdiagram, where all the rings are subrings of k(x0, x1, x2), the ring of rationalfunctions in three indeterminates over k, and all the ring homomorphismsare inclusions:

k[x1x0 ,x2x0

]

((QQQQQQQQQQQQ

++XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX k[x0x1 ,x2x1

]

vvmmmmmmmmmmmm

((QQQQQQQQQQQQk[x0x2 ,

x1x2

]

vvmmmmmmmmmmmm

ssffffffffffffffffffffffffffffff

k[x0x1 , (x2x1

)±1]

((QQQQQQQQQQQQk[x1x2 , (

x0x2

)±1]

��

k[x2x0 , (x1x0

)±1]

vvmmmmmmmmmmmm

k[(x1x0 )±1, (x2x0 )±1]

3. Weak factorization systems

The coming section deals with rather abstract category theory which willbe of use later on in the construction of cotorsion pairs and model structures.The highlight is a version of Quillen’s small object argument, specialized toGrothendieck categories. We omit several technical steps in proofs and referto the monographs [8] and [7]. Although all the arguments are included inthese references, the term “weak factorization system” itself is not. This issince the term and some notation has been taken from [1].

We start with an orthogonality relation on morphisms in an arbitrarycategory C.

8 JAN STOVICEK

Definition 3.1. Given morphisms f : A→ B and g : X → Y in C, we writef � g if for any commutative square given by the solid arrows

A //

f

��

X

g

��B //

??

Y,

a morphism depicted by the diagonal dotted arrow exists such that both thetriangles commute. We stress that we require only existence, not uniquenessof such a morphism.

Given f , g such that f � g, we say that f has the left lifting property forg and g has the right lifting property for f .

Now we can define the central concept of the section.

Definition 3.2. Let C be a category and (L,R) be a pair of classes ofmorphisms in C. We say that (L,R) is a weak factorization system if

(1) L and R are closed under retracts. That is, given any commutativediagram

A −−−−→ X −−−−→ A

h

y f

y h

yB −−−−→ Y −−−−→ B

such that f ∈ L and the rows compose to the identity morphisms,then h ∈ L as well. We require the same for R.

(2) f � g for all f ∈ L and g ∈ R.(3) For every morphism h : X → Y in C, there is a factorization

Xh //

f @@@

@@@@

Y

Z

g

??~~~~~~~

with f ∈ L and g ∈ R.

Remark 3.3. Although the factorization as in (3) is typically not unique inthe cases we are concerned with, we can “compare” different factorizationsusing the lifting property. Namely, given h = gf = gf such that f, f ∈ Land g, g ∈ R, Definition 3.2(2) ensures that there is a morphism depictedby the dotted arrow making the diagram commutative:

Z g

$$

��

X

f11

f --

Y

Zg

;;

Example 3.4. A well-known example of a weak factorization system in anabelian category A is (E ,M), where E is the class of all epimorphisms and


M is the class of all monomorphisms. However, this example is rather mis-leading in our context. In Section 4, we will see weak factorization systems(L,R), where rather L is a class of monomorphisms and R is a class ofepimorphisms.

Another fact is that (E ,M) satisfies a stronger version of the lifting prop-erty from Definition 3.1: the diagonal morphism is unique for every pairf ∈ E and g ∈ M, ensuring in view of Remark 3.3 that the factorizationsas in Definition 3.2(3) are also unique. This is a strong property which ourweak factorization systems typically do not enjoy.

A relatively easy observation concerning weak factorization systems is,that the two classes of morphisms determine each other.

Lemma 3.5. Let (L,R) be a weak factorization system in a category C.Then

L = {f | f � g for all g ∈ R} and R = {g | f � g for all f ∈ L}.

Proof. Clearly L ⊆ {f | f � g for all g ∈ R} and we must prove the otherinclusion. Take any h : X → Y such that h� g for all g ∈ R and consider afactorization

Xh //

f @@@

@@@@

Y

Z

g

??~~~~~~~

with f ∈ L and g ∈ R. Since h� g, the dotted arrow making the followingdiagram commutative exists

Xf //

h��

Z

g

��Y

>>

s

Y

Placing the morphisms we have considered so far into the following diagram

X X X

h

y f

y h

yY

s−−−−→ Zg−−−−→ Y,

we observe that gs = 1Y . Thus, h is a retract of f and as such it mustbelong to L.

