Homological Algebra

Andrew Kobin

Fall 2014 / Spring 2017 Contents Contents

Contents

0 Introduction 1

1 Preliminaries 4 1.1 Categories and Functors ...... 4 1.2 Exactness of Sequences and Functors ...... 10 1.3 Tensor Products ...... 17

2 Special Modules 24 2.1 Projective Modules ...... 24 2.2 Modules Over Noetherian Rings ...... 28 2.3 Injective Modules ...... 31 2.4 Flat Modules ...... 38

3 Categorical Constructions 47 3.1 Products and Coproducts ...... 47 3.2 Limits and Colimits ...... 49 3.3 Abelian Categories ...... 54 3.4 Projective and Injective Resolutions ...... 56

4 Homology 60 4.1 Chain Complexes and Homology ...... 60 4.2 Derived Functors ...... 65 4.3 Derived Categories ...... 70 4.4 Tor and Ext ...... 79 4.5 Universal Coefficient Theorems ...... 90

5 Ring Homology 93 5.1 Dimensions of Rings ...... 93 5.2 Hilbert’s Syzygy Theorem ...... 97 5.3 Regular Local Rings ...... 101 5.4 Differential Graded Algebras ...... 103

6 Spectral Sequences 110 6.1 Bicomplexes and Exact Couples ...... 110 6.2 Spectral Sequences ...... 113 6.3 Applications of Spectral Sequences ...... 119

7 Group Cohomology 128 7.1 G-Modules ...... 128 7.2 Cohomology of Groups ...... 130 7.3 Some Results for the First Group Cohomology ...... 133 7.4 Group Extensions ...... 136 7.5 Central Simple Algebras ...... 139 7.6 Classifying Space ...... 142

i Contents Contents

8 Sheaf Theory 144 8.1 Sheaves and Sections ...... 144 8.2 The Category of Sheaves ...... 150 8.3 Sheaf Cohomology ...... 156 8.4 Cechˇ Cohomology ...... 161 8.5 Direct and Inverse Image ...... 170

ii 0 Introduction

0 Introduction

These notes are taken from a reading course on homological algebra led by Dr. Frank Moore at Wake Forest University in the fall of 2014, with some additions made during a course taught by Dr. Benjamin Webster at the University of Virginia in Spring 2017. The companion text for both courses is Rotman’s An Introduction to Homological Algebra, but many concepts were introduced to the discussion from outside and developed by students. For example, most of the material on spectral sequences (Chapter 6) and group cohomology (Chapter 7) follows Weibel’s An Introduction to Homological Algebra. The main topics covered in this course are: ˆ Basic category theory

ˆ Examples of functors including Hom and tensor products

ˆ Free resolutions of modules, projectives, injectives and flat modules

ˆ Homology of a complex

ˆ Derived functors

ˆ Ext and Tor

ˆ Spectral sequences

ˆ Group cohomology

ˆ Other topics from commutative algebra Basic prerequisites for reading these notes are familiarity with groups, rings and modules. Consider the following motivating example. Let R be a ring and let R-Mod be the collec- tion of all R-modules. (This is an example of an abelian category, which will be defined in subsequent chapters.) Fix two modules M and N in R-Mod and define the set of extensions of M by N:

Ext(M,N) := {short exact sequences of R-modules0 → N → K → M → 0}/ ∼ where ∼ is the equivalence relation of short exact sequences exhibited by the following diagram:

0 N K M 0

id id

0 N K0 M 0

Ext(M,N) has a natural “distinguished” element, which is an equivalence class represented by the canonial split exact sequence:

0 → N → N ⊕ M → M → 0.

1 0 Introduction

p Definition. A splitting of a short exact sequence 0 → N −→i K −→ M → 0 is a map q : M → K (or equivalently, j : K → N) such that p ◦ q = idM (or j ◦ i = idN ). For example, the canonical projections are splittings of 0 → N → N ⊕ M → M → 0. One can show that a short exact sequence has a splitting if and only if it is equivalent to the canonical split exact sequence.

Example 0.0.1. Let M = Z/pZ (p prime) and N = Z, considered as modules over Z. An element of Ext(Z/pZ, Z) is represented by a short exact sequence of the form

0 → Z → K → Z/pZ → 0.

Besides the split exact sequence (with K = Z ⊕ Z/pZ), there is a family of short exact sequences,

p 0 → Z −→ Z → Z/pZ → 0 1 7→ a−1

for each a ∈ (Z/pZ)×. Suppose we have two such extensions that are equivalent:

p a−1 0 Z Z Z/pZ 0

id ϕ id p b−1 0 Z Z Z/pZ 0

Since ϕ is an isomorphism of Z-modules that commutes with the identity on Z, it must in fact be the identity. But since the rest of the diagram commutes, this implies a = b. We have thus exhibited a bijection

Ext(Z/pZ, Z) ←→ Z/pZ. It turns out that Ext(M,N) is always an abelian group. In the above example, the correspondence between a ∈ (Z/pZ)× and 1 7→ a−1 is actually an abelian group isomorphism. p To understand the structure of Ext(M,N), take two extensions 0 → N −→i K −→ M → 0 and 0 p0 0 → N −→i K0 −→ M → 0. A natural composition would be to take the direct sum of the two sequences, but this would yield an extension of M ⊕ M by N ⊕ N. In fact, there is a universal object (called pullback) making the following diagram commute:

K

Ke M

K0

2 0 Introduction

Explicitly Ke = {(k, k0) ∈ K × K0 : p(k) = p0(k0)}. Then we define the space K00 := K/e {(n, −n): n ∈ N} gives an extension of R-modules which is the natural composition of the two initial extensions:

0 → N → K → M → 0 + 0 → N → K0 → M → 0 = 0 → N → K00 → M → 0.

It’s obvious that this construction is commutative, but associativity and inverses require some proof.

Example 0.0.2. Given two extensions

p a−1 p b−1 0 → Z −→ Z −−→ Z/pZ → 0 and 0 → Z −→ Z −−→ Z/pZ → 0,

we have the composite space K00 = {(x, y) ∈ Z ⊕ Z | bx = ay (mod p)}/h(p, −p)i. One can show that this extension represents the class in Ext(Z/pZ, Z) corresponding to a + b in Z/pZ. One question going forward will be: how do we compute these Ext groups in a reasonable fashion?

3 1 Preliminaries

1 Preliminaries

1.1 Categories and Functors

Category theory is a way of describing ‘classes’ of things in mathematics and the relations between them in a general way. Definition. A category C is a class of objects obj(C) and for every pair of objects A, B in obj(C) a set of morphisms HomC(A, B) with a composition law:

HomC(B,C) × HomC(A, B) −→ HomC(A, C) (g, f) 7−→ g ◦ f = gf.

For every A ∈ obj(C) there is an identity morphism 1A ∈ HomC(A, A) such that for all f ∈ HomC(A, B), f ◦ 1A = f and likewise for HomC(B,A). Examples.

1 Veck is a category consisting of vector spaces over a field k as the objects and linear transformations as the morphisms. This is the same category as k-Mod (see below).

2 For a ring R with identity, R-Mod is the category of (left) R-modules with R-linear maps as the morphisms. In this case we will denote HomR-Mod(A, B) as HomR(A, B). 3 Groups (or Grps) is the category consisting of groups together with group homomor- phisms. Even more generally, we can define the category Sets: objects are sets and morphisms are just regular functions. Other algebraic categories arise in a similar fash- ion: AbGrps or Ab is the category of abelian groups with the usual homomorphisms; Rings is the category of rings with ring homomorphisms, where we assume all rings have unity; likewise, ComRings are commutative rings. 4 For a topological space X, we can form a category by taking the open subsets of X as objects and for any open U, V ⊂ X, ( ∅ if U 6⊆ V HomX (U, V ) = {iU,V } if U ⊆ V

where iU,V is the inclusion U,→ V . This category can be generalized to any space X which has a partial ordering . 5 As another topological example, Top is the category of topological spaces together with continuous functions between them.

How do we compare two categories? Definition. A functor F from C to D is an assignment

F : obj(C) −→ obj(D) HomC(A, B) −→ HomD(F (A),F (B)) A 7−→ F (A) f : A → B F (f): F (A) → F (B) such that F (gf) = F (g)F (f).

4 1.1 Categories and Functors 1 Preliminaries

Remark. The functor defined above is sometimes called a covariant functor: the image of a morphism f : A → B is a morphism F (A) → F (B). On the other hand, a contravariant functor takes f : A → B to F (f): F (B) → F (A), and F (gf) = F (f)F (g).

Examples.

1 The forgetful functor is useful in many contexts. For example,

Forget : Veck −→ Ab V 7−→ (V, +) f 7−→ f

takes a vector space V and “forgets” its vector space structure. We can go further and forget a group structure, giving a functor Ab → Sets.

2 For any category C, the identity functor 1C : C → C takes A 7→ A and f 7→ f.

3 In topology, the fundamental group is a functor:

π1 : Top −→ Grps

X 7−→ π1(X)

∗ f : X → Y f : π1(X) → π1(Y ).

3 The nth homology group of a topological space is a functor Hn(·): Top → Ab.

4 Let C be the category of field extensions K/k with morphisms ϕ : K → L such that

ϕ|k = idk, i.e. the set of k-homomorphisms. Then Aut is the functor defined by

Aut : C −→ Grps K/k 7−→ Aut(K/k).

Another category can be formed by taking obj(C) to be all Galois extensions K/k. Then all automorphism groups are called Galois groups, and the functor is

Gal : C −→ Grps

K/k 7−→ Gal(K/k) ϕ : K → L Gal(ϕ) : Gal(L/k) → Gal(K/k) where the last map is the quotient map. Notice that both of these are contravariant functors.

5 1.1 Categories and Functors 1 Preliminaries

5 For the category Veck we have the contravariant functor

∗ () : Veck −→ Veck

V 7−→ Homk(V, k) f : V → W f ∗ : W ∗ → V ∗ g 7→ gf

6 Abelianization is a functor:

(·)ab : Groups −→ Ab

G 7−→ G/G0 where G0 is the commutator subgroup f f.¯

The main functors of interest in this course will be for the categories R-Mod:

ˆ HomR(−,N), a contravariant functor

ˆ HomR(M, −), a covariant functor

ˆ − ⊗R N, a covariant functor.

It is usually better to think of everything in terms of covariant functors. To explain why, we have the following generalization of a concept from ring theory.

Definition. Given a category C, its opposite category Cop consists of obj(Cop) = obj(C) and HomCop (A, B) = HomC(B,A). Then a contravariant functor from C is simply a covariant functor from Cop. For example, we see that the category of left R-modules is isomorphic (to be defined) to the category of op op right R -modules, where R = R as sets and a ·Rop b = b ·R a. Proposition 1.1.1. If T : A → B is a covariant functor, define T op : Aop → B by T op(A) = T (A) for all A ∈ obj(A) and T op(f op) = T (f) for all morphisms in A. Then T op is a contravariant functor.

Proof. First, if C ∈ obj(Aop), then T op(C) = T (C) ∈ obj(B). Next, T op(f op): T op(A) → T op(A0) since T op(f op) = T (f): T (A) → T (A0), and these objects are equal in both maps. Now to show that T op composes, take f : A → A0 and g : A0 → A00 so that f op : A0 → A and gop : A00 → A0. Then

T op(gop): T op(A00) → T op(A0) and T op(f op): T op(A) → T op(A0)

6 1.1 Categories and Functors 1 Preliminaries

so we see that T op(gopf op) = T op((fg)op) = T (fg) = T (f)T (g) since T is covariant. But this last part is just T op(f op)T op(gop) so we have shown that T op is contravariant. op op Finally T (1A) = T (1A) = 1T (A) = 1T op(A) so in particular T is a contravariant functor.

Just as homomorphisms compare algebraic objects and functors compare categories, there is a way to compare functors themselves.

Definition. For covariant functors S,T : A → B, a natural transformation τ : S → T is a family of morphisms in B:

τ = {τA : S(A) → T (A)}A∈obj(A)

such that the following diagram commutes for all morphisms f : A → A0.

τA S(A) T (A)

S(f) T (f)

S(A0) T (A0) τA0

We can define natural transformations between contravariant functors in a similar way, or just treat them as covariant functors from Aop.

Definition. A natural isomorphism is a natural transformation such that every τA is an isomorphism.

Proposition 1.1.2. If σ : S → T and τ : T → U are natural transformations of functors S,T,U : A → B, then τσ : S → U defined by (τσ)A = τAσA is a natural transformation. Proof. Easy to check.

Example 1.1.3. For any functor T : A → B, the identity natural transformation ωT : T → T is defined by (ωT )A = 1T (A) : T (A) → T (A). Proposition 1.1.4. A natural transformation τ : S → T is a natural isomorphism if and only if there is a natural transformation σ : T → S such that στ = ωS and τσ = ωT . Proof omitted.

Definition. For two functors F,G : A → B, define

Nat(F,G) = {natural transformations τ : F → G}.

7 1.1 Categories and Functors 1 Preliminaries

∼ Theorem 1.1.5. If A = B then HomC(A, −) and HomC(B, −) are naturally isomorphic ∼ functors. Similarly, HomC(−,A) = HomC(−,B). Proof. Let A ∼= B via the isomorphism α : A → B and let α−1 be its inverse. Define a transformation τ : HomC(A, −) → HomC(B, −) by

−1 ∗ τC = (α ) : HomC(A, C) −→ HomC(B,C) f 7−→ fα−1 for each C ∈ obj(C). Note that fα−1 : B → C so this map is well-defined. Next we verify that for any morphism f : C → C0, the following diagram commutes:

τC Hom(A, C) Hom(B,C)

f f

Hom(A, C0) Hom(B,C0)

τC0

If h ∈ Hom(A, C) then it maps h 7→ hα−1 7→ fhα−1 via the top map, and h 7→ fh 7→ fhα−1 via the bottom map, so indeed the diagram commutes. Thus τ is a natural transformation. Finally we need to check that each τC is an isomorphism. To do this, construct a trans- formation σC : HomC(B,C) → HomC(A, C) taking g 7→ gα. Then for any f ∈ HomC(A, C) and g ∈ HomC(B,C) we have

−1 τC (σC (g)) = τC (gα) = gαα = g and −1 −1 σC (τC (f)) = σC (fα ) = fα α = f.

Hence τC and σC are inverse morphisms, so τC is an isomorphism. This proves that τ is a ∼ natural isomorphism. The proof that HomC(−,A) = HomC(−,B) is similar.

We will see that the converse of this statement is also true. That is, HomC(−,A) com- pletely determines the object A itself (as does HomC(A, −)). But what’s the meaning be- op hind statements like this? For an object A ∈ obj(C), the functor HomC(−,A): C → Sets gives us a lot of information about A. For example, if C is the category of groups and HomC(Z/2Z,A) contains a nontrivial element, this tells us that A has at least one element of order 2. Likewise, in the category Top of topological spaces, elements of Hom(S1,A) encode the information of closed loops in A. So “probing” A by different objects B, i.e. considering HomC(B,A), tells us different sorts of information about A. By probing A with all objects in C, would it be possible to know all information about A? The answer is yes, and the formal statement of this is called Yoneda’s Lemma. Lemma 1.1.6 (Yoneda’s Lemma). Let C be a category, A ∈ obj(C) and G : C → Sets a covariant functor. Then there is a bijection

Nat(HomC(−,A),G) ←→ G(A).

8 1.1 Categories and Functors 1 Preliminaries

Proof. Let τ : HomC(−,A) → G be a natural transformation. Then for each B ∈ obj(C) and morphism ϕ : A → B, the following diagram commutes: τ Hom(A, A) A G(A)

ϕ∗ G(ϕ)

Hom(B,A) G(B) τB

where ϕ∗ : f 7→ ϕ ◦ f. In particular, 1A ∈ HomC(A, A) maps to τA(1A) ∈ G(A). Define the map Nat(HomC(−,A),G) → G(A) by sending τ 7→ τA(1A). If σ : HomC(−,A) → G is another natural transformation such that σA(1A) = τA(1A), then the above diagram shows that for all ϕ ∈ HomC(B,A),

σB(ϕ) = σBϕ∗(1A) = G(ϕ)τA(1A) = G(ϕ)σA(1A) = τBϕ∗(1A) = τB(ϕ). Thus σ = τ, so the assignment is one-to-one. We next show every element x ∈ G(A) is induced by such a natural transformation τ : HomC(−,A) → G. For each B ∈ obj(C) and f ∈ HomC(B,A), define τB(f) = G(f)(x). Then for any B → C, the following diagram commutes: τ Hom(A, B) B G(B)

Hom(A, C) G(C) τC

Hence τ is a natural transformation and by construction, τA(1A) = G(1A)(x) = 1G(A)(x) = x. This proves the bijection. ∼ Corollary 1.1.7. If HomC(−,A) and HomC(−,B) are naturally isomorphic, then A = B.

Proof. Suppose τ : HomC(−,A) → HomC(−,B) is a natural isomorphism. Applying Yoneda’s Lemma to G = HomC(−,B), we get a unique element x ∈ G(A) = HomC(A, B). Applying Yoneda to the inverse of τ yields a unique inverse to x, proving A ∼= B.

op For any category C, there is a functor C → Fun(C , Sets) given by A 7→ HomC(−,A). The Yoneda Lemma shows that this functor is fully faithful, i.e. it embeds C as a full subcategory of Fun(Cop, Sets). More precisely:

op Corollary 1.1.8 (Yoneda Embedding). There is a functor C → Fun(C , Sets),A 7→ HomC(−,A), which is an isomorphism on HomC(B,C) for all B,C ∈ C. Lemma 1.1.9 (Yoneda’s Lemma, Covariant Version). Let C be a category, A ∈ obj(C) and G : C → Sets a covariant functor. Then there is a bijection

Nat(HomC(A, −),G) ←→ G(A)

τ 7−→ τA(1A).

9 1.2 Exactness of Sequences and Functors 1 Preliminaries

Proof. Similar to the proof of Lemma 1.1.6.

Corollary 1.1.10 (Yoneda Embedding, Covariant Version). There is a fully faithful functor C → Fun(C, Sets),A 7→ HomC(A, −).

1.2 Exactness of Sequences and Functors

f g Definition. A sequence A −→ B −→ C in a category C is exact at B if ker g = im f.

Definition. An exact sequence of the form

f g 0 → A −→ B −→ C → 0

is called a short exact sequence. In particular, this means that f is injective, g is surjective and ker g = im f. This is sometimes called an “extension of A by C”.

f g If 0 → A −→ B −→ C → 0 is a short exact sequence of R-modules, the first isomorphism theorem for modules tells us that A ∼= im f and B/ im f ∼= C. Example 1.2.1. For a commutative ring R, take the set S of all R-module short exact sequences: S = {0 → A → B → C → 0 exact | A, B, C ∈ R-Mod} Then there is an equivalence relation ∼ such that

∼ 1 S/ ∼= ExtR(C,A).

Definition. Let 0 → A → B → C → 0 be an exact sequence in a category C and let F : C → D be a covariant functor. F is left exact if

0 → F (A) → F (B) → F (C)

is exact, and F is right exact if

F (A) → F (B) → F (C) → 0 is exact. If it is both left and right exact, then F is simply called exact.

Note that if F is a contravariant functor, left exactness looks like

0 → F (C) → F (B) → F (A) and right exactness is analogous.

Proposition 1.2.2. HomR(X, −) is a left exact functor on the category R-Mod.

10 1.2 Exactness of Sequences and Functors 1 Preliminaries

f g Proof. Let 0 → A −→ B −→ C → 0 be an exact sequence of left R-modules and take X ∈ obj(R-Mod). If we apply HomR(X, −), the sequence becomes

f∗ g∗ 0 → HomR(X,A) −→ HomR(X,B) −→ HomR(X,C).

We need to show that f∗ is one-to-one and ker g∗ = im f∗. First, take h ∈ HomR(X,A) such that fh = 0. Note that f is one-to-one ⇐⇒ there ˜ ˜ exists a morphism f : im f → A such that ff = idA. In this case ˜ ˜ h = idAh = ffh = f0 = 0

which shows that h = 0, i.e. f∗ is one-to-one. Now let j ∈ HomR(X,B) such that gj = 0, i.e. j ∈ ker g∗. We need to construct a function k : X → A such that fk = j. Let x ∈ X. Then g(j(x)) = 0 so j(x) ∈ ker g. But ker g = im f so there exists a (unique) element a ∈ A such that f(a) = j(x). Now we can define k : X → A by mapping x to this unique a. Then j(x) = f(a) = f(k(x)) so j = fk as desired. Hence HomR(X, −) is left exact.

Note that HomR(X, −) may not always be right exact. In fact, even in the category of abelian groups, HomZ(X, −) is not right exact, as the following example shows. Example 1.2.3. Consider the exact sequence

0 → 2Z → Z → Z/2Z → 0.

Let X = Z/2Z. Then Proposition 1.2.2 tells us that

0 → Hom(Z/2Z, 2Z) → Hom(Z/2Z, Z) → Hom(Z/2Z, Z/2Z) is exact. But up to isomorphism, this sequence is

0 → 0 → 0 → Z/2Z.

Clearly adding a zero on the right makes the sequence not exact, since the kernel of Z/2Z → 0 is necessarily Z/2Z. Two important examples in the theory of modules and their homological properties are

Definition. An R-module P is projective if HomR(P, −) is exact.

Definition. An R-module E is injective if HomR(−,E) is exact. The proof of Proposition 1.2.2 may be adapted to show that the contravariant functor HomR(−,Y ) is always left exact, so proving that a module is projective or injective comes down to proving that the associated Hom is right exact. Example 1.2.3 shows that Z/2Z is not a projective Z-module. We will explore these special modules in Chapter 2. Proposition 1.2.4. P is a projective module if and only if for every surjection f : A → B, f∗ : HomR(P,A) → HomR(P,B) is surjective. This in turn is equivalent to saying for every diagram of the form shown below, where the bottom row is exact, there exists a g˜ that makes the diagram commute.

11 1.2 Exactness of Sequences and Functors 1 Preliminaries

P g˜ g

A B 0 f

Definition. A short exact sequence of R-modules

f g 0 → A −→ B −→ C → 0

is split if there exists a map h : C → B with gh = 1C . Proposition 1.2.5. If a short exact sequence of R-modules

f g 0 → A −→ B −→ C → 0

is split, then B ∼= A ⊕ C.

Proof. We will show that B = im f ⊕ im h where h : C → B satisfies gh = 1C . If b ∈ B, then g(b) ∈ C and b − h(g(b)) ∈ ker g. To see this, note that

g(b − h(g(b))) = g(b) − g(h(g(b))) = g(b) − g(b) = 0 since gh = 1C . By exactness of the sequence, there exists an element a ∈ A such that f(a) = b − h(g(b)). Then B = im f + im h. To show the product is direct, it suffices to show that im f ∩ im h = {0}. Suppose x = f(a) = h(c). Then g(x) = g(f(a)) = 0 because gf = 0. On the other hand, g(x) = g(h(c)) = c because gh = 1C . Hence x = 0, and so we have proven B ∼= A ⊕ C.

Lemma 1.2.6 (3×3 Lemma). Consider the following commutative diagram in R-Mod having exact columns.

12 1.2 Exactness of Sequences and Functors 1 Preliminaries

0 0 0

0 A0 A A00 0

0 B0 B B00 0

0 C0 C C00 0

0 0 0

If the bottom two rows are exact, the top row is exact, and if the top two rows are exact, the bottom row is exact.

Proof. Picking a00 ∈ A00, let b00 be its image in B00. Then b00 must map to zero by exactness of the third column. Thus there is a b ∈ B mapping to b00 since the map is surjective; let c be the image of b in C. By commutativity, c 7→ 0. Therefore there is some c0 ∈ C0 mapping to c, using exactness of the third row. Now by surjectivity of B0 → C0, there is some b0 ∈ B0 mapping to c0; denote its image in B by ¯b. By exactness, this ¯b maps to 0 in B00. This further implies that b − ¯b 7→ b00. Since the lower left square commutes, ¯b 7→ c =⇒ b − ¯b 7→ 0 =⇒ for some a ∈ A, a 7→ b − ¯b. Let a map toa ¯00 ∈ A00. Then since A00 ,→ B00 is injective,a ¯00 = a00 so it is guaranteed that there will be an a mapping to a00. Hence A → A00 is surjective. Next, take some a0 ∈ A0 which maps to 0 in A. Then a0 7→ b0 ∈ B0 and by commutativity around the square, b0 7→ 0 ∈ B. But A0 ,→ B0 is injective, so a0 = 0 and A0 → A is seen to be injective. Now pick another a0 ∈ A0 with image a ∈ A and subsequently a 7→ x ∈ A00. We need to show x = 0. Well a0 maps to some b0 ∈ B0 and b0 7→ b ∈ B, so by commutativity a 7→ b. Since b is in the image of B0 → B and the second row is exact, b 7→ 0 ∈ B00. By commutativity of the upper right square, x 7→ 0 ∈ B00. But since this map is injective, this forces x = 0. We have thus shown that the top row is exact. The second statement is proven similarly.

The proof above is an example of a variety of proofs which are common to homological algebra, called diagram chasing. Another example of this is

Lemma 1.2.7. Consider the following commutative diagram in R-Mod having exact rows and columns.

13 1.2 Exactness of Sequences and Functors 1 Preliminaries

A0 A A00 0

B0 B B00 0

C0 C C00 0

0 0 0

If A00 → B00 and B0 → B are injections, then C0 → C is an injection. Similarly, if C0 → C and A → B are injections, then A00 → B00 is an injection. In other words, if the last column and the second row are short exact sequences, then the third row is a short exact sequence, and similarly, if the bottom row and the second column are short exact sequences, then the third column is a short exact sequence.

Proof. First suppose A00 ,→ B00 and B0 ,→ B are injections. The following diagram serves as a useful blueprint for the diagram chase to follow.