The argument for R = {g | f � g for all f ∈ L} is dual. �

Before proceeding further, we must define what is meant by transfinitecompositions.

Definition 3.6. Let λ be an ordinal number and (Xα, fαβ)α<β<λ be a directsystem indexed by λ in a category C:

X0f01 //

f02

66

f03

88

f0ω

88

f0,ω+1

88X1f12 // X2

f23 // X3// · · · // Xω

fω,ω+1// Xω+1// · · ·

10 JAN STOVICEK

Such a system is called a λ-sequence if for each limit ordinal µ < λ, theobject Xµ together with the morphisms fαµ : Xα → Xµ, α < µ, is a colimitof the direct subsystem (Xα, fαβ)α<β<µ. From now on, if we are going todepict a λ-sequence, we are going to draw only the morphism of the formfα,α+1.

The composition of the λ-sequence is just the colimit morphism

X0 −→ lim−→α<λXα.

Finally, if I is a class of morphisms of C, then a transfinite composition ofmorphisms of I is defined as the composition of a λ-sequence (Xα, fαβ)α<β<λwith fα,α+1 ∈ I for every α+ 1 < λ.

Now we can deduce closure properties of the left orthogonal of a class ofmorphisms with respect to � . Note that in particular the left hand sideclass of any weak factorization system (L,R) has these properties.

Lemma 3.7. Let R be a class of morphisms in a category C and denote

L = {f | f � g for each g ∈ R}.

Then the following hold for L:

(1) L is closed under pushouts. That is, if we are given a diagram

A −−−−→ A

f

yB

with f ∈ L and if the pushout

A −−−−→ A

f

y yfB −−−−→ B

exists in C, then also f ∈ L.(2) L is closed under transfinite compositions. That is, given a λ-

sequence (Xα, fαβ)α<β<λ with fα,α+1 ∈ L for every α + 1 < λ, thecomposition X0 → lim−→α<λ

Xα, if it exists, belongs to L, too.

Proof. This is an easy exercise in manipulating with colimits. �

Inspired by the previous lemma, let us state one more definition.

Definition 3.8. Given a set I of morphisms of a category C, we define arelative I-cell complex as a transfinite composition of pushouts of morphismsfrom I. The class of all relative I-cell complexes will be denoted by I-cell.

We are now ready to state and outline the proof of the highlight of thesection—Quillen’s small object argument. Our version is concerned onlywith Grothendieck categories, but argument works in much broader gener-ality. We encourage the reader to consult [8] or [7] for more details.


Theorem 3.9. Let G be a Grothendieck category and let I be a set (not aproper class!) of morphisms in G. Put

R = {g | f � g for all f ∈ I},L = {f | f � g for all g ∈ R}.

Then (L,R) is a weak factorization system in G and L consists precisely ofretracts of relative I-cell complexes.

Proof. Here we give only a part of the proof. We refer to [8, Theorem 2.1.14]or [7, Proposition 10.5.16] for the missing technical parts.

Clearly, conditions (1) and (2) of Definition 3.2 are satisfied for (L,R), sowe only have to prove condition (3). That is, given an arbitrary morphismh : X → Y in G, we must find a factorization

Xh //

f @@@

@@@@

Y

Z

g

??~~~~~~~

with f ∈ L and g ∈ R. In order to do that, we construct for a sufficientlylarge regular cardinal λ a λ-sequence

X = E0 //E1 //E2 // · · · //Eω //Eω+1 // · · ·together with morphisms gα : Eα → Y such that the following is satisfied:

(1) g0 = h;(2) the diagram (involving also all the morphisms in the λ-sequence

which are not depicted)

X //

h''PPPPPPPPPPPPPPP E1 //

g1

AAA

AAAA

A E2 //

g2

��

· · · //Eω //

gω

vvnnnnnnnnnnnnnnn Eω+1 //

gω+1

tthhhhhhhhhhhhhhhhhhhhhhh · · ·

Y

commutes; and(3) for each α + 1 < λ, the morphisms Eα → Eα+1 is a pushout of a

coproduct∐fi :

∐Ai →

∐Bi of a collection (fi : Ai → Bi)i∈I of

morphisms from I.

Note that if we manage to construct such a system, a few standard obser-vations apply. Firstly, the pushout of a coproduct of maps from I is a relativeI-cell complex by [8, Lemma 2.1.13]. Secondly, a transfinite composition ofrelative I-cell complexes is again a relative I-cell complex; see [8, Lemma2.1.12]. Hence, the composition f : E0 → lim−→α<λ

Eα of our λ-sequence will

be a relative I-cell complex. If we are lucky and the colimit map

g : lim−→α<λ−→ Y

of the collection of morphisms gα : Eα → Y , α < λ, belongs to R, we aredone since then we have the factorization

Xh //

f $$IIIII

IIIII Y

lim−→α<λEα

g

::uuuuuuuuuu

12 JAN STOVICEK

with f ∈ I-cell and g ∈ R and by Lemma 3.7 we know that I-cell ⊆ L.Now we are going to explain how to perform the construction of the

λ-sequence (Eα)α<λ so that we indeed obtain g : lim−→α<λEα → Y which

belongs to R. We will first discuss the construction of the morphisms Eα →Eα+1 and then the choice of λ.

The morphisms gα are constructed inductively. We put g0 = h as requiredand at limit ordinals we take the colimits of the previously constructedmorphisms. At an ordinal successor α + 1 < λ, suppose we have alreadyconstructed gα : Eα → Y . We consider all commutative squares of the form

Ai −−−−→ Eα

fi

y ygαBi −−−−→ Y

with Ai → Bi in I. Since I is a set, all these squares form a set as well.Hence we can take the coproduct of the all the left vertical maps in all thesquares and consider the coproduct square:∐

iAi −−−−→ Eα∐i fi

y ygα∐iBi −−−−→ Y

The object Eα+1 and the morphism Eα → Eα+1 is defined by the pushoutsquare: ∐

iAi −−−−→ Eα∐i fi

y y∐iBi −−−−→ Eα+1

The pushout property allows us to define gα+1 : Eα+1 → Y as the uniquecompletion of the following diagram:∐

iAi∐i fi��

// Eα

��

// Eα+1

gα+1

||∐iBi

//

66lllllllllllllllY

Regarding λ, it suffices to choose its value to be a regular cardinal so thatthe domains A of all morphisms f : A→ B in I are λ-small. In this context,this means that the representable functors

G(A,−) : G −→ Ab

commute with colimits of λ-sequences; see [8, Theorem 2.1.14] or [7, Propo-sition 10.5.16]. For a Grothendieck category, we can always find such λ,since the Popescu-Gabriel theorem [14, Theorem X.4.1] guarantees that fora sufficiently big regular cardinal λ,

(1) G identifies with a full subcategory of Mod-R for some ring R suchthat G is closed in Mod-R under taking colimits of λ-sequences, and


(2) under this identification, all the domains A of f : A→ B with f ∈ Ican be given by fewer than λ generators and relations as R-modules,so that the functors

HomR(A,−) : Mod-R −→ Ab

commute with colimits of λ-sequences.

In particular, the assumptions of [8, Theorem 2.1.14] and [7, Proposition10.5.16] are satisfied and if we construct the λ-sequence (Eα)α<λ togetherwith the maps gα : Eα → Y following the recipe above, the colimit mapg : lim−→α<λ

Eα → Y will belong to R.

Finally, we must show that the morphisms in L are precisely retracts ofrelative I-cell complexes. To see that, consider any h ∈ L and a factorization

Xh //

f @@@

@@@@

Y

Z

g

??~~~~~~~

with f ∈ I-cell and g ∈ R as constructed above. Since h� g, a similarargument as in the proof of Lemma 3.5 applies to show that h is a retractof f . This concludes the proof. �

Remark 3.10. The factorization of morphisms with respect to (L,R) fromTheorem 3.9 is typically not unique. However, it can be made functorial.Namely, observe that the cardinal λ in the construction does not dependon the morphism h which we want to factor—it depends only on the set I.That is, we can use the same λ for factorizing all morphisms in G and it isnot hard to show that in this case the factorization actually gives a functor

MorG −→ L×R

(Xh→ Y ) 7−→ (X

f→ lim−→α<λEα, lim−→α<λ

Eαg→ Y )

Here, MorG is the category whose objects are morphisms of G and whosemorphisms are commutative squares, and we view L and R as full subcate-gories of MorG.