0 00 A (7) A A 0 a0 a a00 = 0 (6)

(8) (4) β0 = B0 B B00 0 (9) b0 b 0 (3) (5)

(2)

0 00 C (1) C C 0 c0 0

0 0 0

Take c0 ∈ C0 such that c0 7→ 0 in C – we must show c0 = 0. By exactness of the first column, there is a b0 ∈ B0 such that b0 7→ c0. Let b be the image of b0 under B0 ,→ B; the diagram commutes, so b0 7→ c0 7→ 0 ∈ C implies b0 7→ b 7→ 0 ∈ C. Therefore b is in the kernel of B → C, and by exactness of the middle column, there exists some a ∈ A such that a 7→ b in B. Denote the image of a along A → A00 by a00. We also have b0 7→ b 7→ 0 since the middle row is exact, and commutativity gives us a 7→ a00 7→ 0 along the top. But A00 ,→ B00 is injective, so a00 = 0. Thus a is in the kernel of A → A00 so by exactness of the top row, there is some a0 ∈ A0 such that a0 7→ a ∈ A. Let the image of a0 under A0 → B0 be denoted β0. Then since a0 7→ a 7→ b ∈ B, β0 is also mapped to b. But B0 ,→ B is injective, so we must

14 1.2 Exactness of Sequences and Functors 1 Preliminaries

have β0 = b0. This shows that b0 is in the image of A0 → B0, so it’s in the kernel of B0 → C0 and hence b0 7→ c0 = 0. This completes the proof that C0 → C is injective. The second statement follows from an identical proof if we reflect the diagram across the line running through A0, B and C00, since this transformation preserves the homological properties of the arrows. Note that the statements about short exact sequences follow immediately from the proofs of injectivity. One of the most famous diagram chasing proofs is the Snake Lemma, which will become useful when we define complexes in Chapter 4. The version below involves a morphism in the category of short exact sequences.

Lemma 1.2.8 (Snake Lemma). If the commutative diagram

α0 α 0 A0 A A00 0

f g h β0 β 0 B0 B B00 0

has exact rows, then there is an exact sequence

0 → ker f → ker g → ker h → coker f → coker g → coker h → 0.

Proof. To begin, note that the left square commutes so α0 restricts to α0 : ker f → ker g. Likewise α restricts to α : ker g → ker h and these both inherit injectivity. Consider the β0 projection B0 −→ B → B/ im g = coker g. For any b0 ∈ im f, there is some a0 ∈ A0 such that f(a0) = b0, but also gα0(a0) = β0(b0) by commutativity. So β0(b0) ∈ im g which induces a map β¯0 : coker f → coker g – in other words β0 factors through f. Likewise, β induces a map β¯ and both β¯0 and β¯ inherit surjectivity. We pause to label the sequence:

0 β¯0 β¯ 0 → ker f −→α ker g −→α ker h −→ð coker f −→ coker g −→ coker h → 0.

The interesting part of the sequence is at ker h → coker f; we have labelled this with the character ð. How can we define ð? Start with a00 ∈ ker h, so that h(a00) = 0. By exactness, there is some a ∈ A such that α(a) = a00. If b is the image of a under g, b 7→ 0 in B00 and so there is some b0 ∈ B0 such that b0 7→ b 7→ 0. With this notation, we define

ð : ker h −→ coker f a00 7−→ b0 + im f.

Since all maps in this diagram are linear, it suffices to check well-definedness. Once we do so, we will have all maps defined and can then proceed to check exactness. 00 0 0 0 0 Suppose we have α(a1) = α(a2) = a . Then there is some a ∈ A such that α (a ) = a1−a2 0 0 0 since this difference lies in ker α. Let b1 and b2 be the unique (by injectivity) elements of B

15 1.2 Exactness of Sequences and Functors 1 Preliminaries

0 0 0 0 0 0 0 such that β (b1) = g(a1) and β (b2) = g(a2). We want to show that b1 − b2 = f(a ). But this 0 is easily seen, since commutativity of the left square gives us a 7→ a1 −a2 7→ g(a1)−g(a2) := 0 0 b1 − b2 around the top, and a 7→ f(a ) 7→ b1 − b2 around the bottom. Then by injectivity, 0 0 0 0 f(a ) must equal b1 − b2 as desired. This shows that ð maps into B / im f = coker f, i.e. it is well-defined. We now have all our maps, so we can proceed to show exactness. First α0α = 0 implies 0 im(α |ker f ) ⊆ ker(α |ker g). For the other containment, take a ∈ ker α that maps to 0 under g. Then by exactness, there exists an a0 7→ a but the left square commutes, so a0 7→ b0 7→ 0 and injectivity of β0 shows b0 = 0. Next suppose b ∈ coker g maps to b00 ∈ coker h = B/ im h. Then there is some a00 mapping to b00 under h, and α is onto to there also exists a ∈ A mapping to a00. Let g(a) = ˜b ∈ B; by commutativity ˜b 7→ b00. Then ˜b − b 7→ 0 ∈ B00 implies there exists b0 ∈ B0 such that β0(b0) = ˜b − b = g(a) − b. So b = β0(b0) − g(a) =⇒ β¯0(b0 + im g) = β0(b0) + im g = b + im g. The hard part is showing exactness around the connecting map ð. For a00 ∈ ker h, we can lift back to some a ∈ A which maps to a00 ∈ A00 and has image g(a) = b. But b 7→ 0 under β so there is some b0 ∈ B0 such that b0 7→ b. Recall that we defined ð(a00) = b0 + im f. Then 00 im(α |ker g) ⊂ ker ð because g(a) = 0. On the other hand, suppose ð(a ) = 0. Then in the diagram chase for the definition of ð, choose a0 ∈ A0 such that f(a0) = b0. Seta ˜ = α0(a0). By commutativity, g(˜a − a) = 0 and by exactness, α(˜a − a) = α(a) = a00. Finally let b0 ∈ B0 such that β¯0(b0 + im f) = im g, so β0(b0) ∈ im g. The fact that im ð ⊂ ker β¯0 is just by definition of the connecting map. Conversely, suppose b0 7→ b such that there is some a ∈ A with g(a) = b. If we set α(a) = a00, we have already constructed the connecting map for a00. Moreover we know b0 7→ b 7→ 0 so a00 must map to 0 along h, i.e. a00 ∈ ker h. This shows exactness at every part of the sequence. In Chapter 4 we will also need

Lemma 1.2.9 (Five Lemma). Suppose the following diagram is commutative with exact rows.

A1 A2 A3 A4 A5

f1 f2 f3 f4 f5

B1 B2 B3 B4 B5

(i) If f2 and f4 are surjective and f5 is injective, then f3 is surjective.

(ii) If f2 and f4 are injective and f1 is surjective, then f3 is injective.

(iii) If f1, f2, f4 and f5 are isomorphisms, then so is f3.

Proof. Commence chase.

16 1.3 Tensor Products 1 Preliminaries

1.3 Tensor Products

In homological algebra, we are often interested in what is called a universal mapping property, which is loosely defined as turning a certain class of maps into linear maps. For example, free objects are a solution to the universal property for set maps. Explicitly, for every set X and map f from X to a module M, there exists a linear map f˜ making the following diagram commute:

X f

FX M f˜

FX is called the free module on X. Tensor products arise a solution to a different universal mapping property, namely the universal property of biadditive maps. First we need to define

Definition. An abelian group A is an R-S bimodule, denoted RAS, if A is a left R-module and a right S-module, and (ra)s = r(as) for all a ∈ A, r ∈ R and s ∈ S.

Definition. Let AR and RB be modules over a ring R. A map ϕ : A × B → G, where G is an object in the category Ab of abelian groups, is biadditive provided

(i) ϕ(ar, b) = ϕ(a, rb)

(ii) ϕ(a + a0, b) = ϕ(a, b) + ϕ(a0, b)

(iii) ϕ(a, b + b0) = ϕ(a, b) + ϕ(a, b0).

Note that the set Biadd(A × B,G) of all biadditive maps A × B → G is an abelian group under pointwise addition and scaling. Our goal is to find a solution to the universal mapping property for biadditive maps, i.e. find an object X ∈ obj(Ab) so that Biadd(A × B,G) = HomAb(X,G). Given a biadditive map f : A × B → G we need to produce a unique group homomorphism f¯ making the following diagram commute:

f A × B G

ϕ f¯ X

where ϕ is a group homomorphism.

17 1.3 Tensor Products 1 Preliminaries

To do this, we mod out by certain relations in A × B. Let W be the (abelian) subgroup of A × B generated by elements of the form

(ar, b) − (a, rb) (a + a0, b) − (a, b) − (a0, b) (a, b + b0) − (a, b) − (a, b0).

One could try to define X = (A × B)/W , but care is required when defining maps out of A × B. For example, if

ψ : A × B −→ G (a, b) 7−→ f(a, b).

Then ψ((a, b) + (a0, b0)) = ϕ(a + a0, b + b0) = ϕ(a, b) + ϕ(a0, b) + ϕ(a, b0) + ϕ(a0, b0) which is NOT equal to ψ(a, b) + ψ(a0, b0) = ϕ(a, b) + ϕ(a0, b0), so ψ is not linear. To remedy the situation, we build X by using the universal mapping property of set maps to embed A × B,→ F , the free module on all elements of A × B. Let S be the subgroup of F generated by the image of W under inclusion, which has the same relations as before. Then X = F/S turns out to be the right object.

Definition. The tensor product of A and B, denoted A ⊗R B, is defined as the abelian group F/S, where F is the free module (or free abelian group) on A × B and S is the submodule of F generated by elements of the form

(ar, b) − (a, rb) (a + a0, b) − (a, b) − (a0, b) (a, b + b0) − (a, b) − (a, b0).

We have the following diagram which summarizes our discussion so far.

h A × B A ⊗R B i π

F f ϕ f˜

G

where f is biadditive, ϕ is induced from the universal mapping property of free abelian groups, π is the natural projection F → F/S, h is the composition π ◦ i and f˜ is our (proposed) solution to the mapping problem for biadditive maps.

18 1.3 Tensor Products 1 Preliminaries

Theorem 1.3.1. Tensor is the unique solution to the universal mapping property of biaddi- tive maps.

Proof. The construction above shows that X = A⊗R B is a solution to the universal mapping ∼ property. If U is another solution, we want to show U = A ⊗R B. Since U satisfies the η property, we have a biadditive map A × B −→ U which induces a commutative diagram

A ⊗R B

B ⊗ η A h

A × B U biadd. η A ⊗ f B

A ⊗R B

where h and f are the linear maps induced from the universal mapping property for A ⊗R B and U, respectively. Since the smaller triangles commute, so does the larger triangle, which ∼ implies fh = idA⊗B and hf = idU . This proves U = A ⊗R B, but notice that we have proven something stronger, namely that A ⊗R B is unique up to a unique isomorphism.

Remark. The set of elements {a ⊗ b | a ∈ A, b ∈ B} is a generating set for A ⊗R B, so every element in the tensor product has the form

N X ni(ai ⊗ bi) i=1

with ni ∈ Z.

If RAS and RB are given, we have

ˆ HomR(RAS,R B) is a left S-module.

ˆ HomR(RB,R AS) is a right S-module.

ˆ RAS ⊗SSB is a left R-module.

ˆ BR ⊗RRAS is a right S-module. In some sense, the tensor A ⊗ B “eats up” the right action of A and the left action of B, so whatever remains is available for an additional module structure. If no action remains, we are just left with an abelian group in every case. Of course in the commutative case, everything works out nicely, since all modules over a commutative ring are bimodules.

19 1.3 Tensor Products 1 Preliminaries

Theorem 1.3.2 (Hom-Tensor Adjointness). Given rings R and S and modules AR,RBS and CS, there is an isomorphism

τA,B,C : HomS(A ⊗R B,C) −→ HomR(A, HomS(B,C)) f 7−→ a 7→ (b 7→ f(a ⊗ b))

i.e. given any f : A ⊗R B → C and an element a ∈ A, we can construct a linear map fa : B → C sending b 7→ f(a ⊗ b). The term adjointness comes from the idea that Hom may be viewed as an inner product on R-Mod: h·, ·i = HomR(−, −):R Mod ×R Mod −→ Ab

which has ⊗R as its adjoint functor. In particular, we have functors

FB := − ⊗R B : ModR −→ ModS

GB := HomS(B, −): ModS −→ ModR.

Then Theorem 1.3.2 says that FB and GB are adjoint functors. FB is sometimes called the left adjoint and GB the right adjoint, and together they are called an adjoint pair. Notice that the above setup uses right actions on A ⊗R B and HomS(B,C). There is a similar setup involving left actions, which is not discussed here.

Proposition 1.3.3. − ⊗R B : ModR → Ab is a right exact functor. Proof. Given f : A → A0 a homomorphism of right R-modules, let ϕ : A × B → A0 ⊗ B be the map sending (a, b) 7→ f(a) ⊗ b. This map is biadditive:

ϕ(a + a0, b) = f(a + a0) ⊗ b = (f(a) + f(a0)) ⊗ b = f(a) ⊗ b + f(a0) ⊗ b = ϕ(a, b) + ϕ(a0, b).

(The other properties are proven similarly.) By the universal mapping property, this induces a linear map

0 f ⊗ 1 : A ⊗R B 7−→ A ⊗R B a ⊗ b 7−→ f(a) ⊗ b.

Therefore − ⊗R B : ModR → Ab is a functor. The proof that A ⊗R − is a functor is similar. To prove that − ⊗R B is right exact, take an exact sequence

p B0 −→i B −→ B00 → 0

of left R-modules. We need to show that the induced sequence is exact:

1⊗p A ⊗ B0 −−→1⊗i A ⊗ B −−→ A ⊗ B00 → 0.

It is clear that im(1 ⊗ i) ⊆ ker(1 ⊗ p) because pi = 0, but it is not as obvious in the other direction. We will prove exactness at A ⊗ B another way in a moment.

20 1.3 Tensor Products 1 Preliminaries

X 00 00 To show 1 ⊗ p is onto, let ai ⊗ bi ∈ A ⊗ B . By exactness of the original sequence at 00 00 00 00 B , for each bi ∈ B there exists some bi ∈ B such that p(bi) = bi . Then

X  X X 00 (1 ⊗ p) ai ⊗ bi = ai ⊗ p(bi) = ai ⊗ bi .

Hence 1 ⊗ p is onto. Finally, notice that if we prove that (A ⊗ B)/ im(1 ⊗ i) is a solution to the universal mapping property in the definition of A ⊗ B00 then this will show that (A ⊗ B)/ im(1 ⊗ i) ∼= A ⊗ B00 which implies exactness at A ⊗ B. Given a biadditive map ϕ : A × B → G, we need to produce a biadditive map ψ and a linear mapϕ ¯ making the following commute: A × B00 ϕ ψ

(A ⊗ B)/ im(1 ⊗ i) G ϕ¯

Define ψ(a, b00) = (a ⊗ b) + im(1 ⊗ i) where p(b) = b00. We need to be carefull about defining 00 maps out of A × B . In particular, if p(b1) = p(b2), we must show that (a ⊗ b1) + im(1 ⊗ i) = 0 0 0 (a⊗b2)+im(1⊗i). But p(b1) = p(b2) implies there exists some b ∈ B such that i(b ) = b1−b2. Tensoring with a yields

0 (a ⊗ b1) − (a ⊗ b2) = a ⊗ (b1 − b2) = a ⊗ i(b ) which lies in im(1 ⊗ i) by exactness. Hence ψ is well-defined. To check that ψ is biadditive,

ψ(a + a0, b00) = (a + a0) ⊗ b + im(1 ⊗ i) = (a ⊗ b) + (a0 ⊗ b) + im(1 ⊗ i) = ((a ⊗ b) + im(1 ⊗ i)) + ((a0 ⊗ b) + im(1 ⊗ i)) = ψ(a, b00) + ψ(a0, b00).

The rest is proven similarly. Now we need to produceϕ ¯ :(A ⊗ B)/ im(1 ⊗ i) → G. We will first define a map A × B → G by (a, b) 7→ ϕ(a, p(b)); it is easy to see that this is biadditive. This induces a linear map Φ : A ⊗ B → G. Moreover, for any b0 ∈ B0,

0 0 Φ(a ⊗ i(b )) = ϕ(a, pi(b )) = ϕ(a, 0) = 0 which shows that ker Φ = im(1⊗i). Therefore we may define the quotient map out of A⊗B:

ϕ¯ :(A ⊗ B)/ im(1 ⊗ i) −→ G which inherits linearity from Φ. This completes the proof of right exactness. Example 1.3.4. For an integer n > 0,

n 0 → Z −→ Z → Z/nZ → 0

21 1.3 Tensor Products 1 Preliminaries is a short exact sequence. Tensoring with the Z-module Z/nZ gives a sequence n Z/nZ −→ Z/nZ → Z/nZ ⊗ Z/nZ → 0.

However, multiplication by n is the zero map on Z/nZ, so this shows that − ⊗ Z/nZ is not left exact. As a bonus, the tensored exact sequence splits into

0 → Z/nZ → Z/nZ ⊗ Z/nZ → 0. ∼ So Z/nZ ⊗ Z/nZ = Z/nZ as abelian groups. Example. Three important examples of tensor products. 1 Let R be a commutative ring and I and ideal of R. Then

F : ModR −→ ModR/I

M 7−→ M/IM (f : M → N) (f¯ : M/IM → N/IN) is a functor. It turns out that this ‘quotient functor’ is naturally isomorphic to the tensor product − ⊗R R/I.

2 Let S be a multiplicative subset of R. The localization S−1R is the set of formal r −1 quotients s , where r ∈ R and s ∈ S. S R is a ring via the operations r r0  rr0 r r0 rs0 + r0s = and + = s s0 ss0 s s0 ss0

r r0 0 0 and the relation s ∼ s0 if and only if there is some t ∈ R so that t(rs − r s) = 0. For any such set S, there is a map

−1 ϕS : R −→ S R r r 7−→ 1 This turns localization into a functor:

−1 S (−): modR −→ modS−1R

M 7−→ S−1M (f : M → N) (S−1f : S−1M → S−1N)

where modR denotes the category of all finite dimensional (right) R-modules. It turns −1 −1 −1 out that S (−) is naturally isomorphic to − ⊗R S R. In fact, S (−) is an exact functor (on the right and the left). For example, consider the exact sequence

2 0 → Z −→ Z → Z/2Z → 0.

22 1.3 Tensor Products 1 Preliminaries

Since Q is a localization of Z via the multiplicative subset of nonzero integers, we can apply − ⊗Z Q to this sequence and obtain another exact sequence.

However, if we instead apply − ⊗Z Z/2Z to the sequence

0 → Z → Q → Q/Z → 0 things don’t work out quite as nicely, since it becomes

0 → Z/2Z → 0 → (Q/Z) ⊗ (Z/2Z) → 0.

This is clearly not exact at Z/2Z. ∼ 3 The identity functor is a tensor product: − ⊗R R = idMod-R.

23 2 Special Modules

2 Special Modules

2.1 Projective Modules

We mentioned projective modules in Section 1.2 as they relate to the covariant Hom functor, but here we will define them in a more module-oriented context. We will see shortly that the two notions coincide. Definition. A left R-module P is projective if for every diagram with exact rows and solid arrows, shown below, there is a dashed arrow making the diagram commute. P

M N 0 In other words, projectives allow you to lift along surjections. We will prove Theorem 2.1.1. Over a ring R, a left R-module P is projective if and only if it is a direct summand of a free R-module. Example 2.1.2. As a Z/6Z-module, Z/6Z is isomorphic to Z/2Z ⊕ Z/3Z which shows that Z/2Z and Z/3Z are projective Z/6Z-modules. However, Z/2Z and Z/3Z are not free (6 kills everything in both submodules) and therefore are not projective in the category of Z/6Z-modules. A long-standing open question, proposed by Serre and proven by Quillen and Suslin, is: If P is a projective module over k[x1, . . . , xn] then must P be free? We now show that the earlier definition of projectives agrees with our definition in this chapter.

Proposition 2.1.3. P is projective ⇐⇒ HomR(P, −) is exact.

Proof. We proved HomR(P, −) is always left exact. For a sequence 0 0 → M 0 −→α M −→α M 00 → 0

applying HomR(P, −) induces an exact sequence with solid arrows: 0 00 0 → Hom(P,M ) → Hom(P,M) → Hom(P,M ) 99K 0. To get the 0 on the right, note that by definition of projective, every f : P → M 00 factors through α : M → M 00: P f˜ f α M 00 M 0

Notice that is equivalent to showing Hom(P,M) → Hom(P,M 00) is surjective. The converse follows by reversing the entire argument.

24 2.1 Projective Modules 2 Special Modules

Perhaps the most important example of a projective module is a free module. This is highly useful since in the future when we wish to build modules out of projectives, we may instead use free modules which are easier to work with.

Proposition 2.1.4. Every free module is projective.

Proof. Suppose F is a free module with basis {fi}i∈I . Let α : M → N be a surjection, i.e. the row of the following diagram with solid arrows is exact. F

α˜ ϕ α M N 0

If ϕ : F → N is an R-linear map, denote ϕ(fi) = ni ∈ N. Since α is surjective, there exists an mi ∈ M such that α(mi) = ni. Then we will defineα ˜(fi) = mi and extend by linearity to all of F . It is clear that αα˜ and ϕ agree on {fi} which is a basis for F ; therefore the diagram commutes. An interesting example of projective modules arises from localization in rings:

Proposition 2.1.5. For a commutative ring R and P a finitely generated R-module, P is projective if and only if for all prime ideals p ⊂ R, the localization Pp is free. A useful property of projectives is

Proposition 2.1.6. P is projective ⇐⇒ every short exact sequence ending in P splits.

Proof. ( =⇒ ) Suppose we have a short exact sequence

0 −→ M −→ N −→ P −→ 0.

where P is projective. The identity on P induces a surjection f: P f id

0 M N P 0

Hence the sequence splits. ( ⇒ = ) Conversely, take a finite presentation of P :

0 −→ K −→ F −→ P −→ 0 where F is free and the map F → R takes a generating set {fi} of F to a generating set ∼ {pi} of P . By hypothesis the sequence splits, so F = P ⊕ K, and by the earlier (unproven) characterization of projectives, P is projective.

25 2.1 Projective Modules 2 Special Modules

In the above proof we used the fact that direct summands of free modules are projective. We will now prove an even stronger result which implies this. Theorem 2.1.7. In the category of R-modules for a ring R, (1) Direct summands of projective modules are projective.

(2) Direct sums of projective modules are projective. Proof. (1) Suppose a projective module P can be written P = P 0 ⊕ P 00. Consider the following diagram with the bottom row exact. P

π i ˜ f P 0 g˜ g

M N 0 f

Given g surjective, we want to construct a mapg ˜ making the smaller triangle commute. Note that P = P 0 ⊕ P 00 induces π : P → P 0 the natural projection and i : P 0 ,→ P the inclusion map. Since P is projective, there exists a map f˜ : P → M making the larger triangle commute. It suffices to show thatg ˜ = fi˜ is what we are looking for, but this is obvious since for any p0 ∈ P 0,

ffi˜ (p0) = ff˜(p0) = g(p0).

Hence P 0 is projective. 0 00 0 00 (2) If P and P are projective then HomR(P , −) and HomR(P , −) are exact functors. Thus we have a natural isomorphism of functors

Hom(P 0 ⊕ P 00, −) ∼= Hom(P 0, −) ⊕ Hom(P 00, −).

Since direct sums of exact functors are exact, this shows Hom(P 0 ⊕ P 00, −) is exact and therefore P 0 ⊕ P 00 is projective. In the infinite case, note that ! M ∼ Y Hom Ai,B = Hom(Ai,B). i∈I i∈I Then the result follows from the fact that direct products of exact functors are exact. Corollary 2.1.8. (a) Every module over a field is projective.

(b) If R is a PID, then every projective R-module is also free.

26 2.1 Projective Modules 2 Special Modules

Proof. (a) A module over a field is a vector space which is free, hence projective by Propo- sition 2.1.4. (b) For modules over a PID, every submodule of a free module is free. Projective modules have many desirable properties.

Proposition 2.1.9. Every projective left R-module P has a free complement, that is, a free left R-module F such that P ⊕ F is free.

Proof. Since P is projective, it’s the direct summand of a free module, i.e. there exists a Q so that P ⊕ Q is free. The trick is to consider

F = Q ⊕ P ⊕ Q ⊕ · · · which is free since each pair in the sum is free. But notice that P ⊕F = P ⊕Q⊕P ⊕Q⊕· · · is free for the same reasons; hence P has a free complement.

Proposition 2.1.10. For a commutative ring R and projective R-modules P and Q, P ⊗R Q is a projective R-module.

Proof. Since P,Q are projective, there exist free modules F1,F2 and modules G1,G2 such that F1 = P ⊕ G1 and F2 = Q ⊕ G2. Tensoring these two expressions gives us

F1 ⊗ F2 = (P ⊕ G1) ⊗ (Q ⊕ G2) = (P ⊗ Q) ⊕ (P ⊗ G2) ⊕ (G1 ⊗ Q) ⊕ (G1 ⊕ G2).

Hence it suffices to prove F1 ⊗ F2 is free which will imply P ⊗ Q is projective. This is easily seen from the fact that tensor distributes across direct sums (we already used this above): ! ! M M M F1 ⊗ F2 = R ⊗ R = (R ⊗ R). i j i,j

Recall that a left R-module M is finitely generated if M is the quotient of a free left R-module F ∼= Rn for n < ∞. We may then write M ∼= F/K where K is a submodule of F .

Definition. A left R-module M is finitely presented if it is finitely generated and M ∼= F/K where K is finitely generated.

This can be summarized with an exact sequence

0 −→ K −→ Rn −→ M −→ 0.

If M is finitely presented we will instead write

Rm −→ Rn −→ M −→ 0.