However, the choice of a suitable λ was not unique, and for differentchoices of λ we may get different functors. Thus, the functoriality of thefactorization is non-canonical, and although considering functorial factor-izations sometimes simplifies ones arguments, the usefulness of this kind offunctoriality seems to be limited in practice.

4. Cotorsion pairs in a Grothendieck category

In order to define cotorsion pairs, we need to have Ext functors at our dis-posal. Therefore, we will first briefly recall two approaches to define them.For the first approach we will assume that our category is a Grothendieckcategory, the second is completely general and works for any abelian cate-gory.

14 JAN STOVICEK

Definition of Ext using injective resolutions. Let G be a Grothendieckcategory and X,Y ∈ G be objects. Consider and exact sequence

0 −→ Y −→ E0Y −→ E1Y −→ E2Y −→ · · ·with EnY injective for all n ∈ Z. When applying the functor G(X,−) tothis sequence and erasing the term G(X,Y ), we get a sequence of abeliangroups

0d−1

−→ G(X,E0Y )d0−→ G(X,E1Y )

d1−→ G(X,E2Y )d2−→ · · ·

This sequence is in general not exact, but it is a complex. That is, dn◦dn−1 =0 for all n ≥ 0. For each n ≥ 0 we can now define the Ext group as

ExtnG(X,Y ) = Ker dn/ Im dn−1.

The Yoneda Ext. Given an abelian category A, let E(X,Y ) be the classof all exact sequences in A which are of the form

ε : 0 −−−−→ Y −−−−→ E1 −−−−→ · · · −−−−→ En −−−−→ X −−−−→ 0.

We define an equivalence relation ∼ on En(X,Y ) by relating ε ∼ ε wheneverwe have a commutative diagram of the form

ε : 0 −−−−→ Y −−−−→ E1 −−−−→ · · · −−−−→ En −−−−→ X −−−−→ 0∥∥∥ y y ∥∥∥ε : 0 −−−−→ Y −−−−→ E1 −−−−→ · · · −−−−→ En −−−−→ X −−−−→ 0

and taking the symmetric and transitive closure.The n-th Yoneda Ext of X and Y is defined as En(X,Y )/ ∼. Although it

is not a priori clear whether the Yoneda Ext is a set (and indeed in generalit may be a proper class even for n = 1, see [3, Exercise 1, p. 131]), we cannevertheless always give En(X,Y )/ ∼ a structure of an abelian group usingthe so-called Baer sums. Considering the case n = 1, the zero element ofE1(X,Y )/ ∼ with respect to this group structure is precisely the class of allsplit short exact sequences.

We refer to [10, Chapter III and §XII.5] for details and further reading.

For a Grothendieck category G, the two definitions of Ext coincide andthey give additive functors. We state this in a theorem, for a proof of whichwe again refer to [10, Chapter III]:

Theorem 4.1. Let G be a Grothendieck category.

(1) For given X,Y ∈ G and n ≥ 0, the group ExtnG is independent of thechoice of the injective resolution for Y , and in fact

ExtnG : Gop × G −→ Ab

is an additive functor.(2) The Yoneda Ext for n ≥ 1 is also an additive functor

En(−,−)/ ∼ : Gop × G −→ Ab,

and it is naturally isomorphic to the functor ExtnG.

(3) There is a natural isomorphism Ext0G(X,Y ) ∼= G(X,Y ) for each

X,Y ∈ G.

Let us now define the central concepts of this section.


Definition 4.2. Let G be a Grothendieck category. For a class S of objectsof G we define

S⊥ = {B ∈ G | Ext1G(S,B) = 0 for all S ∈ S},

⊥S = {A ∈ G | Ext1G(A,S) = 0 for all S ∈ S}.

A pair (A,B) of full subcategories of G is called a cotorsion pair providedthat

A = ⊥B and A⊥ = B.Finally, a cotorsion pair (A,B) is said to be complete if every X ∈ G

admits so-called approximation sequences; that is, short exact sequences ofthe form

0 −→ X −→ BX −→ AX −→ 0 and 0 −→ BX −→ AX −→ X −→ 0

with AX , AX ∈ A and BX , B

X ∈ B.

Example 4.3. In every Grothendieck category G, there is always a trivialcomplete cotorsion pair. Namely, denote by Inj the full subcategory of allinjective objects. Then (G, Inj) is a complete cotorsion pair, where for givenX ∈ G we have the following approximation sequences:

0 −→ X −→ EX −→ EX/X −→ 0 and 0 −→ 0 −→ X −→ X −→ 0.

We denote by X → EX an injective envelope of X (cf. Theorem 1.8).

We will employ Quillen’s small object argument (Theorem 3.9) and showthat there is a general construction giving plentitude of complete cotorsionpairs, but we first need a few preparatory statements and definitions. Westart with formalizing transfinite extensions in a similar way as we formalizedtransfinite compositions in Definition 3.6. In fact, a specialization of thefollowing definition to module categories is equivalent to the concept of afiltration from [4, Definition 3.1.1].

Definition 4.4. Let S be a class of objects in a Grothendieck category. Byan S-filtration we mean a λ-sequence

0 = X0f01 //X1

f12 //X2f23 //X3

// · · · //Xωfω,ω+1//Xω+1

// · · ·

such that X0 = 0 and for each α+1 < λ, the morphism fα,α+1 is a monomor-phism whose cokernel belongs to S. That is, we have short exact sequences

0 −→ Xαfα,α+1−→ Xα+1 −→ Sα −→ 0

with Sα ∈ S. Informally, an S-filtration is just a transfinite extension ofobjects from S.

An object X ∈ G is called S-filtered if 0 → X is the composition (in thesense of Definition 3.6) of some S-filtration.

There is an analogue of Lemma 3.7(2) for the orthogonality with respectto Ext1

G . For modules, the corresponding result is called the Eklof lemma,see [4, Lemma 3.1.2]. We will present a rather different proof, however,which is taken from [13, Proposition 2.12].

16 JAN STOVICEK

Proposition 4.5 (Eklof lemma). Let B be a class of objects in a Grothendieckcategory G and denote A = ⊥B. Then A is closed under filtrations. That is,any A-filtered object belongs to A.

Proof. Consider an A-filtered object X and Y ∈ B. We must prove that anyfixed extension

ε : 0 −→ Yj−→ E

p−→ X −→ 0

splits. Let us also fix an S-filtration

0 = X0f01 //X1

f12 //X2f23 //X3

// · · · //Xωfω,ω+1//Xω+1

// · · ·

for X and let λ be the ordinal by which this filtration is indexed. To facilitatethe notation, we put Xλ = X and denote the colimit morphisms Xα → Xλ =X by fαλ. That is, we consider X as the last term in the filtration.

For each α ≤ λ we construct the pullback short exact sequence

εα : 0 −−−−→ Y −−−−→ Eα −−−−→ Xα −−−−→ 0∥∥∥ jαλ

y yfαλε : 0 −−−−→ Y

j−−−−→ Ep−−−−→ X −−−−→ 0

Morally, εα should be thought of as

0 −→ Y −→ p−1(Xα) −→ Xα −→ 0,

since Xα is a subobject of X. If G = Mod-R for a ring, this will be indeedthe case.

Notice that the pullback diagrams

εα : 0 −−−−→ Y −−−−→ Eα −−−−→ Xα −−−−→ 0∥∥∥ jαβ

y yfαβεβ : 0 −−−−→ Y −−−−→ Eβ −−−−→ Xβ −−−−→ 0

for α < β ≤ λ make (εα)α≤λ into a direct system of short exact sequences.Even better, the exactness of direct limits in G implies that (εα)α≤λ is a(λ+1)-sequence of short exact sequences. Indeed, one can see that from thefact that the pullback objects Eα can be computed as the kernels of

(p, fαλ) : E ⊕Xα −→ X

and that taking kernels is exact. Finally, observe that ελ = ε and, up toisomorphism, ε0 is of the form

ε0 : 0 −→ Y1Y−→ Y −→ 0 −→ 0.