27 2.2 Modules Over Noetherian Rings 2 Special Modules

Proposition 2.1.11. Every finitely generated projective left R-module P is finitely presented.

Proof. Since P is finitely generated, we have an exact sequence

0 −→ K −→ F −→ M −→ 0

where F is free. P projective implies the sequence splits, so F ∼= P ⊕ K. Then we see that K is finitely generated since it is a direct summand, and hence an image, of F a finitely generated module. Therefore P is finitely presented. Finitely generated modules are the most important type of module in any module cate- gory. In the next section we will see that finitely generated modules are intimately related to conditions on chains of submodules.

2.2 Modules Over Noetherian Rings

The proofs in this section are for the most part taken from Stein’s Introduction to Algebraic Number Theory.

Definition. An R-module M is noetherian if every submodule of M is finitely generated. Likewise, a ring R is noetherian if every ideal of R is finitely generated, i.e. R is noetherian as a module over itself.

Remark. Every submodule of a noetherian module is also noetherian.

Definition. An R-module M satisfies the ascending chain condition if every chain M1 ⊂ M2 ⊂ M3 ⊂ · · · of submodules of M eventually stabilizes, i.e. there is some N such that Mn = MN for all n ≥ N. Theorem 2.2.1. For an R-module M, the following are equivalent:

(1) M is noetherian.

(2) M satisfies the ascending chain condition.

(3) Every nonempty collection of submodules of M contains a maximal element.

Proof. (1 =⇒ 2) Suppose M1 ⊂ M2 ⊂ M3 ⊂ · · · is a chain of submodules of M. Then ∞ [ M∞ := Mn is also a submodule of M. Since M is noetherian, there is a finite set n=1 a1, . . . , am of generators of M∞. Each ai must be contained in some Mj so there must be an n such that a1, . . . , am ∈ Mn. But then Mk = Mn for all k ≥ n so M has the ascending chain condition. (2 =⇒ 3) Suppose there exists a nonempty collection S of submodules of M that does not contain a maximal element. We will show that this results in an infinite ascending chain that does not stabilize. Note that S is infinite, otherwise it would necessarily contain a maximal element. Take M1 ∈ S. Since M1 is not maximal, there exists a distinct M2 ∈ S

28 2.2 Modules Over Noetherian Rings 2 Special Modules

so that M1 ⊂ M2. Likewise, M2 is not maximal so there exists some M3 ∈ S such that M2 ⊂ M3 and so forth. Thus we have a chain of submodules M1 ⊂ M2 ⊂ M3 ⊂ · · · where each containment is proper, contradicting the ascending chain condition. (3 =⇒ 1) Suppose M contains a submodule M 0 that is not finitely generated. Let T be the collection of all submodules of M 0 that are finitely generated. Then T has a maximal element L, and L is finitely generated so there is an element a ∈ M 0 such that a 6∈ L. Then L0 = L + Ra is an element of T and properly contains L, contradicting maximality. Hence all submodules of M must be finitely generated, so M is noetherian.

f g Lemma 2.2.2. If 0 → L −→ M −→ N → 0 is an exact sequence of R-modules then M is noetherian if and only if L and N are noetherian.

Proof. (=⇒) First suppose M is noetherian. The sequence is exact, so ker(f) = 0 which means L is a submodule of M. By the remark, L is noetherian. Next, take a submodule N 0 of N. Then g−1(N 0) is a submodule of M so it is finitely generated, and its image in N is finitely generated. Thus N is noetherian.

0 0 (⇐=) Now suppose L and N are noetherian. If M is a submodule of M, then M0 = f(L)∩M is isomorphic to a submodule of L, so M0 is finitely generated, say by the elements a1, . . . , an. 0 0 The quotient M /M0 is isomorphic via g to a submodule of N, so M /M0 is also finitely 0 generated, say by b1, . . . , bm. For each 1 ≤ i ≤ m, let ci be the lift of bi to the M -module 0 M0. Then the elements a1, . . . , an, c1, . . . , cm generate M . Hence M is noetherian. Proposition 2.2.3. Suppose R is a noetherian ring. Then an R-module M is noetherian if and only if M is finitely generated.

Proof. If M is noetherian, then every submodule of M is finitely generated, including M itself. On the other hand, suppose M = RX where X = {a1, . . . , an} for a finite number n. n Then the map R → M given by (0,..., 0, 1, 0,..., 0) 7→ ai, where the 1 is in the ith spot, is a surjective homomorphism. Consider induction on exact sequences

0 → R → R ⊕ R → R → 0

which show by Lemma 2.2.2 that Rn is noetherian. Lemma 2.2.2 also shows that the homo- morphic image of a noetherian module is noetherian, so M is noetherian.

Proposition 2.2.4. If ϕ : R → S is a surjective ring homomorphism and R is noetherian, then S is noetherian.

Proof. The kernel of ϕ is an ideal of R, so the sequence of R-modules

0 → ker(ϕ) → R → S → 0

is exact, so by Lemma 2.2.2, S is a noetherian R-module. Suppose J is an ideal of S. Since J is a submodule of S, J is also an R-module. Then by Proposition 2.2.3, J is finitely generated. Since R acts on J via S, the R-generators of J are also S-generators, so J is finitely generated as an ideal of S. Hence S is noetherian.

29 2.2 Modules Over Noetherian Rings 2 Special Modules

Theorem 2.2.5 (Hilbert’s Basis Theorem). If R is a noetherian ring and S is finitely generated as a ring over R, then S is noetherian.

Proof omitted.

Corollary 2.2.6. If R is a noetherian ring, the polynomial ring R[x1, . . . , xn] for any n, and any of its quotients, is noetherian.

Proof. Follows immediately from Hilbert’s Basis Theorem.

Example 2.2.7. Z is a noetherian ring because every ideal is generated by a single element (Z is a PID). As a result, any finitely generated abelian group is noetherian, as are their subgroups.

A local ring is a ring which has a unique maximal ideal. Nakayama’s Lemma gives a result about local rings which are noetherian:

Lemma 2.2.8 (Nakayama’s Lemma). Let R be a local noetherian ring and let A be a proper ideal of R. Then for any finitely generated R-module M,

(1) AM = M =⇒ M = 0.

(2) If N is a submodule of M such that N + AM = M, then N = M.

Proof. (1) Suppose AM = M but M 6= 0. Choose a minimal set of generators of M, call them e1, . . . , en. Write e1 = a1e1 + ... + anen for some ai ∈ A. Then (1 − a1)e1 = a2e2 + ... + anen. Since 1−a1 6∈ M, it must be a unit. Thus e2, . . . , en generate M, contradicting our minimality assumption. Therefore M = 0. (2) Suppose N +AM = M. We will show that A(M/N) = M/N (⊃ is obvious). Consider X m + N for any m ∈ M. Then m = n + aimi with ai ∈ A and mi ∈ M. So

X X m + N = (aimi + N) = ai(mi + N) which implies m+N ∈ A(M/N). Hence A(M/N) = M/N and by (1), this implies M/N = 0. Therefore M = N. ∞ \ Corollary 2.2.9. If R is a local noetherian ring with maximal ideal I, then In = 0. n=1 For a commutative ring R and finitely generated R-modules A and B, the tensor prod- uct A ⊗R B is easily seen to be finitely generated. However the same cannot be said of HomR(A, B). It turns out that we must have a noetherian ring for Hom to preserve the finitely generated property.

Proposition 2.2.10. If R is commutative and noetherian and A, B are finitely generated R-modules, then HomR(A, B) is a finitely generated R-module.

30 2.3 Injective Modules 2 Special Modules

Proof. Since R is noetherian, it suffices to show that HomR(A, B) is a submodule of a finitely generated R-module. Note that since A is finitely generated, we have an exact sequence

0 −→ K −→ Rn −→ A −→ 0.

Applying the contravariant functor HomR(−,B), which is left exact, to this sequence pro- duces another exact sequence

n 0 −→ HomR(A, B) −→ HomR(R ,B) −→ HomR(K,B).

n n So HomR(A, B) is a submodule of HomR(R ,B), which is isomorphic to B and this is clearly finitely generated since B is. Hence HomR(A, B) is finitely generated. It takes some effort to produce an example of finitely generated modules A and B with Hom(A, B) not finitely generated. The next example really shows how nice it is to work over noetherian rings.

Example 2.2.11. Let V be an infinite dimensional Fp-vector space and define a ring R = Z ⊕ V with multiplication defined by (m, v)(m0, v0) = (mm0, mv0 + m0v). Then V is an ideal of R and it is easy to see that V is not a finitely generated ideal. Next define A = (R/V )/p(R/V ); this is quite obviously isomorphic to Z/pZ = Fp. To produce a counterexample to Proposition 2.2.10 in the non-noetherian case, we will show ∼ that HomR(A, R) = V which is not finitely generated. Suppose f : A → R is R-linear. For any a ∈ Fp mapping to (m, v) ∈ R = Z ⊕ V we must have (0, 0) = f(0) = f(pa) = pf(a) = p(m, v) = (pm, pv).

But Z is torsion-free so this forces m = 0. Thus f : a 7→ (0, v) so we should define S : Hom(A, R) −→ V f 7−→ f(1) T : V −→ Hom(A, R)

v 7−→ (fv : 1 7→ v).

It is easy to check S and T are R-linear. By construction they are inverses, so we have an R-isomorphism between Hom(A, R) and V . We proved V is not finitely generated, so this counterexample shows that Hom does not always preserve finitely generated when R is not noetherian.

2.3 Injective Modules

There is another special type of module that shows up frequently in homological algebra.

Definition. A left R-module E is injective if for every diagram with exact rows and solid arrows, shown below, there is a dashed arrow making the diagram commute.

31 2.3 Injective Modules 2 Special Modules

E

0 A B

Notice the difference between the diagrams for projectives and injectives. We remarked that projectives allow you to lift along surjections. Similarly, injectives allow you to extend injections. In Section 1.2 we defined injectives in terms of exactness of the contravariant Hom functor. We now prove that these two definitions coincide.

Proposition 2.3.1. A left R-module E is injective ⇐⇒ HomR(−,E) is exact. Proof. Suppose we have an exact sequence

p 0 → A −→i B −→ C → 0.

We must show that the sequence

p∗ i∗ 0 → HomR(C,E) −→ HomR(B,E) −→ HomR(A, E) → 0

is exact. We proved in Proposition 1.2.2 that HomR(−,E) is always left exact, so it remains ∗ to show exactness at HomR(A, E). In other words we must prove i is surjective if and only if i is injective. On one hand, if f ∈ Hom(A, E) there exists g ∈ Hom(B,E) with f = i∗(g) = gi; that is, E

f g

0 A B i

commutes, showing E is injective. Conversely, if E is injective then for any f : A → E there exists g : B → E making the above diagram commute. Then we see that f = gi = i∗(g) ∈ im i∗ so i∗ is surjective. This proves Hom(−,E) is an exact functor. Similar to projectives, we have Proposition 2.3.2. E is injective ⇐⇒ every short exact sequence beginning with E splits. Proof. ( =⇒ ) Consider the diagram E

id g

0 E B i

32 2.3 Injective Modules 2 Special Modules

Since E is injective, the identity on E induces g : B → E such that gi = idE. Hence the sequence splits. ( ⇒ = ) Suppose E → M → M 00 → 0 is exact, so that by hypothesis M ∼= E ⊕ M 00. We will see shortly that direct summands of injectives are injective, so it follows that E is injective. There is a useful characterization of injectives that allows us to restrict our attention to ideals of R, rather than all module injections A,→ B. Before stating and proving Baer’s Criterion, we need

Zorn’s Lemma. Let X be a poset. If every chain in X has an upper bound in X then X has maximal elements.

Zorn’s Lemma is of immense consequence in set theory, for it is logically equivalent to the Axiom of Choice. Zorn’s Lemma is often invoked to prove the existence of maximal ideals in a ring, and in particular a non-noetherian ring since the ascending chain condition guarantees the existence of maximal ideals when R is noetherian. We will sometimes say a set is Zornable (not an official term) if it satisfies the conditions of Zorn’s Lemma.

Theorem 2.3.3 (Baer’s Criterion). Let E be a left R-module. E is left injective ⇐⇒ every R-linear map f : I → E, where I is an ideal of R, can be extended to R:

E

f g

0 I R i

Proof. ( =⇒ ) is obvious by the definition of injectives. To prove ( ⇒ = ) suppose E

f

0 A B i

has an exact row. Let X = {(A0, g0) | A ⊆ A0 ⊆ B and g0 extends f}. We define a partial ordering on X by (A0, g0)  (A00, g00) if A0 ⊆ A00 and g00 extends g0. By construction X is 0 0 0 0 bounded above, so this makes X into a Zornable set. Suppose (A1, g1)  (A2, g2)  · · · is 0 [ 0 0 0 0 0 0 0 a chain in X. Define A = Ai and g (a ) = gi(a ) where a ∈ Ai. This defines an upper i≥1 bound on the chain, so by Zorn’s Lemma there exists a maximal element in the chain. Let (A0, g0) be such a maximal element. We claim A0 = B. Suppose not; then there is some b ∈ B r A0. Define a left ideal of R by I = {r ∈ R | rb ∈ A0} – this is sometimes called a colon ideal, denoted (bR : A0). We construct a map h : I → E by defining h(r) = g0(rb). This map may be lifted by

33 2.3 Injective Modules 2 Special Modules

E

h h∗

0 I R i

∗ Let A1 = A0 + Rb and define g1(a0 + rb) = g0(a0) + rh (1). Note that g1 may not be well- defined since a0 + rb is not necessarily the unique way to write an element of A1. However, if g1 is well-defined, we will have shown (A0, g0) ≺ (A1, g1) contradicting maximality of 0 0 (A0, g0). Thus to finish the proof we show well-definedness. Suppose a0 + rb = a0 + r b. 0 0 0 Then (r − r )b = a0 − a0 ∈ A =⇒ r − r ∈ I. This gives us

0 0 0 g0(a0 − a0) = g0((r − r )b) = h(r − r ).

Now we can’t take the constant r − r0 out of h, since this would leave 1 behind, and 1 6∈ I. But we do have h(r − r0) = h∗(r − r0) and h∗ is defined on R, so h∗(r − r0) = (r − r0)h∗(1). It follows easily that g1 is well-defined. We also have

Proposition 2.3.4. Direct sums and direct summands of injectives are injective.

Proof. Similar to Theorem 2.1.7.

Definition. A module M over a domain R is divisible if for all m ∈ M and nonzero r ∈ R, there is some m0 ∈ M such that m = rm0.

Informally, this says that in a divisible R-module, you can ‘divide’ by R.

Example 2.3.5. Q is a divisible Z-module. In fact, this holds for the field of fractions of any domain.

Theorem 2.3.6. Every injective left R-module is divisible.

Proof. Consider the inclusion nZ ,→ Z. Then divisibility of an R-module A is equivalent to completing the diagram A y x - ↑ 1 n 0 nZ Z Of course this is immediate when A is injective. The converse holds when R is a PID:

Theorem 2.3.7. Let R be a PID. Then

34 2.3 Injective Modules 2 Special Modules

(1) Every divisible R-module is injective.

(2) Quotients of injectives are injective.

Proof. (2) follows from (1), while the proof of (1) is similar to the first part of the proof of Baer’s Criterion.

Example 2.3.8. By Example 2.3.5 we see that Q is an injective Z-module. Moreover, by (2) of Theorem 2.3.7, Q/Z is also an injective Z-module. Our next goal is to show that every left R-module can be realized as a submodule of an injective left R-module. We begin by proving this for Z-modules (abelian groups). Let M ∼ be a Z-module. Then M = F/S where F is some free abelian group and S is the module ∼ M L of relations. By the fundamental theorem of abelian groups, M = Z/S and Z can be i∈I L embedded in Q so we have a composition ∼ M M M = Z/S ,→ Q/S. i∈I i∈I L L Now Q is divisible so Q is also divisible. Then by Theorem 2.3.7, Q/S is injective. Before proving the property for R-modules in general, we will need

Proposition 2.3.9. Let ϕ : R → S be a ring homomorphism and E an injective left R- module. Then HomR(S,E) is an injective left S-module.

Proof. First let’s convince ourselves that HomR(S,E) is a left S-module at all. The map ϕ : R → S induces a left R-action on S given by r · s = ϕ(r)s. S is also a right module over itself, so there is an available action for S to act on HomR(S,E) on the left. Now, note that HomR(S,E) is an injective left S-module ⇐⇒ HomS(−, HomR(S,E)) is an exact functor. By Hom-Tensor Adjointness, ∼ ∼ HomS(−, HomR(S,E)) = HomR(S ⊗S −,E) = HomR(−,E) and HomR(−,E) is exact precisely when E is injective. In fact we have proven that the converse of the proposition holds as well.

Corollary 2.3.10. For any divisible abelian group D, HomZ(R,D) is an injective left R- module.

∗ Definition. For an abelian group A, the group A := HomZ(A, Q/Z) is called the dual of A.

Theorem 2.3.11. Let A be an abelian group and A∗ its dual. Then

(1) If A is a left (resp. right) R-module, then A∗ is a right (resp. left) R-module.

(2) Given a surjection (resp. injection) of R-modules A → B, the induced map B∗ → A∗ is injective (resp. surjective).

35 2.3 Injective Modules 2 Special Modules

(3) If P is projective, P ∗ is injective. Likewise, if E is injective then E∗ is projective.

One might hope for a “duality theorem” that says A ∼= A∗ for any abelian group A. Unfortunately, this is false in general. We will however prove that A embeds into A∗∗.

Example 2.3.12. If A is finite, then A ∼= A∗. Indeed, since Hom splits over finite direct sums, it suffices to consider A = Z/nZ, in which case the map

Z/nZ −→ HomZ(Z/nZ, Q/Z) 1
1 7−→ 1 7→ n is an isomorphism. Hence there is a canonical isomorphism between A and its double dual:

∗∗ ∗ A −→ A = HomZ(A , Q/Z) a 7−→ (ϕ 7→ ϕ(a)).

∗ ∗∗ On the other hand, if A = Z, we have Z = Q/Z so that Z = HomZ(Q/Z, Q/Z). One ∼ can show that HomZ(Q/Z, Q/Z) = Zb, the profinite completion of Z. There is a natural embedding Z ,→ Zb. Lemma 2.3.13. For any abelian group A, there is an embedding A,→ A∗∗.

∗∗ Proof. Define Φ : A → A by a 7→ ϕa, where ϕa(f) = f(a) for any f : A → Q/Z. Take any nonzero a ∈ A. We must show that there exists some f : A → Q/Z such that f(a) 6= 0. Let B = Za. Then there is a map ψ : B → Q/Z given by a 7→ ψ(a) 6= 0. Now since Q/Z is injective, ψ extends along the inclusion B,→ A to a map f : A → Q/Z which by construction has f(a) 6= 0. Hence Φ is injective.

Theorem 2.3.14. For any left R-module M, there is an embedding M,→ E where E is an injective left R-module.

∗∗ ∗ Proof. By Proposition 2.3.9, M = HomZ(M , Q/Z) is injective. Therefore the embedding M,→ M ∗∗ realizes M as a submodule of an injective module.

This shows we can embed a module into some injective module, but HomZ(R,D) is often way bigger than necessary. The natural follow-up question is: What is the smallest injective E such that M is a submodule of E? To answer this, first we need

Definition. E is an essential extension of M if there exists an injective map α : M,→ E such that for all nonzero submodules S ⊆ E, α(M) ∩ S 6= 0.

Example 2.3.15. If M = M 0 ⊕ M 00 then M is not an essential extension of M 0 or M 00.

The most important essential extensions are the maximal ones. Essential extensions are related to injectives by

Proposition 2.3.16. An R-module M is injective ⇐⇒ M has no proper essential exten- sions.

36 2.3 Injective Modules 2 Special Modules

Proof. ( =⇒ ) Suppose α : M,→ E is an essential extension. Since M is injective, the sequence 0 → M −→α E splits, so E ∼= M ⊕ N for some N. If N 6= 0, α : M,→ E is not essential so we must have N = 0 from which it follows that E ∼= M. ( ⇒ = ) Embed M −→α E where E is injective; this is possible by Theorem 2.3.14. Note that α is not an essential extension, so there exists a submodule S ⊂ E such that α(M) ∩ S = 0. The set of such S is Zornable so pick N to be a maximal submodule of E with this property. We will show M −→α E −→π E/N is essential. The kernel of the projection π is exactly N, so since α(M) ∩ N = 0, the composite π ◦ α is one-to-one. Let S/N be any nonzero submodule of E/N, i.e. N ( S ⊆ E. By maximality of N, α(M) ∩ S 6= 0, but this implies (S/N) ∩ (α(M)/N) 6= 0 and thus π ◦ α is an essential extension. By hypothesis, M has no essential extensions so E = α(M) + N. Further, α(M) ∩ N = 0 so E = α(M) ⊕ N. Then α(M) is injective since it is the direct summand of an injective. It follows that M is injective.

Theorem 2.3.17. The following are equivalent for M ⊆ E.

(i) E is a maximal essential extension M.

(ii) E is an essential extension of M and E is injective.

(iii) E is injective and there does not exist an injective module E0 such that M ⊆ E0 ( E. An extension E satisfying (i) – (iii) is called the injective hull of M and it is unique up to isomorphism.

Proof omitted.

Examples.

1 If R is a domain, the field of fractions Q of R is the injective hull of R.

2 Let R = k[x1, . . . , xn] and consider the ideal m = (x1, . . . , xn). Let E(R/m) denote the −k1 −kn injective hull of R/m. E(R/m) has a k-basis given by monomials x1 ··· xn where ki ≥ 0. Further, R acts on E(R/m) via

( −k1 −kn −k1 −kn rx ··· x if for all i, powers of xi in rx ··· x are ≤ 0 r · (x−k1 ··· x−kn ) = 1 n 1 n 1 n 0 otherwise.

−2 −1 For example, x2 · x2 = 0 and xi · 1 = 0 for all i. The embedding

R/m ,→ E(R/m) 1¯ 7→ 1

shows that E(R/m) is an essential extension of R/m.

37 2.4 Flat Modules 2 Special Modules

2.4 Flat Modules

As projectives and injectives arose from the Hom functor, there is a third type of module that arises from the tensor product.

Definition. For a ring R, a right R-module A is flat if A⊗R − is an exact functor. Flatness of left R-modules is defined similarly for the covariant functor − ⊗R A. Equivalently, a right R-module A is flat if and only if for every injection of left R-modules i : B → C, 1A ⊗ i : A ⊗R B → A ⊗R C is also an injection. ∼ Example 2.4.1. For any ring, R is a flat R-module. To see this, recall that R ⊗R A = A ∼ and R ⊗R B = B and so we have a commutative diagram i A B

∼= ∼=

R ⊗R A R ⊗R B 1A ⊗ i showing 1A ⊗ i is injective. Proposition 2.4.2. For a ring R, (i) Every projective right (or left) R-module is flat.

(ii) A direct sum of right (or left) R-modules is flat if and only if each summand is flat. Proof. (i) follows from (ii) since a free right R-module is the direct sum of copies of R, which is flat. Moreover, P is projective if and only if it is the direct summand of a free module, which, combined with the first statement says that projectives are always flat. M To prove (ii), consider a direct sum Mk of right R-modules Mk. For any family of R- k M M linear maps {fk : Mk → Nk} there is a map f : Mk → Nk taking (mk) 7→ (fk(mk)), k k and clearly f is injective if and only if each fk is injective. So if i : A → B is a left R-module injection, there is a commutative diagram

L 1 ⊗ i L ( Mk) ⊗R A ( Mk) ⊗R B

∼= ∼= L L (Mk ⊗R A) (Mk ⊗R B) f where ϕ is defined by (mk ⊗ a) 7→ (mk ⊗ i(a)). By the above, 1 ⊗ i is an injection ⇐⇒ each M 1Mk ⊗ i is an injection. Hence Mk is flat ⇐⇒ each Mk is flat. k

38 2.4 Flat Modules 2 Special Modules

Lemma 2.4.3. For a left R-module M, the following are equivalent:

(a) M is flat.

0 0 (b) Every inclusion 0 → N → N of R-modules induces an inclusion 0 → N ⊗R M → N ⊗R M. (c) Every inclusion 0 → N 0 → N of finitely generated R-modules induces an inclusion 0 0 → N ⊗R M → N ⊗R M. Corollary 2.4.4. If every finitely generated submodule of a right R-module M is flat, then M is flat.

However, the converse to Corollary 2.4.4 is false in general:

Example 2.4.5. For a field k, let R = k[x, y]. Then R is flat as a module over itself (it is free) and the ideal M = (x, y) is a finitely generated submodule of R, but M is not flat.

Proposition 2.4.6. If R is an integral domain and A is a flat R-module, then A is torsion- free.

Proof. Let Q be the field of fractions of R. Since A is flat, the functor − ⊗R A is exact, so the exact sequence 0 → R → Q induces an exact sequence

0 → R ⊗R A → Q ⊗R A. ∼ We know R ⊗R A = A, and Q ⊗R A is a vector space over Q, so it is torsion-free and it follows that A is torsion-free. The converse holds over a PID:

Proposition 2.4.7. If R is a PID, then every torsion-free R-module B is flat.

Proof. The theory of modules over a PID says that every finitely generated R-module M that is torsion-free is also free. Thus every finitely generated submodule M ⊆ B is free, hence projective, hence flat. By Corollary 2.4.4, this implies that B is also flat.

Example 2.4.8. For any multiplicative set S ⊆ R, the localization S−1R is a flat module.

Similar to a property for injectives, we have

Theorem 2.4.9. A right R-module A is flat if and only if the sequence

1A⊗i 0 → A ⊗R I −−−→ A ⊗R R

is exact for every left ideal I ⊂ R, where i : I → R is the inclusion map.

Definition. A left R-module M is faithfully flat provided that for all left R-modules A and B, the sequence 0 → A → B is exact if and only if 0 → A ⊗R M → B ⊗R M is exact. One can define faithful flatness similarly for right modules.

39 2.4 Flat Modules 2 Special Modules

Lemma 2.4.10. A left R-module M is faithfully flat if and only if M is flat and for all left R-modules A, A ⊗R M = 0 implies A = 0. ∼ L L ∼ L Example 2.4.11. If F is free, say F = i∈I R, then A ⊗R i∈I R = i∈I A which is nonzero precisely when A is nonzero. Hence free modules are faithfully flat.

Example 2.4.12. Any localization S−1R of a ring R is flat by Example 2.4.8, but is not faithfully flat in general. For example, if R is an integral domain with field of fractions K – the localization of R at the multiplicative set R r {0} – then for any nonzero torsion module M, M ⊗R K = 0. Example 2.4.13. In general, projective modules are not faithfully flat. For instance, sup- pose R has a nontrivial idempotent element e ∈ R, that is, e2 = e. Then e(1 − e) = 0 and 1−e is also an idempotent, and we have R ∼= eR⊗(1−e)R. So eR is a projective R-module, 2 but it is not faithfully flat since (1 − e)R ⊗R eR = (1 − e)R ⊗R e R = e(1 − e)R ⊗R eR = 0. Theorem 2.4.14. Let R be a commutative ring and M an R-module. Then the following are equivalent:

(a) M is flat.

(b) For all prime ideals p ⊂ R, the localization Mp is flat as an R-module (equivalently, as an Rp-module).

(c) For all maximal ideals m ⊂ R, the localization Mm is flat as an R-module (equivalently, as an Rm-module. Recall that an R-module M is finitely presented if there exists a short exact sequence 0 → K → Rn → M → 0 with n finite.

Proposition 2.4.15. Let M be a finitely generated R-module. Then

(a) If M is projective, then it is finitely presented.

(b) If M is finitely presented and 0 → L → Rm → M → 0 is any short exact sequence with m finite, then L is finitely generated.

Proof. (1) Let 0 → K → Rn → M → 0 be any presentation of M. Then since M is projective, the sequence splits, so K is a direct summand of Rn, thus a quotient of Rn and therefore finitely generated. (2) Assuming M is finitely presented, there is some short exact sequence 0 → K → Rn → M → 0 with n finite and K finitely generated. Consider the diagram 0 K Rn M 0

f g id

0 L Rm M 0

40 2.4 Flat Modules 2 Special Modules

Then since Rm is free, we get a map g : Rn → Rm making the right square commute. Call f the restriction of g to K, which gives a commutative square on the left. By the Snake Lemma (1.2.8), 0 = ker id → coker f −→∼ coker g → coker id = 0 is exact, so the middle arrow is an isomorphism. Now coker g is finitely generated as it is a quotient of Rm, so this implies coker f = L/f(K) is finitely generated. Since K was finitely generated, f(K) is also finitely generated and it now follows that L is finitely generated. We now relate finite presentability to flatness. First, we need a lemma.

Lemma 2.4.16. Suppose 0 → N → F → M → 0 is a short exact sequence of left R-modules, where M and F are flat. Then

(a) For any ideal I ⊆ R, N ∩ IF = IN. P (b) If F is free with R-basis {yi}i∈I , then for any n = riyi ∈ N ⊆ F , there exist P i∈I elements ni ∈ N such that n = i∈I rini.

(c) Suppose F is free. For any finite set of elements {n1, . . . , nr} ⊆ N, there exists a morphism f : F → N such that f(ni) = ni for all 1 ≤ i ≤ n. Proof. (a) Tensoring with I gives an exact sequence

I ⊗R N → I ⊗R F → I ⊗R M → 0. ∼ ∼ Notice that since M,F are flat, I ⊗R F = IF and I ⊗R M = IM, and the kernel of the resulting map IF → IM is precisely N ∩ IF . Therefore we have a commutative diagram with exact rows

I ⊗R N I ⊗R F I ⊗R M 0

ψ ∼= ∼=

0 N ∩ IF IF IM 0

This induces a map ψ : I ⊗R N → N ∩IF which by a quick diagram chase is an isomorphism. This identifies N ∩ IF with im ψ = IN ⊆ IF . (b) Consider the ideal I = (ri)i∈I ⊆ R. Then by (a), X n ∈ N ∩ IF = IN = riN i∈I and the statement follows. (c) We induct on r. When r = 1, let n = n = P r y for some finite I ⊆ I and some 1 i∈I0 i i 0 r ∈ R. Applying (b), we get n = P r n0 for some n0 ∈ N. For each y in a basis of F , i i∈I0 i i i i set ( 0 ni, if i ∈ I0 f(yi) = 0, if i 6∈ I0.

41 2.4 Flat Modules 2 Special Modules

Then this extends uniquely to an R-linear map f : F → N such that ! X X X 0 f(n) = f riyi = rif(yi) = rini = n. i∈I0 i∈I0 i∈I0 To induct, assume the statement holds for all finite subsets of N consisting of r −1 elements. For a given {n1, . . . , nr} ⊆ N, the inductive hypothesis constructs for us two maps f1, f2 : F → N satisfying

f1(n1) = n1 and f2(ni − f1(ni)) = ni − f1(ni) for all 2 ≤ i ≤ r.

Set f = f1 + f2 − f2 ◦ f1, where f2 ◦ f1 is defined by changing targets, i.e. considering f1 : F → N ⊆ F . Then f(n1) = f1(n1) = n1 and for each 2 ≤ i ≤ r,

f(ni) = f1(ni) + f2(ni − f1(ni)) = f1(ni) + ni − f1(ni) = ni. Therefore the statement holds for all r by induction. Theorem 2.4.17. For a left R-module M, the following are equivalent: (1) M is finitely generated and projective. (2) M is finitely presented and flat.

(3) M is finitely presented and for all maximal ideals m ⊂ R, the localization Mm is free. Proof. (1) =⇒ (2) is Propositions 2.4.15 and 2.4.2. (2) =⇒ (1) If M is finitely presented, then there exists a short exact sequence 0 → N → F → M → 0 where F is finitely generated and free and N is finitely generated, say by n1, . . . , nr ∈ N. Then since M is flat, Lemma 2.4.16 says there is a morphism f : F → N with f(ni) = ni for all i, but since the ni generate N, this means f|N = idN . Hence the short exact sequence splits, so in particular M is a direct summand of F and thus projective. (1) =⇒ (3) By the above, M is finitely presented (and flat), so there exists a presentation m n R → R → M → 0 with m, n finite. Flatness implies that tensoring with Rm for any maximal ideal m yields an exact sequence of Rm-modules m n Rm → Rm → Mm → 0.

Therefore Mm is finitely presented. Theorem 2.4.14 also shows Mm is flat. By finite presenta- tion, Mm/m is a finite dimensional vector space over the residue field κ(m) = Rm/m. Choose a basisx ¯1,..., x¯r of Mm/m which are the images of x1, . . . , xr ∈ Mm. Then by Nakayama’s Pr Lemma (2.2.8), Mm = i=1 Rmxi. Thus Mm has a finite presentation of Rm-modules

N → F → Mm → 0

where F is free with basis y1, . . . , yr whose images in Mm are x1, . . . , xr. Suppose n = Pr Pr Pr i=1 riyi ∈ N. By exactness, n maps to i=1 rixi = 0 in Mm, so i=1 r¯ix¯i = 0 in Mm/m. Since thex ¯i are k(m)-linearly independent, this implies eachr ¯i = 0, that is, ri Pr lies in ker(Rm → κ(m)), and so ri ∈ m. Hence we see that n = i=1 riyi ∈ mF . By Lemma 2.4.16(a), N = N ∩ mF = mF , so by Nakayama’s Lemma (2.2.8), N = 0. This ∼ shows Mm = F . (3) =⇒ (1) will be proven later.

42 2.4 Flat Modules 2 Special Modules

We now turn our attention back to characterizing faithfully flat R-modules. Lemma 2.4.18. For an R-module M, the following are equivalent: (a) M is faithfully flat. (b) M is flat and for every maximal ideal m ⊂ R, mM 6= M. Proof. (a) =⇒ (b) For any maximal ideal m ⊂ R, R/m 6= 0, so if M is faithfully flat, ∼ M/mM = R/m ⊗R M 6= 0. Hence mM 6= M. (b) =⇒ (a) Take an R-module N 6= 0. By Lemma 2.4.10, it suffices to show N ⊗RM 6= 0. Choose a nonzero element x ∈ N and consider the module homomorphism ϕ : R → N defined by setting ϕ(1) = x. Let I = ker ϕ ⊂ R. Then R/I is a submodule of N and I 6= R (since x 6= 0), so there is some maximal ideal m ⊂ R containing I. Thus IM ⊆ mM 6= M so ∼ R/I ⊗R M = M/IM 6= 0. Now by flatness, R/I ⊗R M is a submodule of N ⊗R M, so this shows N ⊗R M 6= 0 as desired. Example 2.4.19. For any ring R, the module M M = Rm m∈MaxSpec(R) is faithfully flat. Indeed, M is flat since it is a direct sum of flat R-modules (use Proposi- tion 2.4.2(ii) and Theorem 2.4.14). On the other hand, suppose N ⊗R M = 0. Then ! M M M 0 = N ⊗R Rm = (N ⊗R Rm) = Nm. m m m

Therefore Nm = 0 for all maximal ideals m ⊂ R, so N = 0. This gives examples of modules which are faithfully flat but not projective, such as the L direct sum p Z(p) over all prime integers p ∈ Z. Lemma 2.4.20. If M is a finitely presented R-module and N is any R-module, then for all multiplicative sets S ⊆ R, there is an isomorphism −1 ∼ −1 −1 S HomR(M,N) = HomS−1R(S M,S N). Proof. The assumption that M is finitely presented means there is an exact sequence Rm → Rn → M → 0. Localization is an exact functor, so S−1Rm → S−1Rn → S−1M → 0 is still exact, meaning S−1M is finitely presented as an S−1R-module, since S−1Rn ∼= (S−1R)n. On the other hand, since HomR(−,N) is left exact we get an exact sequence

n n m m 0 → HomR(M,N) → HomR(R ,N) = N → HomR(R ,N) = N . Localizing this sequence gives the top row in the following diagram:

−1 −1 n −1 m 0 S HomR(M,N) S N S N

ϕ ∼= ∼=

−1 −1 −1 n −1 m 0 HomS−1R(S M,S N) (S N) (S N)

43 2.4 Flat Modules 2 Special Modules

−1 −1 −1 This induces a morphism ϕ : S HomR(M,N) → HomS−1R(S M,S N) and a quick diagram chase shows that it is an isomorphism. Let f : A → B be a ring homomorphism. This induces two functors on the corresponding categories of modules:

ModB −→ ModA N 7−→ N, with a · n = f(a)n,

and ModA −→ ModB

M 7−→ MB := B ⊗A M.

Lemma 2.4.21. If M is a (faithfully) flat A-module, then MB is a (faithfully) flat B-module. Definition. A ring homomorphism f : A → B is called (faithfully) flat if B is (faithfully) flat as an A-module.

Corollary 2.4.22. If f : A → B is a (faithfully) flat morphism and S ⊆ A is a multiplicative set with T = f(S), then the induced morphism S−1f : S−1A → T −1B is (faithfully) flat.

Proof. This follows from the fact that there is a canonical isomorphism of S−1A-modules −1 ∼ −1 T B = S A ⊗A B. On the other hand, we have:

Lemma 2.4.23. Suppose f : A → B is faithfully flat and M is a faithfully flat B-module. Then M is also (faithfully) flat as an A-module.

f g Corollary 2.4.24. If A −→ B −→ C are faithfully flat ring homomorphisms, then so is g ◦ f : A → C.

Proposition 2.4.25. Let f : A → B be a faithfully flat homomorphism. Then

(a) For every A-module M, the map M → M ⊗A B is injective. In particular, f itself is injective.

(b) If I ⊂ A is an ideal, then IB ∩ A = I.

(c) The induced morphism f ∗ : Spec B → Spec A is surjective.

Proof. (a) Suppose x ∈ M r {0}. Then Ax is a nonzero submodule of M, so because B is a faithfully flat A-module, 0 6= Ax⊗A B,→ M ⊗A B is an inclusion of a nonzero submodule. On the other hand, Ax⊗A B = (x⊗10b, so x⊗1 6= 0 in Ax⊗A B and hence x⊗1 6= 0 in M ⊗A B. This shows that x is not an element of ker(M → M ⊗A B), so ker(M → M ⊗A B) = 0. (b) Apply (a) to the A-module M = A/I to see that ∼ A/I ,→ A/I ⊗A B = B/IB.

In general, the kernel of this map is (IB∩A)/I but because the map is injective, (IB∩A)/I = 0, that is, IB ∩ A = I.

44 2.4 Flat Modules 2 Special Modules

(c) Take a prime ideal p ⊂ A. We want to find a prime ideal q ⊂ B with p = q ∩ A. Let Ap be the localization at p, which is a local ring with maximal ideal pAp. Since A → B is faithfully flat, Corollary 2.4.22 shows that Ap → Bp is also faithfully flat, and hence injective by (a). Applying (b) to the ideal I = pBp ⊂ Bp, we get pBp ∩ Ap = pAp 6= Ap, so pBp 6= Bp. Thus there is some maximal ideal m ⊂ Bp containing pBp. Now m ∩ Ap ⊇ pAp but 1 6∈ m, so m ∩ Ap is a proper ideal of Ap containing its unique maximal ideal pAp; hence m ∩ Ap = pAp. By the correspondence theorem for primes in a localization, there is a prime ideal q ⊂ B extending to m ⊂ Bp, and by construction q ∩ A = pAp ∩ A = p. Remark. It turns out that f : A → B is faithfully flat if and only if f is flat and f ∗ : Spec B → Spec A is surjective. Theorem 2.4.26 (Generic Faithful Flatness). Let A be an integral domain with field of fractions K, B/A a finitely generated ring extension and assume B,→ K ⊗A B is injective. Then there exist nonzero elements a ∈ A and b ∈ B such that the induced morphism Aa → Bb is faithfully flat.

Proof. Suppose B = A[b1, . . . , bn] with bi ∈ B. By Noether’s normalization lemma, there ex- ist elements x1, . . . , xm ∈ BK = K ⊗A B such that BK /K[x1, . . . , xm] is an integral extension. After multiplying through by a common denominator, we may assume x1, . . . , xm ∈ B. In

particular, b1, . . . , bn are integral over K[x1, . . . , xm], so they are integral over Aa0 [x1, . . . , xm]

for some nonzero element a0 ∈ A. This implies Ba0 /Aa0 [x1, . . . , xm] is a finite ring exten-

sion. Thus we may replace A with Aa0 and B with Bb0 to have a finite ring extension B/A[x1, . . . , xm]. Set C = A[x1, . . . , xm], E = K(x1, . . . , xm) and consider the diagram

B BK E ⊗C B

C K[x1, . . . , xm] E

After inverting an element of C, we may assume BK → E ⊗C B is injective. Note that the columns in the diagram are all finite ring extensions; also, the right column is injective 0 0 because E is a flat C-module. Choose an E-basis {z1, . . . , zr} of E ⊗C B and let z1, . . . , zs be Ps 0 0 generators of B as a C-module, so that B = i=1 Czi. Then each zi is a linear combination Pr of the zj, so there is a nonzero polynomial q ∈ C with Bq = j=1 Cqzj. Observe that the map

r ϕ : Cq −→ Bq r X (c1, . . . , cr) 7−→ cjzj j=1 is surjective. Then since E ⊗ − is right exact, the base change ϕ : E ⊗ Cr → E ⊗ B = Cq b Cq q Cq q r E ⊗C B is also surjective. But both E ⊗Cq Cq and E ⊗C B are E-vector spaces of dimension r, so ϕ is also injective. Thus E ⊗ ker ϕ = 0, which implies ker ϕ = 0 since E is faithfully b Cq r ∼ flat. Hence Cq = Bq as Cq-modules.

45 2.4 Flat Modules 2 Special Modules

Choose a nonzero coefficient a in the polynomial q ∈ C = A[x1, . . . , xm] and set b = aq. Then Aa ,→ Bb is injective. We claim that it is faithfully flat. Consider the tower of ring extensions Aa ,−→ Aa[x1, . . . , xm] ,→ Aa[x1, . . . , xm]q ,→ Bb.

The above paragraph shows that Aa[x1, . . . , xm] is a free Aa-module, so in particular it is faithfully flat by Example 2.4.11. Likewise, Bb is a free module over Aa[x1, . . . , xm]q, so the third map is also faithfully flat. Finally, by Corollary 2.4.24 we need only show the middle map is faithfully flat to complete the proof. Note that Aa[x1, . . . , xm]q is flat over Aa[x1, . . . , xm] by Example 2.4.8, so by Lemma 2.4.18 we need only check that if m ⊂ Aa is a maximal ideal, mAa[x1, . . . , xm]q 6= Aa[x1, . . . , xm]q. Since a is a coefficient of q, clearly mAa[x1, . . . , xm]q cannot contain q. Also, mAa[x1, . . . , xm] 6= Aa[x1, . . . , xm] since Aa ,→ Aa[x1, . . . , xm]q is faithfully flat (this follows from Lemma 2.4.18). Together, these imply that k mAa[x1, . . . , xm]q 6= Aa[x1, . . . , xm]q, since otherwise q ∈ mAa[x1, . . . , xm]q for some k ≥ 1, but m is radical, so we would have q ∈ mAa[x1, . . . , xm]q. Hence Aa ,→ Aa[x1, . . . , xm]q is flat.

46 3 Categorical Constructions

3 Categorical Constructions

3.1 Products and Coproducts

Category theory requires us to think outside the box when it comes to our preconceived notions about set theory. Since a category is determined in terms of objects (which may be sets) and morphisms, often ignoring elements completely, it will be advantageous for us to better understand certain set theoretic constructions in the language of categories. We begin by redefining the disjoint union. In set theory, two subsets A and B of a larger, ambient set can be forced to be disjoint by taking the product (A ∪ B) × {0, 1} and defining subsets A0 = A × {0} and B0 = B × {1}. Clearly A0 ∩ B0 = ∅. Definition. The union A0 ∪ B0 is called the formal disjoint union of A and B. We denote this by A t B.

There are two functions associated with this construction, α : A → A0 ∪ B0 which takes a 7→ (a, 0), and β : B → A0 ∪ B0 which maps b 7→ (b, 1). In fact, given functions f : A → X and g : B → X, there is a unique function θ : A t B → X that extends both f and g: ( f(a) if u = (a, 0) ∈ A0 θ(u) = f(b) if u = (b, 1) ∈ B0.

This is a well-defined map since A0 ∩ B0 = ∅. This describes disjoint unions in terms that are easily compatible with categories; notice that we didn’t use an element at all.

Definition. If A, B ∈ obj(C), their coproduct is a triple (A t B, α, β) for some A t B ∈ obj(C) and morphims α : A → A t B and β : B → A t B, called injections, such that for every X ∈ obj(C) and morphisms f : A → X and g : B → X, there exists a unique morphism θ : A t B → X making the diagram commute.

A f α θ A t B X β g B

This shows that the coproduct is a solution to a universal mapping problem. Below we see that some familiar objects together with their most closely-associated morphisms can be realized as coproducts. Examples.

1 We showed above that in the category Sets, the coproduct of two sets A and B is their disjoint union.

47 3.1 Products and Coproducts 3 Categorical Constructions

2 In the category R-Mod of left R-modules, two modules A and B have a coproduct: their direct sum A ⊕ B together with the natural injections α : a 7→ (a, 0) and β : b 7→ (0, b).

3 Let k be a commutative ring and suppose A and B are commutative k-algebras. Then the tensor product A⊗k B is their coproduct in the category of commutative k-algebras. 4 Coproducts may not exist in every category. For example, let X be a set and define C to be the category of objects as subsets of X and morphisms as inclusion maps between them. For an object A ( X, its complement is Ac = X r A. Then A and Ac do not have a coproduct. Definition. An object A in a category C is an initial object if for every object X in C, there is a unique morphism A → X. Examples.

5 The empty set ∅ is an initial object in the category Sets. 6 The zero module {0} is an initial object in the category R-Mod.

7 Not all categories have initial objects. For example, the the partially-ordered set of integers Z does not have an initial object. Proposition 3.1.1. If A and A0 are initial objects in a category C, then they are isomorphic and hence initial objects, if they exist, are unique. Proof. By definition there is only one morphism f : A → A0 and only one in the other 0 direction, g : A → A. It’s easy to see that gf = idA and fg = idB, so f and g are isomorphisms. Corollary 3.1.2. For objects A, B ∈ obj(C), if the coproduct A t B exists it is unique up to isomorphism. Proof. Show A t B is an initial object. See Rotman for details. The dual notion of a coproduct is a product, which we construct next. Definition. Let A, B ∈ obj(C). Their product is a triple (A u B, p, q) for some object A u B in C and morphisms p : A u B → A and q : A u B → B, called projections, such that for every object X ∈ C and every pair of morphisms f : X → A and g : X → B, there exists a unique morphism θ : X → A u B making the diagram commute.

A f p θ X A t B β g B

48 3.2 Limits and Colimits 3 Categorical Constructions

Examples.

8 For two sets A, B ∈ Sets, their product is the regular cartesian product A×B, together with the usual coordinate projections.

9 If R is a ring and A and B are left R-modules, their product is the direct product A ⊕ B. Thus the product and coproduct coincide in R-Mod.

10 In Groups, the coproduct of two groups G and H is called the free product, denoted G∗H, which is related to the construction of free groups. In any case, G∗H is distinct from G × H, which is the product in the category Groups.

11 Recall that in topology, the fundamental group is a functor π1 : Top → Groups. On one side, the coproduct of two topological spaces X and Y is their wedge, X ∨ Y . On the ∼ other side, the Seifert-van Kampen Theorem tells us that π1(X ∨ Y ) = π1(X) ∗ π1(Y ). So this functor preserves coproducts.

As we did with initial objects and coproducts, we use terminal products to prove that products are unique (up to isomorphism).

Definition. An object B in a category C is a terminal object if for every object X in C, there is a unique morphism X → B.

Proposition 3.1.3. If B and B0 are terminal objects in a category C, then they are isomor- phic and hence terminal objects, if they exist, are unique.

Proof. Similar to the proof for initial objects; just reverse the arrows!

Corollary 3.1.4. For objects A, B ∈ obj(C), if the product A u B exists it is unique up to isomorphism.

Proof. Again, see Rotman.

3.2 Limits and Colimits

Products and coproducts are special cases of a more general notion in category theory. Let C be a diagram category, i.e. a category whose objects are the vertices of a directed graph and whose morphisms are in bijection with the directed edges of this graph (there may be multiple loops on a given vertex, but every vertex is assumed to possess a distinguished ‘identity’ loop).

Definition. A direct (resp. inverse) system in a category A is a covariant (resp. con- travariant) functor C → A where C is a diagram category.

Definition. Let F : C → A be an inverse system in A. The limit (also called the projective or inverse limit) of F is an object lim F ∈ obj(A) such that for all objects C ∈ C, there are morphisms ϕC : lim F → F(C) making the diagrams

49 3.2 Limits and Colimits 3 Categorical Constructions

lim F

ϕC ϕC0

F(C) F(C0)

0 commute whenever HomA(F(C), F(C )) 6= ∅, and such that lim F is universal among all such objects in A.

Limits are sometimes also written lim F. By the universal property, limits are unique up ←− to unique isomorphism.

Example 3.2.1. Let C be the category consisting of two objects {1, 2} and only identity morphisms. An inverse system F : C → A is just defined by specifying two objects, F(1) = a1 and F(2) = a2. Then lim F is the product of these elements, a1 u a2. Definition. Let F : C → A be a direct system in A. The colimit (also called the injective or direct limit) of F is an object colim F ∈ obj(A) such that for all C ∈ C, there are morphisms ψC : F(C) → colim F making the diagrams

F(C) F(C0)

ψC ψC0

colim F

0 commute whenever HomA(F(C), F(C )) 6= ∅, and such that colim F is universal among all such objects in A.

Example 3.2.2. Let C be the same category as in Example 3.2.1. Then a direct system F : C → A is once again defined by specifying F(1) = a1 and F(2) = a2, but colim F is the coproduct of these elements, a1 t a2. There are plenty of other important examples of limits and colimits in category theory.

Definition. Let C be the diagram category ˆ ˆ ˆ

The colimit of a direct system F : C → A is called the pushout of F and the limit of an inverse system F : C → A is called the pullback of F.

B Explicitly, if a direct system F has image A then we write the pushout Q = C colim F as a square diagram

50 3.2 Limits and Colimits 3 Categorical Constructions

A B

C Q

B Similarly, if F is an inverse system with image A then we write the pullback P = lim F C as a square diagram P B

C A

Definition. Suppose A is a category with an object 0 ∈ obj(A) that is both initial and f B terminal. For any morphism f : A → B, the pushout of the diagram A is called 0 the cokernel of f, written coker f. Similarly, for such a morphism f, the pullback of the 0 diagram B is called the kernel of f, written ker f. A f Example 3.2.3. The category AbGps of abelian groups has the trivial group 0 as a zero object, i.e. an object that is both initial and terminal. Then for a homomorphism of abelian groups f : A → B, the kernel and cokernel of f coincide exactly with these notions from abstract algebra: ker f = {a ∈ A | f(a) = 0} and coker f = {[b] | b ∈ B and [b] = [b0] if b0 = b + f(a)}.