Now we will inductively construct a collection of morphisms gα : Eα → Ysuch that g0 = 1Y and for each α < β ≤ λ, the triangle

Eαjαβ //

gα

��

Eβ

gβ

}}||||

||||

Y


commutes. For α = 0 and β = λ, this precisely says that ε splits since thenj0λ = j and we have a commutative diagram

0 // Yj //

1Y��

Ep //

gλ~~~~

~~~~

~X // 0

Y

Thus, once we have succeed with the construction, we are finished with theproof.

Regarding the induction, we put g0 = 1Y as required and at limit steps,we construct the gα as the colimit map of (gγ)γ<α. For ordinal successorsα + 1 ≤ λ, suppose we have constructed gα : Eα → Y . Notice that by theconstructions above we have a pullback diagram of the form

0 −−−−→ Eαjα,α+1−−−−→ Eα+1 −−−−→ Aα −−−−→ 0y y ∥∥∥

0 −−−−→ Xαfα,α+1−−−−→ Xα+1 −−−−→ Aα −−−−→ 0

where Aα ∈ A since the morphism fα,α+1 comes from an S-filtration of X.If we consider the pushout of

0 −→ Eαjα,α+1−→ Eα+1 −→ Aα −→ 0

along the map gα : Eα → Y , we get an exact sequence of the form

0 −→ Y −→ E −→ Aα −→ 0.

Such a sequence must split since we assume Ext1G(Aα, Y ) = 0 as Aα ∈ A

and Y ∈ B. It is a standard and easily seen fact that then there must exista factorization

0 // Eαjα,α+1//

gα

��

Eα+1//

||

Aα // 0

Y

It only remains to call the dotted morphism gα+1. �

One result is due before turning to the main result of the section. Namely,we need to make a first step to relate weak factorization systems to cotorsionpairs. We will tell more about this relation in Section 7.

Proposition 4.6. Let G be a Grothendieck category with a generator G,and let S be a set of objects of G. Then there exists a set I of morphismsof G with the following properties:

(1) Each f ∈ I is a monomorphism with a cokernel in S and it is of the

form f : K → G(I). In other words, there is a short exact sequence

0 −→ Kf−→ G(I) −→ S −→ 0 with S ∈ S.

18 JAN STOVICEK

(2) Every morphism h : X → Y in G which is a monomorphism with acokernel in S is a pushout of some f ∈ I. In other words, for eachsuch h there exists f ∈ I so that we have a commutative diagramwith exact rows of the form

0 −−−−→ Kf−−−−→ G(I) −−−−→ S −−−−→ 0y y ∥∥∥

0 −−−−→ Xh−−−−→ Y −−−−→ S −−−−→ 0

In particular, I-cell consists precisely of monomorphisms with an S-filteredcokernel.

Proof. Note that if we have I with properties (1) and (2), the description ofrelative I-cell complexes is a straightforward consequence of Definition 3.8and properties of pushouts in abelian categories. Thus, we will focus onlyon constructing I with the required properties.

In fact, we give a direct construction of I and then prove it has theproperties we require. For any S ∈ S, consider sets I ⊆ G(G,S) such thatthe canonical morphism

pS,I : G(I) −→ S,

is an epimorphisms. To be more specific, pS,I is defined by the property that

pS,I ◦ ji = i for all i, where ji : G → G(I) is the coproduct injection to thecomponent indexed by i. Note that since G is a generator, for each S ∈ Swe have at least one such I. In fact I = G(G,S) always works, but we willneed to consider all pairs (S, I) with the property above.

Given such S and I, we can complete pS,I to a short exact sequence

0 −→ KS,IkS,I−→ G(I) pS,I−→ S −→ 0.

We claim that the set

I = {kS,I : KS,I −→ G(I)},where S runs over all objects of S and I runs over all subsets of G(G,S)such that pS,I is an epimorphism, satisfies properties (1) and (2) from thestatement.

Since (1) is obviously satisfied, we focus on (2) and consider a mapf : X → Y which fits into a short exact sequence

0 −→ Xf−→ Y

p−→ S −→ 0

with S ∈ S. G being a generator, there is an epimorphism

G(J) g−→ Y −→ 0.