This also holds in ModR for any ring R, and indeed in any abelian category as we shall see. In the category ModR, notice that the functor HomR(C, −) does not preserve colimits since it doesn’t preserve cokernels (this functor is not right exact; see Example 1.2.3). However, by Proposition 1.2.2, HomR(C, −) does preserve kernels. We will show in general that this condition is nearly equivalent to the property of preserving all limits. Similarly, recall (Proposition 1.3.3) that C⊗R− is a right exact functor, so it preserves cokernels, but does not preserve kernels. We will also show that the property of right exactness is nearly equivalent to preserving all colimits. Definition. Let F : A → B and G : B → A be functors. Then (F, G) is called an adjoint pair if there is a natural isomorphism ∼ HomB(F−, −) = HomA(−, G−). In this case F is called a left adjoint of G and G is a right adjoint of F.

51 3.2 Limits and Colimits 3 Categorical Constructions

Example 3.2.4. Let R and S be rings and fix a left R-module M, a left S-module N and an (S,R)-bimodule L. Then by Theorem 1.3.2, there is a natural isomorphism ∼ HomS(L ⊗R M,N) = HomR(M, HomS(L, N)).

This shows (L ⊗R −, HomS(L, −)) is an adjoint pair of functors. Example 3.2.5. In representation theory, Frobenius reciprocity is a special version of Hom- tensor adjointness used to study adjoint pairs of representations. Lemma 3.2.6. Let F : A → B and G : B → A be a pair of functors. Then (F, G) is an adjoint pair if and only if there exist natural transformations η : idA → GF and ε : FG → idB such that the compositions Gε ◦ ηG and εF ◦ Fη are both the identity natural transformation. Proof. ( =⇒ ) Suppose (F, G) is an adjoint pair. Then for any objects X ∈ A and Y ∈ B, there is an isomorphism

∼ ΦX,Y : Hom(FX,Y ) −→ Hom(X, GY ). ∼ Applying this to Y = FX, we have ΦFX,FX : Hom(FX, FX) −→ Hom(X, GFX) which takes the identity idFX to some morphism ηX = ΦFX,FX (idFX ): X → GFX. This defines the natural transformation η : idA → GF. On the other hand, applying the natural isomorphism ∼ Φ to X = GY gives an isomorphism ΦFGY,Y : Hom(FGY,Y ) −→ Hom(GY, GY ) under which −1 the identity idGY is mapped to by some εY = ΦFGY,Y (idGY ): FGY → Y . This defines the natural transformation ε : FG → idB. To check the identity conditions, apply Φ to the idFGY εY sequence εY : FGY −−−→FGY −→ Y to get:

η GY −−→GFGGY Y −→GGε Y.

Thus Gε◦ηGY = ΦFGY,Y (εY ) which by definition equals idGY . The proof of the other identity is similar. ( ⇒ = ) Given natural transformations η : idA → GF and ε : FG → idB, we define a natural transformation ΦX,Y : Hom(FX,Y ) → Hom(X, GY ) by sending α : FX → Y −1 to G(α) ◦ η : X → GFX → GY . Similarly, define its inverse ΦX,Y : Hom(X, GY ) → Hom(FX,Y ) by sending β : X → GY to ε ◦ F(β): FX → FGY → Y . To see that these are natural inverses, fix X ∈ A,Y ∈ B and α : FX → Y and consider

−1 −1 ΦX,Y ΦX,Y (α) = ΦX,Y (G(α) ◦ η) = ε ◦ F(G(α) ◦ η) = ε ◦ FG(α) ◦ Fη since F is a functor = εF(G(α))Fη = εF ◦ Fη(α) = α

by the identity εF ◦ Fη = id. Similarly, for β : X → GY , we have

−1 ΦX,Y ΦX,Y (β) = ΦX,Y (ε ◦ F(β)) = G(ε ◦ F(β)) ◦ η = Gε ◦ GF(β) ◦ η since G is a functor = Gε ◦ ηG(β) = β

−1 by the identity Gε ◦ ηG = id. Hence ΦX,Y and ΦX,Y form a natural isomorphism.

52 3.2 Limits and Colimits 3 Categorical Constructions

Definition. For an adjoint pair (F, G), the natural transformations η : idA → GF and ε : FG → idB are called the unit and counit of the adjunction, respectively. The conditions Gε ◦ ηG = id and εF ◦ Fη = id are called the triangle identities.

Theorem 3.2.7. If (F, G) is an adjoint pair of functors, then F preserves colimits and G preserves limits.

Proof. Assume F : A → B and G : B → A. Let X : C → A be a direct system and for each C ∈ C, let ϕC : X(C) → colim X be the induced morphism making the relevant diagrams commute. We must prove that F(colim X) is the colimit of the direct system F ◦ X : C → A → B. For any C ∈ C and B ∈ B, there is an isomorphism ∼ HomB(F(X(C)),B) = HomA(X(C), G(B)).

Thus for any diagram F(X(C)) B

colim(F ◦ X)

the corresponding diagram can be completed: X(C) G(B)

colim X

On the other hand, applying F to colim X gives a diagram F(colim X) B

F ◦ ϕC

F(X(C))

In other words, every map F(X(C)) → B which is compatible with the F ◦ ϕC will factor through F(colim X). Hence by the universal property of colimits, F(colim X) = colim(F ◦ X). The proof that G preserves limits is dual.

Corollary 3.2.8. For any (S,R)-bimodule L, L ⊗R − preserves colimits and HomS(L, −) preserves limits.

Theorem 3.2.9. Let F : ModR → AbGps be an additive functor. Then the following are equivalent: ∼ (1) F = L ⊗R − for some R-module L.

53 3.3 Abelian Categories 3 Categorical Constructions

(2) F preserves colimits. (3) F is right exact and preserves direct sums. (4) F has a right adjoint. Proof. (1) =⇒ (4) is Corollary 3.2.8. (4) =⇒ (2) is Theorem 3.2.7. (2) =⇒ (3) is immediate since a direct sum is a colimit. (3) =⇒ (1) Consider L = F(R), which is a right R-module under right multiplication by elements of R. Then for any R-module M, we have a presentation of M by free R-modules:

ϕ RY −→ RX → M → 0,

where X and Y are some sets. Note that coker ϕ = M. Since F preserves direct sums, Y Y Y ∼ Y X ∼ X F(R ) = (F(R)) = L = L ⊗R R and likewise F(R ) = L ⊗R R . Then applying F to the exact sequence above gives the top row in the following diagram with exact rows:

LY LX F(M) 0

∼= ∼=

Y X L ⊗R R L ⊗R R L ⊗R M 0

Then there is a map F(M) → L⊗R M which by the Snake Lemma (1.2.8) is an isomorphism. Naturality is easy to check.

Theorem 3.2.10. Let F : ModR → AbGps be an additive functor. Then the following are equivalent: ∼ (1) F = HomR(L, −) for some R-module L. (2) F preserves limits. (3) F is left exact and preserves direct products. (4) F has a left adjoint. Proof. Similar.

3.3 Abelian Categories

Abelian categories are the preferred setting for working with derived functors, which are the main tools used in homological algebra. Definition. C is an additive category if

ˆ HomC(A, B) is an abelian group for all A, B ∈ obj(C). Note that we normally only require Hom to be a set.

54 3.3 Abelian Categories 3 Categorical Constructions

ˆ For all a ∈ A, b ∈ B, f ∈ Hom(A, B) and g ∈ Hom(B,A),

b(f + g) = bf + bf and (f + g)a = fa + ga.

ˆ There exists a 0 object, which is both initial and terminal, meaning for all A ∈ obj(C), there exist maps 0 → A and A → 0.

ˆ C has finite products (Q) and coproducts (L) and they agree on finite-index sets. The last condition, that Q and L agree on finite sets, can be proven using the other axioms. We can also define additive functors between additive categories, which are functors that preserve the additive structure of the category. To fully understand abelian categories, it requires us to redefine our concept of kernels and cokernels. These are best understood in terms of monics and epics. Definition. A monomorphism, or monic, in a category C is a morphism u : B → C so that if there are maps f, g : A → B such that uf = ug, then f = g.

f u A B C g

Definition. An epimorphism, or epic, is the dual notion, that is π is an epic if for every f, g : B → C, fπ = gπ implies f = g.

f π A B C g

In general, if u is one-to-one then u is monic, and if π is onto then π is epic, but the converse does not hold in a general category. The converse does hold however for abelian categories, which we take to be our definition. Definition. A category C is abelian if every one-to-one morphism is monic and every onto morphism is epic.

Example 3.3.1. Let Comm be the category of commutative rings, with ring homomorphisms as morphisms. Consider the map i : Z → Q. This is an epimorphism in C but it is clearly not f onto. To see that it is epic, suppose we have Q R so that fi = gi, i.e. f(n) = g(n) for g all n ∈ Z. Clearly if f and g agree on Z they agree on Q, so i is an epimorphism. However, i cannot be onto because Q is much bigger than Z. This shows that Comm is not an abelian category.

Next we relate kernels and cokernels to monics and epics. Proposition 3.3.2. Let u : A → B be a morphism in an additive category A.

55 3.4 Projective and Injective Resolutions 3 Categorical Constructions

(i) If ker u exists, then u is monic ⇐⇒ ker u = 0.

(ii) Likewise, if coker u exists then u is epic ⇐⇒ coker u = 0. Proof omitted. From this we define Definition. An additive category A is abelian if (1) Every morphism has a kernel and a cokernel.

0 (2) If f : M → N is monic, then M is the kernel of the diagram coker f N

M (3) If f is epic, then N is the cokernel of the diagram ker f 0 Note that condition (2) implies that if f is monic, then there exists a short exact sequence

f 0 → M −→ N −→ coker f → 0.

Likewise, condition (3) implies that if f is epic, there exists a short exact sequence

f 0 → ker f −→ M −→ N → 0.

Example 3.3.3. For any ring R, the category ModR is an abelian category. If R is (left) noetherian, then the subcategory modR of finitely generated (left) R-modules is also an abelian category.

Theorem 3.3.4. Any abelian category is naturally isomorphic to a full subcategory of ModR for some ring R.

3.4 Projective and Injective Resolutions

Free resolutions are easy to motivate, and give rise naturally to the idea of projective resolu- tions (since free and projective are closely related). For an R-module A, a free resolution of A is an exact sequence

· · · → Fn → Fn−1 → · · · → F1 → F0 → A → 0 where each Fi is a free R-module. This can be thought of in terms of generators and relations: in the first step, we let F0 be a free module with submodule K0 = ker(F0 → A) such that A is the quotient of F0 by K0. Then K0 is the module of relations that describe A and F0/K0 is a presentation of A. Next we map F1 → K0 and let K1 be the module of relations for this presentation, and so forth. Free resolutions give a natural way of thinking about equations in A in terms of sequences of equations in free modules.

56 3.4 Projective and Injective Resolutions 3 Categorical Constructions

Proposition 3.4.1. If the short exact sequence 0 → M 0 → M → M 00 → 0 is split, and N is another R-module, then the sequences

0 → M 0 ⊗ N → M ⊗ N → M 00 ⊗ N → 0 (1) 0 → Hom(M 00,N) → Hom(M,N) → Hom(M 0,N) → 0 (2) 0 → Hom(N,M 0) → Hom(N,M) → Hom(N,M 00) → 0 (3) are all exact.

Proof. (1) Tensor is right exact so it suffices to show i ⊗ 1 : M 0 ⊗ N → M ⊗ N is injective. Notice that (π ⊗ 1)(i ⊗ 1) = πi ⊗ 1 = 0 ⊗ 1 = 0 so i ⊗ 1 has a left inverse and is therefore injective. (2) By Proposition 1.2.2, we only need to check that i∗ : Hom(M,N) → Hom(M 0,N) is 0 π ∗ ∗ ∗ ∗ ∗ surjective. But the splitting M ←− M gives i π = (πi) = (idM 0 ) = 1 so i has a right inverse and is therefore surjective. (3) Similarly, covariant Hom is always left exact so it’s enough to show j∗ : Hom(N,M) → 00 f 00 Hom(N,M ) is surjective. The other splitting M ←− M gives us j∗f∗ = (jf)∗ = (idM 00 )∗ = 1, so j∗ has a right inverse and is surjective. In Proposition 2.1.4 we showed that every free module was projective. This gives rise to a more general notion of projective resolutions.

Definition. A projective resolution of an object A in an abelian category A is an exact sequence P = · · · → P2 → P1 → P0 → A → 0

where each Pi is projective.

Definition. For a projective resolution P = · · · → P2 → P1 → P0 → A → 0 of A, we define the nth syzygy of P to be Kn = ker(Pn → Pn−1). Definition. We say an abelian category A has enough projectives if projective resolutions exist for every object A ∈ obj(A); equivalently there exists a projective P ∈ obj(A) and an epic P → A.

Lemma 3.4.2. (a) Every module over a field has a projective resolution of length 0.

(b) Every module over a PID has a projective resolution of length at most 1.

Proof. (a) If V is a module over a field, it is a vector space and therefore free and projective. Hence 0 → V −→id V → 0 is a projective resolution. (b) Let P be a projective module with a surjection P → M, so that P is also free by Corollary 2.1.8. Then the kernel K of this map is a submodule of P , so it’s free and therefore projective by Corollary 2.1.8 again. Hence 0 → K → P → M → 0 is a projective resolution of M. Before proving some results for projective resolutions, we need

57 3.4 Projective and Injective Resolutions 3 Categorical Constructions

Proposition 3.4.3 (Schanuel’s Lemma). Given exact sequences

p 0 → K −→i P −→ M → 0 0 p0 0 → K0 −→i P 0 −→ M → 0 where P and P 0 are projective, there is an isomorphism K ⊕ P 0 ∼= K0 ⊕ P . Proof. See Rotman. This generalizes nicely to long exact sequences of modules. Proposition 3.4.4. Let A be a left R-module and suppose we have two exact sequences

i pn pn−1 p2 p1 p0 0 → K −→ Pn −→ Pn−1 −−−→· · · −→ P1 −→ P0 −→ A → 0 and 0 0 i qn qn−1 q2 q1 q0 0 → K −→ Qn −→ Qn−1 −−→· · · −→ Q1 −→ Q0 −→ A → 0,

where the Pi and Qi are projective. Then ∼ 0 K ⊕ Qn ⊕ Pn−1 ⊕ · · · = K ⊕ Pn ⊕ Qn−1 ⊕ · · · Proof. For all j = 0, . . . , n we will show that there are short exact sequences

0 → ker pj → Pj ⊕ Qj−1 · · · → ker pj−1 ⊕ Qj−1 ⊕ · · · → 0 and

0 → kerqj → Qj ⊕ Pj−1 · · · → ker qj−1 ⊕ Pj−1 ⊕ · · · → 0. We prove this using induction. For the base case,

i p0 0 → ker p0 −→ P0 −→ B → 0 and 0 i q0 0 → ker q0 −→ Q0 −→ B → 0 ∼ By Schanuel’s Lemma (Proposition 3.4.3), we have ker p0 ⊕ Q0 = ker q0 ⊕ P0, proving the ∼ base case. Next suppose the sequences above are exact for k < n, i.e. ker pk ⊕Qk ⊕Pk−1 ··· = ker qk ⊕ Pk ⊕ Qk−1 ··· . Consider

0 → ker pk+1 → Pk+1 ⊕ Qk · · · → ker pk ⊕ Qk ⊕ · · · → 0 and

0 → kerqk+1 → Qk+1 ⊕ Pk · · · → ker qk ⊕ Pk ⊕ · · · → 0.

In each case the kernel of the first map is contained in Pk+1 ⊕ Qk (flip these for the bottom sequence) so the inductive step follows easily. Now since the original sequence is exact at K, K embeds into Pn via i and therefore K = ker pn. Hence ∼ 0 K ⊕ Qn ⊕ Pn−1 ⊕ · · · = K ⊕ Pn ⊕ Qn−1 ··· as desired.

58 3.4 Projective and Injective Resolutions 3 Categorical Constructions

Definition. If A is a left R-module and · · · → P2 → P1 → P0 → A → 0 is a projective resolution of A, we say A has projective dimension n if there is a smallest n such that Kn is projective. Otherwise A has infinite projective dimension (or no projective dimension). If A has projective dimension n, we write pd(A) = n. The following result tells us that the integer n does not depend on the projective resolution chosen for A. Proposition 3.4.5. If one projective resolution of a module A has a projective nth syzygy, then the nth syzygy of every projective resolution of A is projective. Proof. Consider the projective resolutions

P = · · · → Pn → Pn−1 → · · · → P1 → P0 → A → 0 and 0 0 0 0 0 P = · · · → Pn → Pn−1 → · · · → P1 → P0 → A → 0. Make a soft truncation at the nth location of each to produce exact sequences

0 → Kn → Pn → Pn−1 → · · · → P1 → P0 → A → 0

0 0 0 0 0 0 → Kn → Pn → Pn−1 → · · · → P1 → P0 → A → 0. 0 ∼ 0 0 Then by Schanuel’s Lemma (Proposition 3.4.3), Kn ⊕Pn ⊕Pn−1 ⊕· · · = Kn ⊕Pn ⊕Pn−1 ⊕· · · . 0 0 0 Let Qn = Pn ⊕ Pn−1 ⊕ · · · and Qn = Pn ⊕ Pn−1 ⊕ · · · . These are both projective since they 0 ∼ 0 are direct sums of projectives, and so we see that Kn ⊕ Qn = Kn ⊕ Qn. Notice that we haven’t yet used the hypothesis that some Kn is projective, so this proof works for all n. 0 ∼ 0 0 Now if Kn is projective, the isomorphism Kn ⊕ Qn = Kn ⊕ Qn shows that Kn is a direct summand of a projective, so it too is projective. Hence if the nth syzygy in a projective resolution of A is projective, the nth syzygy is projective in all projective resolutions of A, reassuring us that the term ‘projective dimension’ is well-defined. There is a dual notion for injectives. Definition. An injective resolution of an object A in an abelian category A is an exact sequence E = 0 → A → E0 → E1 → E2 → · · · such that each En is injective. Notice that when the indices increases along the arrows, we write them as superscripts, and when they decrease along arrows we write them as subscripts. This notational convention will be seen throughout homological algebra. Definition. As a dual to the definition of syzygies, in an injective resolution E = 0 → A → n E0 → E1 → E2 → · · · of A we call V = coker(En−1 → En) the nth cosyzygy of E. Definition. An abelian category A is said to have enough injectives if injective resolutions exist for every object A ∈ obj(A); equivalently there exists an injective E ∈ obj(A) and a monic A → E.

59 4 Homology

4 Homology

4.1 Chain Complexes and Homology

The environment in which homological algebra is performed is abelian categories. In general, abelian categories may be realized as the category of left R-modules for some ring R, so we will often reduce proof to the case where C = R-Mod. Definition. A chain complex, usually shortened to complex, in an abelian category A is a chain of differentials ∂n,

∂n+1 ∂n A• = · · · → An+1 −−−→ An −→ An−1 → · · ·

such that for all n, ∂n∂n+1 = 0. Definition. If A is an abelian category, the category of complexes over A is denoted Comp(A). Proposition 4.1.1. If A is an abelian category then Comp(A) is also abelian. Proof omitted.

Definition. If A• and B• are complexes in Comp(A), a chain map f• : A• → B• is a sequence of morphisms (fn : An → Bn) making the following commute

∂n+1 ∂n ··· An+1 An An−1 ···

fn+1 fn fn−1

··· Bn+1 B Bn−1 ··· 0 n ∂n+1 ∂n

2 Note that the condition ∂n∂n+1 = 0 (sometimes denoted ∂ = 0) is equivalent to im ∂n+1 ⊆ ker ∂n. We will usually write ker ∂n = Zn(A•), called the n-cycles of A•. Likewise, we denote im ∂n+1 = Bn(A•), called the n-boundaries of A•. Then we have subcomplexes Z(A•) and B(A•) of A• given by

0 0 0 Z(A•) = ··· −→ Zn(A•) −→ Zn−1(A•) −→· · · 0 0 0 B(A•) = ··· −→ Bn(A•) −→ Bn−1(A•) −→· · ·

Note that B(A•) ⊆ Z(A•). This allows us to define the central object in homological algebra:

Definition. Let A• be a complex in an abelian category A. The nth homology of A• is the quotient Hn(A•) := Zn(A•)/Bn(A•), which is an object in A. Intuitively, the nth homology measures the failure of exactness of a complex at the nth location in the chain. Since A is abelian, we also have H(A•) = Z(A•)/B(A•). Notice that A• is exact at An ⇐⇒ Hn(A•) = 0, and the entire complex A• is exact ⇐⇒ H(A•) = 0.

60 4.1 Chain Complexes and Homology 4 Homology

Proposition 4.1.2. For every abelian category A, the nth homology Hn : Comp(A) → A is an additive functor for each n ∈ Z. Proof. We prove this for A = Ab, the category of abelian groups. The definition above defines the functor Hn : Comp(Ab) → Ab on objects by A• 7→ Hn(A•) so it remains to define homology on morphisms. Let f : C → C0 be a chain map and define

0 Hn(f): Hn(C) −→ Hn(C ) 0 zn + Bn(C) 7−→ fn(zn) + Bn(C ).

We must show that fn(zn) is a cycle and that Hn(f) is well-defined, i.e. independent of the choice of cycle zn. Since f is a chain map, we have the following commutative diagram

∂n+1 ∂n ··· Cn+1 Cn Cn−1 ···

fn+1 fn fn−1

··· C0 C0 C0 ··· n+1 0 n n−1 ∂n+1 ∂n

0 showing (by a quick diagram chase) that indeed fn(zn) ∈ Zn(C ), and if zn + Bn(C) = 0 wn + Bn(C) then fn(zn − wn) must land in Bn(C ). Next we prove Hn is a functor. Obviously Hn(1C ) must be the identity on Hn(C). If f and g are chain maps such that gf is defined in some Hom set, then for all z ∈ Zn(C) we have

0 0 Hn(gf)(z) = (gf)n(z) + Bn(C ) = gn(fn(z) + Bn(C )) 0 = Hn(g)(fn(z) + Bn(C )) = (Hn(g)Hn(f))(z).

Finally, if f, g : C → C0 are chain maps, then

0 Hn(f + g)(z) = (fn + gn)(z) = (fn(z) + gn(z)) + Bn(C ) = (Hn(f) + Hn(g))(z).

Hence Hn is a functor. The first big result we will prove is a statement for short exact sequences of complexes. We will make extensive use of the Snake Lemma (1.2.8). Note that by the equivalence of abelian categories and module categories R-Mod for rings R, we are able to state things in terms of R-modules.

p Theorem 4.1.3 (The Long Exact Sequence). Let 0 → C0 −→i C −→ C00 → 0 be an exact sequence in Comp(R), the category of complexes of left R-modules. Then there is a long exact sequence in R-Mod

0 00 0 00 · · · → Hn(C ) → Hn(C) → Hn(C ) → Hn−1(C ) → Hn−1(C) → Hn−1(C ) → · · ·

Proof. First we expand the short exact sequence:

61 4.1 Chain Complexes and Homology 4 Homology

......

i p 0 n+1 n+1 00 0 Cn+1 Cn+1 Cn+1 0

0 00 ∂n+1 ∂n+1 ∂n+1 p 0 in n 00 0 Cn Cn Cn 0

0 00 ∂n ∂n ∂n i p 0 n−1 n−1 00 0 Cn−1 Cn−1 Cn−1 0

......

0 How do we define Hn(i): Hn(C ) → Hn(C)? In the proof of the Snake Lemma (1.2.8), we 0 in proved that Zn(C ) −→ Zn(C) induces

0 in Zn(C ) −→ Zn(C) → Hn(C)

0 so we just need to check that if z ∈ Bn(C ) then in(z) ∈ Bn(C). This follows from the commutativity of the left squares in the diagram above. The dual argument works for pn 00 0 Hn(C) → Zn(C) −→ Zn(C ). Hence we have well-defined maps Hn(i): Hn(C ) → Hn(C) 00 and Hn(p): Hn(C) → Hn(C ). The Snake Lemma (1.2.8) is also a good place to consult when defining a connecting map 00 0 ð : Hn(C ) → Hn−1(C ). In that proof, we saw that there is a map 00 0 0 0 Zn(C ) −→ Cn−1/Bn−1(C ) = coker ∂n 00 0 0 z 7−→ c + im ∂n. 0 0 By the Correspondence Theorem, the submodules of Cn−1/Bn−1(C ) are in correspondence 0 0 0 with submodules of Cn−1 containing Bn−1, and one of these if Zn−1(C ). Thus we want to 0 0 0 0 0 0 0 0 prove c + im ∂n = z + im ∂n for some z ∈ ker ∂n−1, which is equivalent to c − z ∈ im ∂n. 0 0 A diagram chase actually reveals that c itself lies in ker ∂n−1. So by the lifting property of kernels, we have a map

00 0 0 0 Zn(C ) −→ Zn−1(C )/Bn−1(C ) = Hn−1(C ) 00 0 0 z 7−→ c + im ∂n−1. 00 00 00 0 0 Another diagram chase shows that if z ∈ Bn(C ) then z 7→ c ∈ Bn−1(C ) in the above 00 0 map. This induces a map ð : Hn(C ) → Hn−1(C ). Now that we have all the maps in hand, it remains to show that the long sequence is exact at each homology. The proof of this is very similar to the latter part of the proof of the Snake Lemma (1.2.8).

62 4.1 Chain Complexes and Homology 4 Homology

Proposition 4.1.4. Let R and A be rings and suppose T : RMod → AMod is an exact additive functor. Then T commutes with homology, that is, for every complex C ∈ Comp(R) and for every n, there is an isomorphism ∼ Hn(T (C)) = T (Hn(C)).

Proof. The nth homology Hn(C) defines two exact sequences in RMod:

0 → Bn(C) → Zn(C) → Hn(C) → 0 (4)

0 → Zn(C) → Cn → Bn−1(C) → 0. (5)

Since T is an additive exact functor, applying it to the sequences above gives two new exact sequences

0 → T (Bn(C)) → T (Zn(C)) → T (Hn(C)) → 0 (6)

0 → T (Zn(C)) → T (Cn) → T (Bn−1(C)) → 0. (7)

∂n ∂n−1 On the other side of things, expand the complex C = · · · → Cn −→ Cn−1 −−−→· · · and apply the exact functor T , giving

T (∂n) T (∂n−1) T (C) = · · · → T (Cn) −−−→ T (Cn−1) −−−−→· · ·

Since T is additive, T (∂n)T (∂n−1) = T (∂n∂n−1) = T (0) = 0 (the zero map) so we see that T (C) is a complex in Comp(A). Therefore we can define its nth homology Hn(T (C)) = Zn(T (C))/Bn(T (C)). As above, this defines two sequences in AMod:

0 → Bn(T (C)) → Zn(T (C)) → Hn(T (C)) → 0 (8)

0 → Zn(T (C)) → T (Cn) → Bn−1(T (C)) → 0 (9)

Now consider the diagram whose rows are exact sequences (6) and (4).

0 Zn(T (C)) T (Cn) Bn−1(T (C)) 0

f id g

0 T (Zn(C)) T (Cn) T (Bn−1(C)) 0

Since the middle arrow is the identity (i.e. a boring isomorphism), if we can show that there exist maps f and g where either one is an isomorphism, then the Five Lemma (1.2.9) proves the other is an isomorphism. This will nearly finish the problem. To prove that there is an isomorphism f, note that any exact additive functor T commutes ϕ with kernels, since applying T to an exact sequence 0 → K → A −→ B produces an exact T (ϕ) sequence 0 → T (K) → T (A) −−→ T (B). Then T (K) is the kernel of T (ϕ), and kernels are unique up to isomorphism so T (ker ϕ) ∼= ker T (ϕ). In our context this proves that ∼ Zn(T (C)) = ker T (∂n) = T (ker ∂n) = T (Zn(C)). Now we can fill one arrow:

63 4.1 Chain Complexes and Homology 4 Homology

0 Zn(T (C)) T (Cn) Bn−1(T (C)) 0

∼= id g

0 T (Zn(C)) T (Cn) T (Bn−1(C)) 0

Clearly the left square commutes since both vertical arrows are isomorphisms. Now we turn our attention to constructing g. If b ∈ Bn−1(T (C)) then exactness of the top row says there is an element c ∈ T (Cn) mapping to b. Since the middle vertical arrow is 0 the identity, c 7→ c ∈ T (Cn) in the bottom row, which has image b ∈ T (Bn−1(C)). We use 0 this to define g : Bn−1(T (C)) → T (Bn−1(C)), b 7→ b . The only concern is that g may not be well defined, so we take a moment to verify this with a diagram chase:

c˜ (1) (2) c z c − c˜ b 0 Zn(T (C)) T (Cn) Bn−1(T (C)) 0

(3) ∼= id g

0 T (Zn(C)) T (Cn) T (Bn−1(C)) 0 z0 c − c˜ 0 (4) (5)

Suppose b pulls back to c andc ˜ in T (Cn). Then their difference c − c˜ is in the kernel of T (Cn) → Bn−1(T (C)) so c−c˜ pulls back to some z ∈ Zn(T (C)). The left vertical arrow takes z 7→ z0 isomorphically, so z0 7→ c − c˜ along the bottom left by commutativity. By exactness of the bottom row, pushing z0 along both maps on the bottom takes z0 7→ c − c˜ 7→ 0, which shows that g is well defined after all. So g exists and by construction it makes the right square commute. We can apply the Five Lemma (1.2.9) to see that g must be an isomorphism, and this holds for any n so if we consider the diagram with (5) and (3) as its rows we can put in isomorphisms on the left and middle arrows.

0 Bn(T (C)) Zn(T (C))) Hn(T (C)) 0

∼= ∼=

0 T (Bn(C)) T (Zn(C)) T (Hn(C)) 0

The proof above generalizes to any diagram of this form, showing there exists an isomorphism ∼ (the dashed arrow) Hn(T (C)) = T (Hn(C)) as desired.

64 4.2 Derived Functors 4 Homology

The dual notion to homology is cohomology, which as with many ‘dual’ concepts in homological algebra, is formed by reversing the arrows. We define things explicitly below. Definition. A co-chain complex in an abelian category A is a chain

∂ A• = · · · ← An+1 ←−−−n+1 An ←−∂n An−1 ← · · ·

where ∂n−1∂n = 0 for all n. Note that the subscripts of homology become superscripts in cohomology, and most n • objects acquire a “co-” prefix. For example, ker ∂n = Z (A ) are called the n-cocycles of • n • • A , im ∂n+1 = B (A ) are the n-coboundaries of A and these allow us to define Definition. The nth cohomology of a cochain complex A• is the quotient

Hn(A•) = Zn(A•)/Bn(A•)

which is an object in A. Therefore Hn is a functor Comp(A) → A.

4.2 Derived Functors

Definition. Suppose T : A → C is a left exact functor between abelian categories. Then a n cohomological δ-functor extending T is a family of additive functors (T : A → C)n∈N such that for any short exact sequence in A,

0 → A → B → C → 0,

there exists a long exact sequence in C:

0 → TA → TB → TC −→δ T 1A → T 1B → T 1C −→δ T 2A → · · ·

Likewise, for a right exact functor T : A → C, a homological δ-functor extending T is

a family (Tn)n∈N such that for every short exact sequence as above, there is a long exact sequence δ δ ··· T2C −→ T1A → T1B → T1C −→ TA → TB → TC → 0. To make notation consistent, we will put T 0 = T .

n Example 4.2.1. We will see below that there is a cohomological δ-functor ExtR(M, −) extending the left exact functor HomR(M, −): ModR → AbGps. Also, there is a cohomolog- n R ical δ-functor ExtR(−,N) extending HomR(−,N) and a homological δ-functor Torn (M, −) extending M ⊗R −. Example 4.2.2. Let p be prime and consider the functor T : AbGps → AbGps,A 7→ A/pA. n 1 Then there is a δ-functor (T ) extending T given by definng T A = pA, the p-torsion of n A, and T A = 0 for all n ≥ 2. Dually, the p-torsion functor S : A 7→ pA is extended by the δ-functor (Sn) with S1A = A/pA and SnA = 0 for n ≥ 2. It turns out that n n T n = Ext (Z/pZ, −) and Sn = Ext (−, Z/pZ) from the previous example.

65 4.2 Derived Functors 4 Homology

Theorem 4.2.3. Let T : A → C be a left (resp. right) exact functor on an abelian category A with enough injectives (resp. projectives). Then there exists a universal cohomological n n (resp. homological) δ-functor (T ) (resp. (Tn)) extending T and such that T (E) = 0 for all injectives E ∈ A (resp. Tn(P ) = 0 for all projectives P ∈ A). Definition. A cohomological δ-functor (T n) is effaceable if for any n ≥ 0 and any object M, there is a monomorphism M,→ N such that the induced map T n(M) → T n(N) is 0. Proposition 4.2.4. If A is an abelian category with enough injectives, then a δ-functor is effaceable if and only if it vanishes on all injectives. Proof. ( =⇒ ) Let (T n) be a δ-functor and E ∈ A an injective. Then given a monomorphism E,→ N, by Proposition 2.3.2 there exists a splitting N → E. Hence N ∼= E ⊕ N 0 for some object N 0, where ⊕ denotes the coproduct. Now by left exactness and Theorem 3.2.10, T n(N) ∼= T n(E) ⊕ T n(N 0), and the map T n(E) → T n(N) ∼= T n(E) ⊕ T n(N 0) is 0 by hypothesis, so T n(E) = 0. ( ⇒ = ) Conversely, suppose T n(E) = 0 for all injectives and take an arbitrary M ∈ A. Then by Theorem 2.3.14, M,→ E for some injective E and therefore T n(M) → T n(E) = 0 must be the zero map. Corollary 4.2.5. Effaceable δ-functors are unique up to unique isomorphism. Proof. Assume (T n) is an effaceable δ-functor. Then by Proposition 4.2.4, for any monomor- phism M,→ N, the map T 1(M) → T 1(N) is 0. Thus in the long exact sequence for T n, we have 0 → T 0(M) → T 0(N) → T 0(N/M) → T 1(M) −→0 T 1(N) → · · · so T 1(M) ∼= T 0(N/M)/ im T 0(N). More generally, for k ≥ 0, T k+1(M) → T k+1(N) is 0 and this implies T k+1(M) ∼= T k(N/M)/ im T k(N) so the objects T n(M) are completely determined up to isomorphism by the objects T 0(M) and the choices of monomorphisms M,→ N in Proposition 4.2.4. Corollary 4.2.6. If (T n) is a cohomological δ-functor on an abelian category A with enough injectives, then there is a natural isomorphism T n ∼= Hn(T 0(−)) for all n ≥ 0. Proof. Take an object M ∈ A. Since A has enough injectives, we may take an injective resolution of M:

M E1 E2 E3 ···

K1 K2

n ∼ n−1 ∼ ∼ 1 Then T (M) = T (K1) = ··· = T (Kn−1) and this last term is

0 0 0 1 ∼ T (Kn) ∼ ker(T (En) → T (En+1)) n 0 T (Kn−1) = 0 = 0 0 = H (T (E•)). im T (En−1) im(T (En−1) → T (En))

This proves Theorem 4.2.3 for left exact functors. Dually, we have:

66 4.2 Derived Functors 4 Homology

Definition. A homological δ-functor (Tn) is coeffaceable if for any n ≥ 0 and any object M, there is an epimorphism N → M such that the induced map Tn(N) → Tn(M) is 0. Proposition 4.2.7. If A is an abelian category with enough projectives, then a δ-functor is coeffaceable if and only if it vanishes on all projectives. Corollary 4.2.8. Coeffaceable δ-functors are unique up to unique isomorphism.

Corollary 4.2.9. If (Tn) is a homological δ-functor on an abelian category A with enough ∼ projectives, then there is a natural isomorphism Tn = Hn(T0(−)) for all n ≥ 0. This proves Theorem 4.2.3 for right exact functors. Definition. Suppose T : A → C is a covariant, additive, right exact functor between abelian categories, where A has enough projectives. The nth left derived functor of T is the universal homological δ-functor of T :

LnT : A −→ C

A 7−→ Hn(T (P•)) where P• is a fixed projective resolution of A. Derived functors in some sense measure the failure of exactness in the nth homological dimension of the functor T .

Definition. In the category R-Mod of left R-modules, if TM = − ⊗R M then its left derived functors are called Tor:

R Torn (M,N) := (LnTM )(N) = Hn(P•,N ⊗R N).

0 Likewise for a right R-module M, the left derived functors of TM = M ⊗R − are called tor:

R 0 torn (N,M) := (LnTM )(N) = Hn(M ⊗R P•,N ). The notation tor is only temporary, as we will show in Corollary 4.4.11. The dual of left derived functors is right derived functors, which we define now. Definition. For a covariant, additive, left exact functor S : A → C between abelian cate- gories, where A has enough injectives, the nth right derived functor of S is the universal cohomological δ-functor of S:

RnS : A −→ C

A 7−→ H−n(S(E•)) where E• is a fixed injective resolution of A. The counterpart to Tor is a right derived functor called Ext.

Definition. In R-Mod, the right derived functors of SM = HomR(M, −) are called Ext:

n n ExtR(M,N) := (R SM )(N) = H−n(HomR(M,E•,N )).

67 4.2 Derived Functors 4 Homology

We summarize the different variances and types of derived functors in the table below. Variance of T Category requirements Derived functor covariant enough projectives LnT covariant enough injectives RnT contravariant enough projectives RnT contravariant enough injectives LnT

R ∼ Example 4.2.10. For all right R-modules A and left R-modules B, Tor0 (A, B) = A ⊗R B. 0 ∼ Likewise, if A and B are both left R-modules then ExtR(A, B) = HomR(A, B). In defining left and right derived functors, we have neglected the fact that we are choosing a particular projective (or injective) resolution of A. The Comparison Theorem says that unique chain maps exist between projective resolutions of M and N when M → N is a n module homomorphism, so we need not worry when defining LnT and R T .

Theorem 4.2.11 (Comparison). Let P• : · · · → P2 → P1 → P0 be a projective chain complex and suppose C• : ··· C2 → C1 → C0 is an acyclic chain complex. Then for any homomorphism ϕ : H0(P•) → H0(C•), there is a chain map f : P• → C• whose induced map on H0 is ϕ, and ϕ is unique up to chain homotopy. Proof. Consider the diagram

∂2 ∂1 ∂0 P2 P1 P0 H0(P•) 0

f2 f1 f0 ϕ 0 0 0 ∂2 ∂1 ∂0 C2 C1 C0 H0(C•) 0

Since P0 is projective, there exists an f0 lifting ϕ to P0 → C0. Inductively, given fn−1 we have a diagram

∂n Pn Pn−1

fn fn−1 0 ∂n Cn Cn−1

0 0 0 Note that fn−1∂ has image lying in ker ∂n ⊆ Cn−1, but C• is acyclic, so ∂n(Cn) = ker ∂n−1. Since Pn is projective, we can lift fn−1∂n to the desired map fn : Pn → Cn. By construction, ∞ f = {fn}n=0 satisfies the desired properties. For uniqueness, suppose g : P• → C• is another chain map restricting to ϕ on H0. Since 0 0 f0 − g0 = 0 on H0, it must be that (f0 − g0)(P0) ⊆ ker ∂0 = im ∂1, so by projectivity of P0 there exists s0 : P0 → C1 making the following diagram commute:

68 4.2 Derived Functors 4 Homology

P0

s0 f0 − g0

0 C1 im ∂1 0

0 Inductively, given s0, . . . , sn−1 satisfying fk − gk = ∂k+1sk + sk+1∂k for all 0 ≤ k ≤ n − 1, we have

0 0 ∂n(fn − gn − sn−1∂n) = (fn−1 − gn−1)∂n − ∂nsn−1∂n since f, g are chain maps 0 0 = (∂nsn−1 − sn−2∂n−1)∂n − ∂nsn−1∂n = 0.

Hence there is a commutative diagram

Pn

sn fn − gn − sn−1∂n

C ker ∂0 n+1 0 n 0 ∂n+1

0 This establishes the chain homotopy s : P• → C• such that fn − gn = ∂n+1sn + sn+1∂n for all n ≥ 0. Hence f is unique up to chain homotopy.

Corollary 4.2.12. Let g : M → N be R-linear and pick projective resolutions P• and Q• of M and N, respectively. Then there exists a chain map f : P• → Q• such that H0(f) = g and f is unique up to chain homotopy.

Proof. Given projective resolutions P•,Q• → M, we have M = H0(P•) = H0(Q•) so let ϕ = idM . Since projective resolutions are acyclic, the comparison theorem gives us a chain map f : P• → Q•. Reversing the roles of P• and Q• gives a chain map in the opposite direction, and uniqueness forces the composition of these maps to be the identity in either direction.

Corollary 4.2.13. For R-modules M and N, the assignments (M,N) 7→ Hn(P• ⊗ N) and n (M,N) 7→ H (Hom(P•,N)) are independent of the deleted projective resolution P• → M and are functorial in both M and N. In particular, Ext and Tor are well-defined functors.

Proof. If P•,Q• → M are two projective resolutions, then by Corollary 4.2.12, P• and Q• are chain homotopy equivalent, and it easily follows that P• ⊗ N and Q• ⊗ N, as well as Hom(P•,N) and Hom(Q•,N), are chain homotopy equivalent as well. 0 0 Next, a map N → N induces a map P• ⊗N → P• ⊗N which is functorial since ⊗ and H• 0 0 are functors to begin with. On the other hand, suppose P• → M and P• → M are projective resolutions. Then given a map M → M 0, the uniqueness portion of Corollary 4.2.12 says that the map is functorial. The proof for Hom is analagous.

69 4.3 Derived Categories 4 Homology

Some of the basic facts about derived functors are described in detail in section 6.2 of Rotman. The following properties of left derived functors have analogs that hold for right derived functors.

Theorem 4.2.14. If T : A → C is an additive covariant functor between abelian categories, and A has enough projectives, then each LnT : A → C is an additive covariant functor.

Proof sketch. Use the fact that Hn is an additive covariant functor Comp(A) → A. Be careful checking that LnT is well-defined on chain maps. Proposition 4.2.15. If T : A → C is an additive covariant functor of abelian categories, then (LnT )A = 0 for all A ∈ obj(A) whenever n is negative. Proof. This is immediate from the definition since the nth term of a resolution P of A is 0 whenever n is negative.

R R R Notice that since Tor• (L, −) is an effaceable δ-functor, Tor1 (L, M) = 0 implies Torn (L, M) = 1 n 0 for n ≥ 2 and for all modules M. Similarly, ExtR(L, M) = 0 implies ExtR(L, M) = 0 for n ≥ 2 and all M.

Definition. The depth of an R-module M, denoted depth(M), is the smallest n for which n ExtR(R,M) is nonzero. The Krull dimension (see Section 5.1) of an R-module M is defined to be the supremum of the lengths of all chains of prime ideals. An important result, which we won’t prove, is

Theorem 4.2.16. The depth of M is bounded above by the Krull dimension of M.

There is a deep connection between projective dimension and depth.

Theorem 4.2.17 (Auslander-Buchsbaum). For an R-module M with finite projective di- mension n, n + depth(M) = depth(R), where R is viewed as a module over itself.

In particular, if the projective dimension of M is finite, we see that

n ≤ depth(R) ≤ dim R.

4.3 Derived Categories

Let A be an abelian category and Ch(A) the category of chain complexes on A:

··· −→d A−1 −→d A0 −→d A1 −→·d · · with d2 = 0

whose morphisms are chain maps.

Definition. A chain map θ : A• → B• is called a quasi-isomorphism if the induced map on cohomology HnA• → HnB• is an isomorphism for all n ∈ Z.

70 4.3 Derived Categories 4 Homology

Example 4.3.1. Let A = ModR be the category of modules over a ring R. Projective resolutions are quasi-isomorphisms, since the following diagram commutes.

··· P2 P1 P0 0

··· 0 0 M 0

−i (Here, we have written P = Pi since a projective resolution is bounded above.) Likewise, injective resolutions are quasi-isomorphisms.

• Example 4.3.2. For any complex A ∈ Ch(A), define the left truncation functor τ<0 : Ch(A) → Ch(A) by

• • −2 −1 0 1 A 7−→ τ<0A = · · · → A → A → ker(A → A ) → 0.

• • i • Then the natural inclusion of complexes τ<0A ,→ A is a quasi-isomorphism if H (A ) = 0 i • ∼ i • i • for all i > 0. Indeed, one has H (τ<0A ) = H (A ) for i ≤ 0 in general, and H (τ<0A ) = 0 • • for i > 0. There is also a right truncation functor τ>0, defined similarly, such τ>0A ,→ A is a quasi-isomorphism. Definition. Two chain maps f, g : A• → B• are chain homotopic, written f ' g, if there is some degree −1 map h : A• → B• for which f − g = dh + hd:

d d ··· An−1 An An+1 ··· h h f g f g f g d d ··· Bn−1 Bn Bn+1 ···

Definition. A homotopy equivalence is a chain map f : A• → B• which admits a • • homotopy inverse, i.e. a chain map g : B → A such that f ◦ g ' idB and g ◦ f ' idA. Definition. For an abelian category A, the homotopy category of A is the category K(A) whose objects are chain complexes on A and whose morphisms are chain maps up to homotopy equivalence. Lemma 4.3.3. Homology is well-defined on K(A), up to isomorphism. Proof. If A• and B• are homotopy equivalent by a pair of chain maps f : A• → B• and • • • • g : B → A such that fg − idB = dh1 + h1d for some degree 1 map h1 : B → B and • • gf − idA = dh2 + h2d for some degree 1 map h2 : A → A , we have

∗ ∗ ∗ ∗ H (f)H (g) − idH∗(B) = H (fg − idB) = H (dh1 + h1d) = 0.

So H∗(f)H∗(g) is the identity on homology; a similar proof shows H∗(g)H∗(f) is the other identity, and thus H∗(f) is an isomorphism.

71 4.3 Derived Categories 4 Homology

This says that H∗ is a homotopy functor, i.e. it factors

H∗ Ch(A) Ab

K(A)

Note that Lemma 4.3.3 also implies that a homotopy equivalence is a quasi-isomorphism.

Definition. Let C be any category with a specified class of morphisms S. Then S is called a localizing class if:

(i) S is closed under compositions and contains all identity morphisms.

(ii) For all morphisms X −→s Y and Z → Y where s ∈ S, there exists T ∈ C and a commutative diagram

s0 T Z

s X Y

with s0 ∈ S. Likewise, for all X → Z, there exists W ∈ C and a commutative diagram

s X Y

s0 Z W

f s (iii) For a pair of maps X Y , there exists Y −→ T, s ∈ S, such that sf = sg if and g 0 only if there exist U −→s X, s0 ∈ S, such that fs0 = gs0.

Example 4.3.4. Let A be an abelian category and C ⊆ A be a subcategory which is closed under subobjects, quotients and extensions (C is called a Serre subcategory). Then the collection of morphisms

SC = {f : A → B | ker f, coker f ∈ C} is a localizing class.

72 4.3 Derived Categories 4 Homology

Definition. Let C be a category with localizing class S. The localization of C at S is the −1 −1 category C[S ] with objects obj(C[S ]) = obj(C) and morphisms HomC[S−1](A, B) given by equivalence classes of diagrams

X s A B with s ∈ S, called roofs, where two roofs

X X0 0 s and s A B A B are equivalent if there exists a roof from X to X0 such that the diagram

Y t t0 X X0

s s0 A B

Composition of morphisms is given by “roof completion”:

Z

X Y

A B C and the identity A → A is given by the trivial roof:

A id id . A A

−1 In other words, each HomC[S−1](A, B) consists of formal fractions s f where s ∈ S. This should feel similar to the localization S−1R in a commutative ring R, where S is a multiplicative subset of R. Under this construction, S−1R consists of formal fractions r s−1r = , r ∈ R, s ∈ S. There is actually a notion of localization in noncommutative rings s which is captured by the categorical construction of localization described above, which explains part of its appeal.

73 4.3 Derived Categories 4 Homology

Lemma 4.3.5. Roof composition is well-defined. A f There is a functor C → C[S−1] sending a morphism A −→ B to the roof id f . A B Under this functor, all morphisms in S become invertible: if A −→s B is in S, then the roof A A id s has inverse s id . Conversely, every invertible morphism in C[S−1] A B B A is, up to composition, an element of S. For a chain complex A•, one obtains another complex by shifting every term to the left; this is denoted A[1], with graded pieces A[1]i = Ai+1 and differential dA[1] = −dA. This can be extended to A[n] for any n ∈ Z. Definition. The mapping cone of a chain map f : A• → B• is the chain complex M Cf = A[1] ⊕ B = (Ai+1 ⊕ Bi) i∈Z −d 0  with differential A . f dB In topology, if f : X → Y is a map of spaces, the cone of f is obtained by gluing a disjoint copy of X to Y along Y : a Cf = (X × [0, 1] Y )/ ∼ where (x, 0) ∼ (x0, 0) and (x, 1) ∼ (f(x), 1) for all x, x0 ∈ X. Then the singular chain complex for the pair (X,Y ) induced by f is quasi-isomorphic to the singular chain complex of Cf. Example 4.3.6. Take a morphism f : A → B in the original category A and view it as a morphism in D(A): ··· 0 A 0 ···

f

··· 0 B 0 ···

f Then Cf is the chain complex · · · → 0 → A −→ B → 0 → · · · and its cohomology is H−1(Cf) = ker f, H0(Cf) = coker f and 0 otherwise. For any chain map f : A• → B•, the natural map B• → Cf extends to an exact sequence A• → B• → Cf → A•[1] which in turn induces a long exact sequence in cohomology: · · · → HiA• → HiB• → HiCf → HiA•[1] → HiB• → HiCf[1] → · · · (This is because Hi+1A• = HiA•[1] for any i.)

74 4.3 Derived Categories 4 Homology

Proposition 4.3.7. A chain map f : A• → B• is a quasi-isomorphism if and only if Cf is acyclic, i.e. H•Cf = 0.

Proposition 4.3.8. Let f : A → B be a chain map and let τ : B → Cf be the induced map to the cone. Then Cτ is quasi-isomorphic to A[1].

Proof. There are maps A[1] ,→ Cτ → A[1] given by

−f  1  0 0 1 0
Ai+1 Bi+1 ⊕ Ai+1 ⊕ Bi Ai+1

  −dB 0 0 −dA  0 −dA 0  −dA 1 f dB

Ai+2 Bi+2 ⊕ Ai+2 ⊕ Bi+1 Ai+2

It is now straightforward to check that these are chain maps and the compositions are homotopy equivalent to the identity. Next, the short exact sequence Cf → Cτ → B[1] induces a commutative diagram

−f Cf A[1] B[1]

id q id

Cf Cτ B[1]

where q is a quasi-isomorphism. Note however that the arrow Cτ → A[1] only gives a diagram that commutes up to homotopy.

Theorem 4.3.9. Let A be an abelian category. Then the quasi-isomorphisms in K(A) form a localizing class.

f q Proof. Given a quasi-isomorphism X −→ Y and any morphism Z −→ Y , we show how to complete a diagram q T Z

g f X Y

75 4.3 Derived Categories 4 Homology

f with a quasi-isomorphism q. Consider the exact sequence X −→ Y −→τ Cf → A[1]. Then g fits into a commutative diagram with exact rows: τg C(τg)[−1] Z Cf C(τg)

g id τ Cτ[−1] Y Cf Cτ

q id id q f τ X Y Cf X[1]

(Here, q is the quasi-isomorphism from above.) Set T = C(τg)[−1]. Then C(T → Z) = Cf which is acyclic, so H•C(τg)[−1] = 0 and hence T → Z is a quasi-isomorphism by Proposition 4.3.7. Showing that squares of the form f X Y

q Z W

can be completed with a quasi-isomorphism is similar, and the proof of axiom (2) is also straightforward.