Consider the composition pg : G(J) → S, which is necessarily an epimor-phism too. Denoting by ì : G → G(J) the coproduct inclusions, we mayget the same compositions pgì : G → S for distinct elements i ∈ J . Wetherefore define an equivalence relation on J by putting

i ∼ i′ for i, i′ ∈ J if pgì = pgì′ .

Let now J ′ ∈ J be any set of representatives for the equivalence classeswith respect to ∼, and define g′ : G(J ′) → Y as the restriction of g to G(J ′).


That is, g′ is determined by the property that g′ì = gì for all i ∈ J ′. Weclaim that the composition

pg′ : G(J ′) −→ S

is still an epimorphism. In fact, if G = Mod-R is a module category, thenthe proof is easy since then

Im pg′ =∑i∈J ′

Im pgì =∑i∈J

Im pgì = Im pg = S.

The same argument works for a general Grothendieck category, if we givea proper meaning to subobjects and the infinite sums. We refer to [14,§IV.2 and Example 3 in §IV.8] and leave checking the details to the reader.Alternatively, a direct proof in a more general setting can be found [13,Lemma A.5].

Having proved the claim, we have a commutative diagram with exact rowsof the form

0 −−−−→ Kk−−−−→ GJ

′ pg′−−−−→ S −−−−→ 0y g′y ∥∥∥

0 −−−−→ Xf−−−−→ Y

p−−−−→ S −−−−→ 0,

In a diagram of this form in any abelian category, the left hand side commu-tative square is necessarily bicartesian, that is, it is a pushout and pullbacksquare at the same time. We conclude the proof by noting that when iden-tifying J ′ with a subset of G(G,S) via the injective mapping J ′ → G(G,S)given by

i 7−→ pg′ì,

the kernel map k belongs (up to isomorphism) to the set I constructedabove. �

Now are ready to tackle the main result of the section, which can bethought of as an analogue of Theorem 3.9 for cotorsion pairs. The argumentcan be traced back to [9, Theorem 6.5], but the presentation here followsmore the one in [13, §2].

Theorem 4.7. Let G be a Grothendieck category and S be a set (not aproper class!) of objects containing a generator for G. Put

B = S⊥ and A = ⊥B.

Then (A,B) is a complete cotorsion pair in G and A consists precisely ofretracts of S-filtered objects.

Stopping for a moment, cotorsion pairs as above will be so important inthe further discussion that they deserve a name.

Definition 4.8. A cotorsion pair in a Grothendieck category is called goodif it arises as in Theorem 4.7. That is, if it is generated from the left by aset containing a generator for G.

20 JAN STOVICEK

Proof of Theorem 4.7. Let G be a generator, which need not belong to S atthe moment, and I be a class of morphisms which satisfies the conditions ofProposition 4.6 for G and S. Further, let (L,R) be the weak factorizationsystem generated by I in the sense of Theorem 3.9. As L consists of retractsof relative I-cell complexes, one immediately sees that the morphisms inL are precisely monomorphisms, whose cokernels are retracts of S-filteredobjects.

Given an object Y ∈ G, we next claim that Y belong to B if and only ifthe map Y → 0 belongs to R. The direct implication is easy and left to thereader. We will focus on the other implication, that is, we will assume thatY → 0 belongs to R. We must prove that any fixed exact sequence

ε : 0 −→ Y −→ E −→ S −→ 0

with S ∈ S splits. By the choice of I using Proposition 4.6, we know thatthere is a commutative diagram

η : 0 −−−−→ Kk−−−−→ G(J) −−−−→ S −−−−→ 0

v

y y ∥∥∥ε : 0 −−−−→ Y

f−−−−→ E −−−−→ S −−−−→ 0

with exact rows and such that k ∈ I. Hence k has the left lifting propertyfor Y → 0 by the very construction of R. We apply this fact to completethe following commutative square:

Kv //

k��

Y

��G(J) //

==

0

In particular, v factors to k, and since ε is a pushout of the short exactsequence η along v, this implies that ε splits. The claim is proved.