Example 4.3.10. Beware that in the category Ch(Ab) of chain complexes of abelian groups, not every square can be completed as above. For example, there is no quasi-isomorphism of chain complexes T • → [0 → Z] completing the diagram

T • [0 → Z]

id 0 [Z −→ Z]

Definition. For an abelian category A, the derived category of A is the localization D(A) := K(A)[Q−1] where Q is the localizing class of quasi-isomorphisms.

Proposition 4.3.11. Let A be an abelian category and D(A) the derived category. Then

(1) D(A) is an additive category.

(2) The homology functors Hn : K(A) → A factor through D(A).

Proof. (1) Consider two roofs in HomD(A)(A, B):

76 4.3 Derived Categories 4 Homology

X Y and A B A B Then there is a roof extension Z q q0 X Y

A B

q with q, q0 ∈ Q. Let the two compositions Z → B be denoted f : Z −→ X → B and q0 g : Z −→ Y → B. Then roof addition is defined by

X Y Z + = f + g A B A B A B

It is easy to check that this is well-defined. (2) is obvious, since quasi-isomorphisms become invertible in D(A). Note that Proposition 4.3.7 can be restated as saying f is a quasi-isomorphism if and only if Cf is isomorphic to 0 in D(A). We next show that Ext can be viewed as a Hom-set in the derived category. To do so, recall that in an abelian category A with enough injectives (or projectives), we have

i i i ExtA(A, B) = H Hom(A, E•) (= H Hom(P•,B))

where E• is an injective resolution of B (or P• is a projective resolution of A). This works because Hi is a homology functor. More generally, when E• is a complex of injectives (or • • ∼ • P is a complex of projectives), there are isomorphisms HomD(A)(A, E ) = HomK(A)(A, E ) • ∼ • (and HomD(A)(P ,A) = HomK(A)(P ,A)). Thus morphisms in the derived category can be computed in the homotopy category. To prove these isomorphisms, we need some lemmas.

Lemma 4.3.12. Suppose C• is an acyclic complex over A and P • is a bounded above complex • • of projectives. Then HomK(A)(P ,C ) = 0. Likewise, for every bounded below complex of • • • injectives E , HomK(A)(C ,E ) = 0. Proof. Standard.

Lemma 4.3.13. Suppose q : A• → P • is a quasi-isomorphism, where P • is a bounded above complex of projectives. Then there is a splitting of q in K(A), that is, a morphism • • f : P → A such that qf ∼ idP • .

77 4.3 Derived Categories 4 Homology

Proof. Since q is a quasi-isomorphism, the cone Cq is acyclic by Proposition 4.3.7. Then the induced map P • → Cq is nullhomotopic by Lemma 4.3.12. Define f : P • → Cq → A• via the natural maps P n → P n ⊕ An+1 → An. The nullhomotopy of P • → Cq gives a homotopy qf ∼ idP . These imply:

Proposition 4.3.14. If f : A → B is a quasi-isomorphism of complexes in A and E• is a complex of injectives, then the induced map

• • HomK(A)(B,E ) → HomK(A)(A, E )

is a natural isomorphism. Likewise, if P • is a complex of injectives,

• • HomK(A)(P ,A) → HomK(A)(P ,B)

is a natural isomorphism.

Corollary 4.3.15. If E• is a complex of injectives, then for any A ∈ D(A), there is a • ∼ • • natural isomorphism HomD(A)(A, E ) = HomK(A)(A, E ). Likewise, if P is a complex of • ∼ • injectives, there is a natural isomorphism HomD(A)(P ,A) = HomK(A)(P ,A). Theorem 4.3.16. Let A be an abelian category with enough injectives. Then for all A, B ∈ A and i ∈ Z, there is an isomorphism i ∼ ExtA(A, B) = HomD(A)(A, B[i]). which is natural in each variable.

Proof. Take an injective resolution B → E0 → E1 → · · · so that B 0 0 ···

E0 E1 E2 ···

Thus we can replace B by E• in the derived category and compute HomD(A)(A, B[i]) as follows: ∼ HomD(A)(A, B[i]) = HomD(A)(A, E•[i]) ∼ = HomK(A)(A, E•[i]) by Corollary 4.3.15 ker(Hom(A, E ) → Hom(A, E )) = i i+1 im(Hom(A, Ei−1) → Hom(A, Ei)) i i = H (A, E•) = ExtA(A, B).

78 4.4 Tor and Ext 4 Homology

4.4 Tor and Ext

In this section, we explore the properties of Tor and Ext. Recall the definition in Section 4.2:

R Torn (M,N) = Hn(P•,N ⊗R N). Theorem 4.4.1. Let R be a ring, A a right R-module and B a left R-module. Then R ∼ Rop (i) For all n ≥ 0, Torn (A, B) = Torn (B,A). R ∼ R (ii) If R is commutative, then for all n ≥ 0, Torn (A, B) = Torn (B,A). Proof. It suffices to prove (i) since (ii) will follow immediately. We know that every left R-module is a right Rop-module and likewise every right R-module is a left Rop-module. Let P• be a projective resolution of A. Then t : P• ⊗R B −→ B ⊗Rop P• is a chain map of Z-complexes with tn : Pn ⊗R B 7−→ B ⊗Rop Pn defined by pn ⊗ b 7→ b ⊗ pn. It is easy to see that each tn is an isomorphism, so the complexes are isomorphic and therefore have the same homology: R ∼ Rop Torn (A, B) = Hn(P• ⊗R B) = Hn(B ⊗Rop P•) = Torn (B,A).

The key property is that Torn vanishes on projective modules of homological degree n ≥ 1, since a projective resolution P• of a projective module P is simply → 0 → 0 → P → 0. Proposition 4.4.2. Let R be a ring and let M and N be R-modules.

R n (a) If M is a free R-module, then Torn (M,N) = ExtR(M,N) = 0 for n > 0. R n (b) If R is a PID, then Torn (M,N) = ExtR(M,N) = 0 for n > 1. Proof. Apply Corollary 2.1.8. Theorem 4.4.3 (Long Exact Sequence for Tor). For every short exact sequence 0 → A0 → A → A00 → 0

R in ModR and R-module B, Torn (−,B) induces an exact sequence R 00 R 0 R R 00 R 0 · · · → Torn+1(A ,B) → Torn (A ,B) → Torn (A, B) → Torn (A ,B) → Torn−1(A ,B) → · · ·

0 00 ending in · · · → A ⊗R B → A ⊗R B → A ⊗R B → 0. Proof. Use the definition of Tor via homology and the long exact sequence from Theo- rem 4.1.3. This theorem is true more generally for any left derived functor, and we will see a similar sequence with Ext and right derived functors. The long exact sequence for Tor is usually summarized in the literature by saying that the Tor sequence repairs loss of exactness on the left of tensoring a short exact sequence. R R The next result says that Torn (A, −) and Torn (−,B) can be computed using projective resolutions (or alternatively, flat resolutions) in either slot.

79 4.4 Tor and Ext 4 Homology

Theorem 4.4.4. For all projective (or flat) resolutions P• of A and Q• of B and for all n ≥ 0, ∼ R ∼ Hn(P• ⊗R B) = Torn (A, B) = Hn(A ⊗R Q•). Proof. See Rotman, section 7.1.

Proposition 4.4.5. If A and some family {Bi}i∈I are R-modules then for all n ≥ 0 there are natural isomorphisms ! R M ∼ M R Torn A, Bi = Torn (A, Bi). i∈I i∈I The same is true for direct sums in the first slot. R Proof. The base case is the natural equivalence between Tor0 (A, −) and A ⊗R − and the full theorem can be proven using induction on n.

Z Example 4.4.6. Let B be an abelian group and consider Tor1 (Z/nZ,B). For every n there is an exact sequence fn 0 → Z −→ Z → Z/nZ → 0

where fn is multiplication by n. Applying − ⊗Z B gives an exact sequence

1⊗fn Tor1(Z,B) → Tor1(Z/nZ,B) → Z ⊗ B −−−→ Z ⊗ B.

Since Z is projective, Tor1(Z,B) = 0. Moreover 1 ⊗ fn is also just multiplication by n. We also know that Z ⊗ B = B since in general R ⊗R − is naturally isomorphic to the identity functor on R-Mod. This shows that Tor1(Z/nZ,B) = {b ∈ B | nb = 0}. In the case when A and B are finitely generated abelian groups, we can now compute Tor1(A, B). Decompose A and B by the fundamental theorem of finitely generated abelian groups: r M s M A = Z ⊕ Ai and B = Z ⊕ Bj

where Ai and Bj are cyclic. Since Tor1 commutes with direct sums, ∼ r s M M Tor1(A, B) = (Tor1(Z,B)) ⊕ (Tor1(A, Z)) ⊕ Tor1(Ai,B) ⊕ Tor1(A, Bj). i j

Since Z is projective, Tor vanishes on the free parts in the first two summands. Then we’re left with Tor of cyclic groups. By the work above, each Tor1(Ai,B) = {b ∈ B | nib = 0} where ni = |Ai|. Thus Tor1(Ai,B) is a cyclic group of order di = gcd(ni, |B|) and the same holds for the Tor1(A, Bj). Example 4.4.7. If R is an integral domain with field of fractions Q and residue field K = Q/R, then Tor captures the idea of torsion submodules. In fact, if A is an R-module and At R ∼ is its torsion submodule, then Tor1 (K,A) = At. Moreover, there is an exact sequence

0 → At → A → Q ⊗R A → K ⊗R A → 0

which shows that A is torsion ⇐⇒ Q ⊗R A = 0. This explains the name Tor for in general, R Torn is a torsion R-module.

80 4.4 Tor and Ext 4 Homology

Lemma 4.4.8 (Horseshoe Lemma). Given a short exact sequence of R-modules 0 → A → B → C → 0 and deleted projective resolutions P• → A and R• → C, there is a deleted projective resolution Q• → B and a short exact sequence of chain complexes 0 → P• → Q• → R• → 0 inducing the maps of the original short exact sequence on H0.

Proof. We must have Q• = P• ⊕R• so that the short exact sequence 0 → P• → Q• → R• → 0 splits. Consider the following diagram with exact columns and exact top and bottom row: 0

ε ··· Pn ··· P1 P0 A 0 i δ0 Qn ··· Q1 Q0 B 0

Φ j

··· Rn ··· R1 R0 C 0

Our goal is to complete the middle row such that the diagram commutes. We begin by constructing δ0 : Q0 → B. Since R0 is projective, there exists a map Φ : R0 → B lifting j, given by the dashed arrow above. Then define δ0 on Q0 = P0 ⊕ R0 by δ0 = (i ◦ ε) ⊕ (−Φ). Having constructed δ0, . . . , δn−1, we have the following portion of the above diagram:

··· Pn Pn−1 Pn−2 ···

δn−1 ··· Qn Qn−1 Qn−2 ···

··· Rn Rn−1 Rn−2 ···

Setting K = ker(Pn−1 → Pn−2),L = ker(Qn−1 → Qn−2) and M = ker(Rn−1 → Rn−2), the above diagram induces a smaller diagram of the same form as the base case:

··· Pn K 0

δn Qn L

··· Rn M 0

Complete the diagram as in the base case to construct δn.

81 4.4 Tor and Ext 4 Homology

Corollary 4.4.9. If 0 → N 0 → N → N 00 → 0 is a short exact sequence of R-modules and M is any R-module, then there is a long exact sequence

0 00 0 → Torn(M,N ) → Torn(M,N) → Torn(M,N ) → Torn−1(M,N ) →

0 0 00 00 Proof. Take projective resolutions P• → N and P• → N ; then Lemma 4.4.8 provides a projective resolution P• for N and a short exact sequence of complexes

0 00 0 → P• → P• → P• → 0.

Since each term in the sequence is a complex of projective modules, applying M ⊗ − yields a short exact sequence again. Then the desired long exact sequence is merely the long exact sequence in homology for this induced short exact sequence.

R Theorem 4.4.10. For each n ≥ 0, there exists a functor Torn : R−Mod×R−Mod → R−Mod which satisfies

R (1) Tor0 (M,N) = M ⊗ N. (2) For any short exact sequence 0 → M 0 → M → M 00 → 0 and any R-module N, there is a long exact sequence

0 00 0 → Torn(M ,N) → Torn(M,N) → Torn(M ,N) → Torn−1(M ,N) →

which is natural in N.

R (3) For any free module F , Torn (F,N) = 0 for all n > 0.

R Moreover, any functor satisfying these three properties is naturally isomorphic to Torn .

Proof. We have proven that Torn satisfies the stated properties so it remains to show that these in fact characterize Torn. We prove this inductively on n. For n = 0, uniqueness follows from the universal property of the tensor product. For n ≥ 1, take modules M and N and a free module F such that there is an exact sequence 0 → K → F → M → 0. Then by (2), there is a long exact sequence

0 = Torn(F,N) → Torn(M,N) → Torn−1(K,N) → Torn−1(F,N) → · · · ∼ When n > 1, Torn−1(F,N) = 0 as well so Torn(M,N) = Torn−1(K,N). When n = 1, ∼ Tor1(M,N) = ker(K ⊗ N → F ⊗ N). In all cases, induction implies that Torn is determined as a functor by Torn−1 so uniqueness holds. Corollary 4.4.11. For any n ≥ 0 and any modules M and N, there is a natural isomorphism ∼ Torn(M,N) = Torn(N,M).

Proof. Consider the assignment (M,N) 7→ torn(M,N) := Torn(N,M). Then ∼ (1) tor0(M,N) = Tor0(N,M) = N ⊗ M = M ⊗ N.

82 4.4 Tor and Ext 4 Homology

(2) torn has a long exact sequence in the first variable because Torn has a long exact sequence in the second variable by Corollary 4.4.9.

(3) For a free module F , torn(F,N) = Torn(N,F ) = 0 for each n > 0 since − ⊗ F preserves exactness.

Hence by Theorem 4.4.10, Torn and torn are naturally isomorphic. Next we shift our focus to Ext. This is in some ways the more interesting of the two derived functors, since in any module category it naturally admits an R-algebra structure. As we saw for Tor, Ext defines a long exact sequence which repairs the loss of exactness of applying the Hom functor to a short exact sequence. Theorem 4.4.12 (Long Exact Sequence for Ext). If 0 → B0 → B → B00 → 0 is a short n exact sequence in RMod then for every left R-module A, ExtR(A, −) induces a long exact sequence

0 00 0 → HomR(A, B ) → HomR(A, B) → HomR(A, B ) 1 0 1 1 00 → ExtR(A, B ) → ExtR(A, B) → ExtR(A, B ) 2 0 2 2 00 → ExtR(A, B ) → ExtR(A, B) → ExtR(A, B ) → · · · Proof. Same as above, except reverse the arrows. Corollary 4.4.13. If 0 → M 0 → M → M 00 → 0 is a short exact sequence of R-modules and N is any R-module, then there is a long exact sequence

→ Extn(M 00,N) → Extn(M,N) → Extn(M 0,N) → Extn+1(M 00,N) →

Proof. Similar to the proof of Corollary 4.4.9.

n Theorem 4.4.14. For each n ≥ 0, there exists a functor ExtR : R−Mod×R−Mod → R−Mod which satisfies

0 (1) ExtR(M,N) = HomR(M,N). (2) For any short exact sequence 0 → M 0 → M → M 00 → 0 and any R-module N, there is a long exact sequence

→ Extn(M 00,N) → Extn(M,N) → Extn(M 0,N) → Extn+1(M 00,N) →

which is natural in N.

n (3) For any free module F , ExtR(F,N) = 0 for all n > 0. n Moreover, any functor satisfying these three properties is naturally isomorphic to ExtR. Proof. Reverse the arrows in the proof of Theorem 4.4.10. Ext is characterized by the property that Extn vanishes on all injective modules whenever n ≥ 1, for the same reason that Torn vanishes on projectives. The next proposition shows that Ext interacts with direct sums and products in the same way that Hom does.

83 4.4 Tor and Ext 4 Homology

Proposition 4.4.15. For R-modules B and {Ai}i∈I and for all n ≥ 0, there are natural isomorphisms ! n M ∼ Y n ExtR Ai,B = ExtR(Ai,B). i∈I i∈I As with Hom, things are slightly different in the second slot.

Proposition 4.4.16. For R-modules A and {Bi}i∈I and for all n ≥ 0, there are natural isomorphisms ! n Y ∼ Y n ExtR A, Bi = ExtR(A, Bi). i∈I i∈I

n Example 4.4.17. Fix n ≥ 0, m ≥ 1 and consider ExtZ(Z/mZ, Z). A simple projective resolution of Z/mZ is the short exact sequence

m 0 → Z −→ Z → Z/mZ → 0.

Applying Hom(−, Z) and deleting Z/mZ, we obtain a complex

m 0 → Hom(Z, Z) = Z −→ Hom(Z, Z) = Z → 0. The homology of this sequence is now easy to calculate:

0 1 Ext (Z/mZ, Z) = 0 and Ext (Z/mZ, Z) = Z/mZ. In general, for an abelian group B and for any n ≥ 2,

1 ∼ ExtZ(Z/nZ,B) = B/nB. (Compare to Example 4.4.6.) As before, one can use this to show that for any finitely 1 1 generated abelian groups A, B, Ext (A, B) is the direct sum of Ext (Ai,Bj) where Ai and Bj are finite cyclic groups. Definition. For R-modules A, C, an extension of A by C is a short exact sequence

p 0 → A −→i B −→ C → 0.

An extension splits if there exists a map s : C → B such that ps = 1C .

1 Lemma 4.4.18. If ExtR(C,A) = 0 then every extension of A by C splits.

i p Proof. Starting with an extension 0 → A −→ B −→ C → 0, apply HomR(C, −) to obtain an exact sequence p Hom(C,B) −→∗ Hom(C,C) −→ð Ext1(C,A) = 0.

So p∗ is surjective, meaning there is some s ∈ Hom(C,B) such that 1C = p∗(s), or 1C = ps. Hence the extension splits.

84 4.4 Tor and Ext 4 Homology

1 The elements in ExtR(C,A) can be thought of as ‘obstructions’ to the splitting property of extensions of A by C. This explains where the name Ext comes from. An important property of Ext is

1 Proposition 4.4.19. A left R-module P is projective ⇐⇒ ExtR(P,B) = 0 for every 1 R-module B. Similarly, a left R-module E is injective ⇐⇒ ExtR(A, E) = 0 for every R-module A.

1 Proof. If P is projective, then ExtR(P,B) = 0 for all B by Proposition 2.1.3. On the other 1 hand, if ExtR(P,B) = 0 for all B then Lemma 4.4.18 shows that every exact sequence ending in P splits and hence P is projective. The second statement is proven similarly. Likewise, we have a homological characterization of flat modules. Proposition 4.4.20. For an R-module M, the following are equivalent: (a) M is flat.

R (b) Torn (A, M) = 0 for all n ≥ 1 and R-modules A. R (c) Tor1 (A, M) = 0 for all R-modules A. Proposition 4.4.21. Let R be a commutative ring, a ∈ R a non-zero-divisor and M an R-module. Write aM = {m ∈ M | am = 0} for the a-torsion part of M. Then (a) R/aR ⊗ M ∼= M/aM. ∼ (b) Tor1(R/aR, M) = aM. ∼ (c) Hom(R/aR, M) = aM. (d) Ext1(R/aR, M) ∼= M/aM. Proof. (a) Define a map φ : R/aR × M → M/aM by (r, m) 7→ rm. If r − r0 ∈ aR, then r − r0 = as for some s ∈ R. Then (r − r0)m = (as)m = a(sm) ∈ aM so the map is well-defined on the quotient R/aR. It is also clearly bilinear. By the universal property of tensor products, this determines a linear map Φ : R/aR ⊗ M → M/aM. If Φ(r ⊗ m) = 0 in M/aM then Φ(r ⊗ m) = rm ∈ aM. Thus rm = am0 for some m0 ∈ M, and since the tensor is over R, we can pass elements of R across the tensor:

r ⊗ m = 1 ⊗ rm = 1 ⊗ am0 = a ⊗ m0 = 0 ⊗ m0 = 0 in M/aM.

So Φ is one-to-one. Clearly Φ is also surjective: 1 ⊗ m 7→ m for any m ∈ M. Hence we have the desired isomorphism. (b) Consider the exact sequence

0 → R −→a R −→ R/aR → 0.

Tensoring with M gives a long exact sequence in Tor:

0 = Tor1(R,M) → Tor1(R/aR, M) → R ⊗ M → R ⊗ M → R/aR ⊗ M → 0.

85 4.4 Tor and Ext 4 Homology

(By Proposition 4.4.2, since R is a free R-module, Tor1(R,M) = 0.) Using the facts that R ⊗ M ∼= M and, from part (a), R/aR ⊗ M ∼= M/aM, we get an isomorphic exact sequence:

a 0 → Tor1(R/aR, M) → M −→ M → M/aM → 0.

Since the map out of Tor1(R/aR, M) is injective and the sequence is exact, we see that a Tor1(R/aR, M) is isomorphic to the kernel of M −→ M, which is clearly aM. Hence ∼ Tor1(R/aR, M) = aM. (c) For each m ∈ M, there is an R-map fm : R → M, r 7→ rm. Moreover, if m ∈ aM, then ¯ fm factors through the quotient R/aR, giving an element fm of Hom(R/aR, M). This de- ¯ termines a map ϕ : aM → Hom(R/aR, M), m 7→ fm. Conversely, each g ∈ Hom(R/aR, M) determines an element g(1) ∈ aM, since ag(1) = g(a1) = g(a) = 0. This gives ψ : Hom(R/aR, M) → aM, and we have ¯ ϕψ(g)(r) = ϕ(g(1)) = fg(1)(r) = rg(1) = g(r) ¯ ¯ and ψϕ(m) = ψ(fm) = fm(1) = 1m = m. ∼ So we see that ϕ and ψ are inverses. Hence Hom(R/aR, M) = aM. (d) Using the same exact sequence

0 → R −→a R −→ R/aR → 0 and applying Hom(−,M), we get a long exact sequence in Ext:

0 → Hom(R/aR, M) → Hom(R,M) → Hom(R,M) → Ext1(R/aR, M) → Ext1(R,M) = 0.

1 (Again since R is free, Ext (R,M) = 0.) Replacing Hom(R/aR, M) with aM and Hom(R,M) with M, we get a 1 0 → aM → M −→ M → Ext (R/aR, M) → 0. By exactness, Ext1(R/aR, M) is isomorphic to the cokernel of the map M −→a M, which is M/aM by definition.

Example 4.4.22. Let G be an abelian group, with torsion part T (G). For another abelian group B, applying − ⊗ B to the short exact sequence 0 → T (G) → G → G/T (G) → 0 gives a long exact sequence in Tor (coefficients in Z):

0 → Tor1(T (G),B) → Tor1(G, B) → Tor1(G/T (G),B) = 0,

with the final 0 coming from Proposition 4.4.2 (using that G/T (G) is torsion-free). Thus one may replace G by T (G) when computing Tor1. Further, consider the short exact sequence 0 → Z → Q → Q/Z → 0. Applying T (G) ⊗ − gives another long exact sequence

Tor1(T (G), Z) → Tor1(T (G), Q) → Tor1(T (G), Q/Z) → T (G) ⊗ Z → T (G) ⊗ Q = 0. ∼ ∼ Further, Tor1(T (G), Q) = 0, so we get Tor1(Q/Z,G) = Tor1(Q/Z,T (G)) = T (G).

86 4.4 Tor and Ext 4 Homology

Example 4.4.23. Let m ≥ 2 and n ≥ 0 be integers. Then using the above results, we can compute ( Z/mZ, n = 0 Torn(Z, Z/mZ) = Torn(Z/mZ, Z) = 0, n > 0  0, n = 0 n  Ext (Z/mZ, Z) = Z/mZ, n = 1 0, n > 1 ( n Z/mZ, n = 0 Ext (Z, Z/mZ) = 0, n > 0.

Example 4.4.24. Let R be a commutative ring and m ⊂ R a maximal ideal. Let k = R/m 1 be the residue field. An interesting Ext group in commutative algebra is ExtR(k, k), as we will see in more detail in Section 5.1. To compute this group, consider the short exact sequence 0 → m → R → k → 0.

Since R is a free module over itself, by Theorem 4.4.14, the long exact sequence in ExtR(−, k) becomes

=∼ 1 0 → HomR(k, k) −→ HomR(R, k) → HomR(m, k) → ExtR(k, k) → 0. ∼ ∼ Moreover, since HomR(k, k) = k and HomR(R, k) = k, the first arrow is an isomorphism. By exactness, the second arrow must be 0, so we get

1 ∼ ExtR(k, k) = HomR(m, k). ∼ For any R-module N, every map N → R/m factors through N/mN, so we see that HomR(N,R/m) = ∼ 2 ∼ HomR(N/mN,R/m). Setting N = m, we then get HomR(m, k) = HomR(m/m , k) = (m/m2)∗, the dual space of m/m2. This is more commonly known as the tangent space to 2 ∗ Spec R at m, Tm Spec R = (m/m ) . Thus tangent spaces have a homological interpretation.

1 Example 4.4.25. For an R-module M, we can similarly consider ExtR(M,M), which con- sists of R-modules Mf for which there is a short exact sequence

0 → M → Mf → M → 0.

There is an endomorphism t : Mf → Mf making Mf into a module over the ring R[ε] = R[t]/(t2); in fact, Mf is free over R[ε]. There is a bijective correspondence between elements 1 ∼ of ExtR(M,M) and equivalence classes of free R[ε]-modules Mf such that M/tf Mf = M, where Mf and Mf0 are equivalent if there exists an isomorphism Mf ∼= Mf0 inducing the identity on M. One can then view M as a ‘point’ in the ‘space’ of all R-modules and Mf as a ‘tangent vector’ to M in this space. (Officially, Mf is called a deformation of M.) The deformation Mf = M[ε] has an explicit R-module structure given by

α : R −→ End(M[ε]), α(r) · m1 + εm2 = rm1 + εrm2.