The pair (A,B) in the statement is easily seen to be a cotorsion pair, butwe still must show that it is complete. That is, we must for any given X ∈ Gfind the two approximation sequences from Definition 4.2.

For the first one, let us factor X → 0 with respect to (L,R). Looking atthe proof of Theorem 3.9, we get a commutative triangle

X //

f AAA

AAAA

0

B

g

??��

with f ∈ I-cell and g ∈ R. By the description of I-cell in from Proposi-tion 4.6, we have a short exact sequence

0 −→ Xf−→ B −→ A −→ 0

such that A is S-filtered. In particular A ∈ A by Proposition 4.5. Moreover,B ∈ B by the previous claim since B → 0 belongs to R.

For constructing the other approximation sequence, we will suppose forthe first time that we have a generator G ∈ S. Then we can take a short


exact sequence

0 −→ K −→ G(I) −→ X −→ 0

and the type of an approximation sequence

0 −→ K −→ B −→ A −→ 0

for K, which we already can construct. We will use the two to form apushout diagram:

0 0y y0 −−−−→ K −−−−→ G(I) −−−−→ X −−−−→ 0y y ∥∥∥0 −−−−→ B −−−−→ A −−−−→ X −−−−→ 0y y

A Ay y0 0

Now let us focus on the middle line. We have B ∈ B by the construction.Further, A can be taken S-filtered, and as G ∈ S, it is not hard to checkthat G(I) is S-filtered and so is A (see e.g. [15, Lemma 1.6]).

Finally, if we apply the latter construction to X ∈ A, the middle row willsplit as then Ext1

G(X,B) = 0. Thus, X will be a retract of the S-filtered

object A. Conversely, any retract of an S-filtered object belongs to A byProposition 4.5. This finishes the description of the objects in A and alsothe proof of the theorem. �

Remark 4.9. A similar comment as in Remark 3.10 applies. Approxima-tion triangles of X can be constructed so that they depend functorially onX, but this functoriality is again non-canonical—there are choices in theconstruction.

5. The derived category of an abelian category

6. Homological algebra for complexes

7. Model categories

Appendix A. Quasi-coherent modules versus sheaves

References

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[2] E. Enochs and S. Estrada. Relative homological algebra in the category of quasi-coherent sheaves. Adv. Math., 194(2):284–295, 2005.

[3] P. Freyd. Abelian categories. An introduction to the theory of functors. Harper’s Seriesin Modern Mathematics. Harper & Row Publishers, New York, 1964.

22 JAN STOVICEK

[4] R. Gobel and J. Trlifaj. Approximations and endomorphism algebras of modules, vol-ume 41 of de Gruyter Expositions in Mathematics. Walter de Gruyter GmbH & Co.KG, Berlin, 2006.

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[7] P. S. Hirschhorn. Model categories and their localizations, volume 99 of MathematicalSurveys and Monographs. American Mathematical Society, Providence, RI, 2003.

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[10] S. Mac Lane. Homology. Die Grundlehren der mathematischen Wissenschaften, Bd.114. Academic Press Inc., Publishers, New York, 1963.

[11] B. Mitchell. The full imbedding theorem. Amer. J. Math., 86:619–637, 1964.[12] A. Neeman. Algebraic and analytic geometry, volume 345 of London Mathematical

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joints and model structures. Adv. Math., 228(2):968–1007, 2011.[14] B. Stenstrom. Rings of quotients. Springer-Verlag, New York, 1975. Die Grundlehren

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[15] J. St’ovıcek. Deconstructibility and Hill lemma in Grothendieck categories. To appearin Forum Math., doi:10.1515/FORM.2011.113, preprint at arXiv:1005.3251v2, 2010.

Charles University in Prague, Faculty of Mathematics and Physics, De-partment of Algebra, Sokolovska 83, 186 75 Praha 8, Czech Republic

E-mail address: [email protected]

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