87 4.4 Tor and Ext 4 Homology

Then other deformations of M come from mapsα ˜ : R → End(M[ε]) such that α ≡ α˜ mod ε. That is,α ˜ = α + εµ for µ : R → End(M). For any r ∈ R, m1, m2 ∈ M,

α˜(r)(m1 + εm2) = rm1 + ε(rm2 + µ(r)µ1).

Thus forα ˜ to define an R-module structure on M[ε], we must haveα ˜(r1)˜α(r2) =α ˜(r1r2), i.e. for any m ∈ M,

α˜(r1)(r2m + εµ(r2)m) = r1r2m + εµ(r1)(r2m) + εr1µ(r2)m

= r1r2m + εµ(r1r2)m =α ˜(r1r2)m. Thus the necessary condition on µ is

µ(r1)r2 + r1µ(r2) − µ(r1r2) = 0. This is obviously a cocycle condition, so it can be measured with homology. On the other hand, when areα ˜ andα ˜0 equivalent? i.e. when do they determine equivalent deformations? Letα ˜0 = α + εµ0. For equivalence to be true, there must be some β = 1 + εγ ∈ End(M[ε]) such that βα˜ =α ˜0β. (Notice that β−1 = 1 − εγ since ε2 = 0.) Then α˜ = β−1α˜0β = (1 − εγ)˜α0(1 + εγ), so the necessary condition on µ and µ0 is µ = µ0 + ε(αγ − γα). This proves there is a bijection {µ : R → End(M) | µ(r )r + r µ(r ) − µ(r r ) = 0} Ext1 (M,M) ←→ 1 2 1 2 1 2 . R hαγ − γα | γ ∈ End(M)i One can also check that this correspondence is compatible with the group operations, where addition on the right is pointwise addition of the maps µ : R → End(M). Now assume R is a finitely presented ring with generators x1, . . . , xn, and M is both an R-module and a finite dimensional vector space over some field k (e.g. k is the residue field of a local ring as in the previous example). A potential R-action on M is specified by a map ∼ Span{x1, . . . , xn} −→ Endk(M) = Mn(k), xi 7−→ Xi such that the relations on the xi are mapped to 0, i.e. there are polynomials pj for which we have pj(X1,...,Xn) = 0. This defines an algebraic set Y , the space of all potential R-actions on M, which has an action of GL(M). The quotient space ModR(M) = Y/GL(M) is called the moduli space of R-actions on M. Under the quotient map Y → Y/GL(M), a tangent vector to M maps to a tangent vector modulo a tangent vector to GL(M), which is just a matrix. These tangent vectors on ModR(M) are precisely given by certain cocycle conditions like the one above. Example 4.4.26. Let Γ be a quiver, that is, an oriented graph. ˆ 3

ˆ ˆ 1 2 ˆ 4

88 4.4 Tor and Ext 4 Homology

The graph algebra of Γ is defined to be the span of all oriented paths in Γ over some commu- tative ring R (usually a field), with multiplication defined by concatenation where possible (and 0 otherwise) and addition given by formal sums over R. This algebra is written Π(Γ). The representation theory of Π(Γ) is quite important in modern representation theory. If 1v P denotes the trivial path at a vertex v ∈ V (Γ), notice that 1 = v∈V (Γ) 1v and 1v1v0 = δvv0 1v, that is, the elements 1v are orthogonal idempotents in Π(Γ). Therefore for any module M over the path algebra, there is a decomposition

∼ M M = 1vM. v∈V (Γ)

Set Mv = 1vM. We can label the vertices of the graph with these submodules:

ˆ M3

ˆ ˆ

M1 M2 ˆ M4

This allows us to study the representations of Π(Γ) in more detail by making choices at each vertex. For example, let R = k be a field and consider the following module:

ˆ 0

ˆ ˆ 0 k ˆ 0

This depicts a simple Π(Γ)-module since it is 1-dimensional. (In fact, if Γ has no oriented cycles, then these are the only simple modules.) Let Sv denote the simple module with k at vertex v, e.g. S1 is depicted above. We are interested in computing the group ExtΠ(Γ)(Sv,Sv0 ) for different v, v0 ∈ V (Γ). Consider the short exact sequence

0 → K → Π(Γ) → Sv → 0.

Here, 1w 7→ δvw for any idempotent 1w ∈ Π(Γ). The kernel K is spanned by all paths of length at least one, together with all 1w for w 6= v. Decompose the path algebra as: M Π(Γ) = Pw w∈V (Γ) ∼ where Pw := Π(Γ)1w. Then each summand Pw is a projective module. Note that HomΠ(Γ)(Pw,M) = ∼ Mw for any module M – in general, this is the fact that HomR(Re, M) = eM for an idem- potent e. Thus we have a short exact sequence

0 → K → Pv → Sv → 0

89 4.5 Universal Coefficient Theorems 4 Homology

which is easier to understand. Applying ExtΠ(Γ)(−,Sv0 ) yields an exact sequence

0 → Hom(Sv,Sv0 ) → Hom(Pv,Sv0 ) → Hom(K,Sv0 ) → Ext(Sv,Sv0 ) → 0

since Pv is projective. Since im Hom(Pv,Sv0 ) = 0, we get an isomorphism ∼ Ext(Sv,Sv0 ) = Hom(K,Sv0 ).

Further, one can see that K is in canonical bijection with the set of paths in Γ starting at any w adjacent to v, counted with multiplicity given by the number of edges v → w. That ∼ L jw is, K = v→w Pw , where jw is this multiplicity of paths. Thus we can write ! ∼ ∼ M jw ∼ `vv0 Ext(Sv,Sv0 ) = Hom(K,Sv0 ) = Hom Pw ,Sv0 = k , v→w

0 where `vv0 represents the number of paths v → v . In practice, these Ext groups contain enough information (number of paths) to reconstruct the quiver from just a set of vertices.

4.5 Universal Coefficient Theorems

Now that we have sufficiently described the functors ⊗ and Hom and their derived functors, it is natural to ask: what is the relation between H•(C• ⊗ M) and H•(C•) ⊗ M for a chain complex C• and module M? Likewise, what is the relation between H•(Hom(C•,M)) and Hom(H•(C•),M)? The universal coefficient theorems give answers to these questions in terms of Tor and Ext.

Theorem 4.5.1 (Universal Coefficient Theorem for Homology with Coefficients). For a free chain complex C• over a PID and a coefficient module M, for each n ≥ 0 there is a short exact sequence

0 → Hn(C•) ⊗ M → Hn(C• ⊗ M) → Tor1(Hn−1(C•),M) → 0.

Moreover, this short exact sequence splits and is natural in C• and M.

Proof. The map g : Hn(C•) ⊗ M → Hn(C• ⊗ M) may be defined by [a] ⊗ m 7→ [a ⊗ m]. Write Zn = ker(∂ : Cn → Cn−1), Bn = im(∂ : Cn+1 → Cn) and Hn = Hn(C•). Then for each n ≥ 0, there are exact sequences

i q 0 → Bn −→ Zn −→ Hn → 0 (1) j ∂ 0 → Zn −→ Cn −→ Bn−1 → 0 (2)

where (2) is split. Tensoring with M, we get a diagram with exact rows and columns:

90 4.5 Universal Coefficient Theorems 4 Homology

0 0 0

i q Bn ⊗ M Zn ⊗ M Hn ⊗ M 0 g j ∂ ∂ Cn+1 ⊗ M Cn ⊗ M Cn−1 ⊗ M f ∂

0 Tor1(Hn−1,M) Bn−1 ⊗ M Zn−1 ⊗ M

0 0 0

The bottom row is the long exact sequence in Tor coming from the short exact sequence (1), but it terminates after Tor1 because Zn−1 is free. Also, since (2) is split we get the zeroes at the top of the middle column and bottom of the right column. By diagram chasing, one can define f and g and show that the sequence is exact. Moreover, the sequence is split due to the fact that sequence (2) is split. Finally, naturality follows from naturality of tensor, Hn and Tor1. Theorem 4.5.2 (Universal Coefficient Theorem for Cohomology). For a free chain complex C• over a PID and a coefficient module M, for each n ≥ 0 there is a short exact sequence 1 n 0 → Ext (Hn−1(C•),M) → H (Hom(C•,M)) → Hom(Hn(C•),M) → 0

which splits and is natural in C• and M. 1 Proof. Replacing − ⊗ M with Hom(−,M), Tor1 with Ext and reversing the arrows in the proof of Theorem 4.5.1 gives a diagram: 0 0 0

i q Hom(Bn,M) Hom(Zn,M) Hom(Hn,M) 0 g j ∂ ∂ Hom(Cn+1,M) Hom(Cn,M) Hom(Cn−1,M) f ∂

1 0 Ext (Hn−1,M) Hom(Bn−1,M) Hom(Zn−1,M)

0 0 0

91 4.5 Universal Coefficient Theorems 4 Homology

The proof is dual to the proof of Theorem 4.5.1.

Lemma 4.5.3. Let G = F (G) ⊕ T (G) be a finitely generated abelian group, where F (G) ∼ is the free part of G and T (G) is the torsion part of G. Then Hom(G, Z) = F (G) and 1 ∼ Ext (G, Z) = T (G). Proof. Use the universal coefficient theorem, Example 4.4.23 and additivity.

Corollary 4.5.4. Suppose C• is a chain complex and M is a module. If Hn(C•; M) and n Hn−1(C•,M) are finitely generated, then so is H (C•,M) and there is an isomorphism

n ∼ H (C•,M) = Fn ⊕ Tn−1, where Fn = F (Hn(C•,M)) and Tn−1 = T (Hn−1(C•,M)). Note that the isomorphism in Corollary 4.5.4 is not canonical.

92 5 Ring Homology

5 Ring Homology

5.1 Dimensions of Rings

Homology, especially derived functors like Ext and Tor, is used to define various dimensions of a ring R, which measure how far R is from having some nice characteristic. We have already introduced projective dimension, which was defined by

Definition. If A is a left R-module and · · · → P2 → P1 → P0 → A → 0 is a projective resolution of A, we say A has projective dimension n if there is a smallest n such that Kn is projective. This is written pd(A) = n. If no such finite n exists, A has infinite projective dimension.

In commutative algebra we are often interested in proving a theorem of the same flavor as the one below.

Theorem 5.1.1. The following are equivalent for a left R-module A.

(1) pd(A) ≤ n.

i (2) ExtR(A, B) = 0 for all left R-modules B and all i ≥ n + 1.

n+1 (3) ExtR (A, B) = 0 for all left R-modules B.

(4) There exists a projective resolution P• of A such that the (n − 1)st syzygy Kn−1 is projective.

(5) In every projective resolution of A the (n − 1)st syzygy is projective.

Proof. (1) =⇒ (2) By assumption there is a projective resolution Pn → · · · → P0 → A → 0 ∗ with Pi = 0 for all i ≥ n + 1. Therefore the maps di : HomR(Pi−1,B) → HomR(Pi,B) i induced by HomR(−,B) are all 0 for i ≥ n + 1. Hence ExtR(A, B) = 0 for all i ≥ n + 1. (2) =⇒ (3) is immediate. (3) =⇒ (4) Let Kn−1 be the (n − 1)st syzygy of a projective resolution of A. Since n+1 ExtR (A, B) = 0 for all B, by properties of right derived functors we have n+1 ∼ 1 ExtR (A, B) = ExtR(Kn−1,B) = 0.

Hence Kn−1 is projective. (4) =⇒ (5) follows from Proposition 3.4.5. (5) =⇒ (1) Let P• = · · · → P1 → P0 → A → 0 be a projective resolution of A. If Kn−1 is the (n − 1)st syzygy of P• then

0 → Kn−1 → Pn−1 → · · · → P1 → P0 → A → 0 is exact. By assumption Kn−1 is projective, so this is a finite projective resolution of A and hence pd(A) ≤ n.

93 5.1 Dimensions of Rings 5 Ring Homology

Proposition 5.1.2. Suppose 0 → M 0 → M → M 00 → 0 is an exact sequence of left R- modules. If two of the modules have finite projective dimension, so does the third. Proof omitted. The proof of the next proposition shows that the long exact Ext sequence can be useful when dealing with projective dimension. Proposition 5.1.3. Let 0 → M 0 → M → M 00 → 0 be an exact sequence of left R-modules. (i) If pd(M 0) < pd(M) then pd(M 00) = pd(M).

(ii) If pd(M 0) > pd(M) then pd(M 00) = pd(M 0) + 1.

(iii) If pd(M 0) = pd(M) then pd(M 00) ≤ pd(M 0) + 1. Proof. Let 0 → M 0 → M → M 00 → 0 be exact, take A to be a left R-module and consider the long exact sequence for Ext:

0 → Hom(M 00,A) → Hom(M,A) → Hom(M 0,A) → Ext1(M 00,A) → Ext1(M,A) → Ext1(M 0,A) → · · · · · · → Extn(M 00,A) → Extn(M,A) → Extn(M 0,A) → Extn+1(M 00,A) → Extn+1(M,A) → Extn+1(M 0,A) → · · ·

(i) Let pd(M) = n and suppose pd(M 0) < n. Then Extn(M 0,A) = 0 and Extn+1(M,A) = 0 so exactness implies Extn+1(M 00,A) = 0. Thus pd(M 00) ≤ n. Notice that since n is the projective dimension of M, Extn(M,A) 6= 0. Then we cannot have Extn(M 00,A) = 0 since the right arrow of then nth row

· · · → Extn(M 00,A) → Extn(M,A) → 0 → · · ·

is a surjection and hence nonzero. Therefore pd(M 00) = n. (ii) Now suppose pd(M 0) > n. Let pd(M 0) = N so that ExtN+1(M 0,A) = 0. Since N > n, ExtN (M,A) and ExtN+1(M,A) are both 0 so the Nth and (N + 1)st rows of the long sequence look like

··· ExtN (M 00,A) → 0 → ExtN (M 0,A) → ExtN+1(M 00,A) → 0 → 0 → · · ·

By exactness, this shows that ExtN (M 0,A) ∼= ExtN+1(M 00,A) and in fact this will hold for Extk(M 0,A) and Extk+1(M 00,A) for all k ≥ N (all of the Exts on M in between will be 0). Hence ExtN+1(M 00,A) 6= 0 and Extk(M 00,A) = 0 for all k > N + 1 so by definition pd(M 00) = N + 1. (iii) Finally, suppose pd(M 0) = n. Then Extk(M 0,A) = 0 = Extk(M,A) for all k ≥ n+1. Consider

· · · → Extn(M 00,A) → Extn(M,A) → Extn(M 0,A) → Extn+1(M 00,A) → 0 → 0 → Extn+2(M 00,A) → 0 → 0 → · · ·

94 5.1 Dimensions of Rings 5 Ring Homology

As before, the 0’s on either side show that Extn+2(M 00,A) = 0 which is enough to prove that pd(M 00) ≤ n + 1. The proofs when any of the projective dimensions are infinite follows from Proposi- tion 5.1.2.

Definition. Let (R, m, k) be a local ring. The Krull dimension of R (often just shortened to dimension and denoted dim R) is

dim R = sup{n | there exists a chain of prime ideals P0 ( P1 ( ··· ( Pn in R}.

Example 5.1.4. If k is a field, dim k[x1, . . . , xn] = n. Theorem 5.1.5. For any local (noetherian) ring R, dim R < ∞.

Definition. The embedding dimension of a local ring (R, m, k) is the minimal number of generators of the maximal ideal m, that is,

2 e dim R = dimk(m/m ).

One can prove that

Proposition 5.1.6. Krull dimension is bounded above by embedding dimension.

An important type of ring arises when dim R = e dim R.

Definition. If dim R = e dim R then R is said to be a regular ring.

Regular local rings are an important object of study in algebraic geometry, as they are in correspondence with smooth points on a scheme X. See Section 5.3 for more details.

Example 5.1.7. For (R, p, k) a local ring, pick a projective resolution P• of k over R. Since − ⊗R Rp is exact, P• ⊗R Rp is a projective resolution of k ⊗R Rp. Then we can compute

i ExtRp (kp, kp) = H−i HomRp (P• ⊗R Rp, kp) ∼ = H−i (HomR(P•, k))p = H−i (HomR(P•, k) ⊗R Rp) ∼ = (H−i HomR(P•, k)) ⊗R Rp since ⊗R commutes with homology i = ExtR(k, k) ⊗ Rp i
= ExtR(k, k) p . Thus Ext localizes well.

For the remainder of the section, assume (R, m, k) is a local ring. We will prove

Theorem 5.1.8. Let M be a finitely generated left R-module.

(1) There is a well-defined notion of a minimal free resolution (MFR) of M.

95 5.1 Dimensions of Rings 5 Ring Homology

(2) If 0 ← M ← F0 ← F1 ← F2 ← · · · is this minimal free resolution of M then for each i ≥ 0, i R rank Fi = dimk ExtR(M, k) = dimk Tori (M, k). Below we define the correct notion of ‘minimal’ to make the above theorem work. We will prove (2) in a moment.

Definition. A complex C• of R-modules is minimal if ∂(C•) ⊆ mC•, i.e. with respect to some basis, all of the entries of the matrices representing the linear maps along the arrows in C• lie in the maximal ideal m.

x 2 ( x x ) 2 ( 1 ) Example 5.1.9. For R = k[x], C• = 0 ← R ←−−−− R ←−− R ← 0 is not minimal since not all of the entries in the matrices along the arrows lie in (x) (1 is the culprit in the first matrix). There is a ‘better’ resolution, namely

(x) 0 ← R ←− R ← 0.

This resolution is minimal. Notice that C• is acyclic and H•(C•) = k.

The next proposition says that if C• is any free resolution of M and P• is a minimal free resolution of M, then P• is (isomorphic to) a subcomplex of C•. The map θ defined below is called a retraction of C• onto P•.

Proposition 5.1.10. Let M be a finitely generated left R-module and suppose C• is any free resolution of M and P• is a minimal free resolution of M. Then there is a chain map 2 ∼ θ : C• → C• such that θ = θ and im θ = P•.

Corollary 5.1.11. If P• and Q• are minimal free resolutions of M, a left module over (R, m, k), there is a chain map P• → Q• that is an isomorphism.

(a ) Example 5.1.12. Let F −−→ij G be a map of left modules over (R, m, k) a local noetherian ring. Tensoring with k yields (¯aij ) F ⊗R k −−→ G ⊗R k.

R ∼ rank Fi Thus for any finitely generated left R-module M, Tori (M, k) = k where Fi is the ith free module in a minimal free resolution of M. Likewise, applying HomR(−,R/m) gives us

(¯aji) HomR(G, k) −−→ HomR(F, k)

i ∼ rank Fi so ExtR(M, k) = k as well. This proves (2) of Theorem 5.1.8. In practice, we apply this to an exact sequence 0 → M 0 → M → M 00 → 0 to produce

0 00 0 → Tor0(M , k) → Tor0(M, k) → Tor0(M , k) 0 00 → Tor1(M , k) → Tor1(M, k) → Tor1(M , k) → · · · 0 00 · · · → Torn(M , k) → Torn(M, k) → Torn(M , k) → 0

96 5.2 Hilbert’s Syzygy Theorem 5 Ring Homology

where n = max{pd(M), pd(M 0), pd(M 00)}. The above shows that the Tor are all vector spaces, and rank is additive along exact sequences, so

n n X i X i 0 00 (−1) βi(M) = (−1) (βi(M ) + βi(M )) i=0 i=0

X i where βi denotes the ith Betti number of the module in its parentheses. The sum (−1) βi(M) is called the Euler characteristic of M, denoted χ(M). We have therefore shown that χ(M) = χ(M 0) + χ(M 00), i.e. Euler characteristic is additive along short exact sequences.

5.2 Hilbert’s Syzygy Theorem

Let R = k[x1, . . . , xn] where k is a field. R is an example of a graded ring, for it can be M written R = Ri where Ri = Spank{monomials of degree i}, R0 = k and RiRj ⊆ Ri+j for i≥0 all i, j. M Definition. For a graded ring R, M is a graded R-module if M = Mi for submodules i∈Z Mi that satisfy RiMj ⊆ Mi+j for all i, j. Definition. A graded homomorphism is a map f : M → N between graded R-modules M and N that preserves grading, i.e. f(Mi) ⊆ Ni for all i. In proving his Syzygy Theorem, Hilbert was trying to answer the following question: If M is a finitely generated graded R-module, is there a recurrence formula for the dimensions of the Mi? Definition. Given a graded R-module M, its Hilbert series is the formal power series

X i HM (t) = (dim Mi)t . i∈Z

Example 5.2.1. If R = k[x1, . . . , xn] as above, then 1 H (t) = 1 + 3t + 6t2 + 10t3 + ... = . R (1 − t)3

Hilbert’s question can be restated by asking if there is a recurrence relation among the coefficients of HM (t). This in turn is equivalent to asking if HM (t) is rational. Proposition 5.2.2. Suppose we have an exact sequence of graded modules

0 → M 0 → M → M 00 → 0 where the arrows are graded homomorphisms. Then HM (t) = HM 0 (t) + HM 00 (t).

97 5.2 Hilbert’s Syzygy Theorem 5 Ring Homology

Proof. For every i, we get a short exact sequence of vector spaces

0 00 0 → Mi → Mi → Mi → 0 and dimension is additive along short exact sequences of vector spaces. More generally, one can show

Proposition 5.2.3. Suppose C• = 0 → M1 → · · · → Mk → 0 is a complex of graded R-modules. Then X i X i i (−1) HMi (t) = (−1) HH (C•)(t). i∈Z i∈Z

Suppose M is a graded left R-module generated by the elements {m1, . . . , m`} which have degrees b01, . . . , b0`. Consider the following projective (free) resolution of M. L L 0 M j R(−b0j) j R(−b1j) ···

K1

0 0

The ring R(−bij) is called a shifting of the graded ring R, which is used to line up the degrees of the mi with the graded pieces of the ring. One can prove

k Lemma 5.2.4. If k ∈ Z then HR(−k)(t) = t HR(t).

tbij In particular, H (t) = tbij H (t) = for all i, j. Hilbert’s proof of the Syzygy R(−bij ) R (1 − t)n Theorem boiled down to showing

X (−1)i+1tbij H (t) = M (1 − t)n i,j

whenever pd(M) ≤ n. In modern language, this can be stated in terms of global dimension.

Definition. For a ring R, its global dimension is

gldim(R) = sup{pd(A) | A is an R-module}.

Theorem 5.2.5. Let R be a ring. Then

n (i) For any R-module A, pd(A) = max{n ≥ 0 | ExtR(A, M) 6= 0 for some M}.

n (ii) gldim(R) = max{n ≥ 0 | ExtR(A, M) 6= 0 for some A, M}.

Proposition 5.2.6. For n ∈ N,

(a) If k is a field of characteristic p | n then k[Z/nZ] has infinite global dimension.

98 5.2 Hilbert’s Syzygy Theorem 5 Ring Homology

(b) Z[Z/nZ] has infinite global dimension. One defines injective and flat dimension in analogy with projective dimension:

Definition. If A is an R-module and 0 → A → E0 → E1 → E2 → · · · is an injective resolution of A, then A has injective dimension id(A) = n if there is a smallest n such that Kn is injective. Likewise, if · · · → F2 → F1 → F0 → A → 0 is a flat resolution of A, then A has flat dimension fd(A) = n if there is a smallest n such that Kn is flat. If no such n exists, we say A has infinite injective/flat dimension.

Theorem 5.2.7. The following are equivalent for a left R-module A:

(1) id(A) ≤ n.

i (2) ExtR(B,A) = 0 for all left R-modules B and all i ≥ n + 1.

n+1 (3) ExtR (B,A) for all left R-modules B.

(4) There exist an injective resolution E• of A such that the (n − 1)st cosyzygy Kn−1 is injective.

(5) In every injective resolution of A the (n − 1)st cosyzygy is injective.

Proof. Similar to Theorem 5.1.1.

Theorem 5.2.8. For any ring R,

max{id(M) | M is an R-module} = max{pd(M) | M is an R-module} = max{pd(R/I) | I ⊆ R is a left ideal} n = max{n ≥ 0 | ExtR(A, B) 6= 0 for some R-modules A, B}.

Proof. By Theorems 5.1.1 and 5.2.7,

n max{pd(M) | M ∈ R-Mod} ≤ n ⇐⇒ max{n | ExtR(A, B) 6= 0} ≤ n ⇐⇒ max{id(M) | M ∈ R-Mod} ≤ n.

Further, max{pd(R/I) | I ⊆ R} ≤ max{pd(M) | M ∈ R-Mod} is trival. To complete the circle, assume e = max{pd(R/I) | I ⊆ R} < id(N) for some R-module N. If

0 → N → E0 → E1 → · · · → Ee−1 → C

is an injective resolution of N of length e − 1 with (e − 1)st cosyzygy C, then C must not n be injective since ExtR(C,N) 6= 0 for some n > e. By Baer’s criterion (Theorem 2.3.3), 1 e+1 ExtR(R/I, C) 6= 0 implies ExtR (R/I, N) 6= 0, which implies pd(R/I) > e, a contradiction. Hence e ≥ id(N) for all R-modules N, so e ≥ max{id(N) | N ∈ R-Mod} = max{pd(M) | M ∈ R-Mod}, completing the proof.

Example 5.2.9. R has global dimension 0 if and only if R is a semisimple ring.

99 5.2 Hilbert’s Syzygy Theorem 5 Ring Homology

Example 5.2.10. R has global dimension 1 if and only if every submodule of a projective R-module is projective. Such a ring is called hereditary. An example of a hereditary ring is a Dedeking domain, e.g. the ring of integers in a number field or the coordinate ring of a smooth algebraic curve.

Theorem 5.2.11 (Hilbert’s Syzygy Theorem). For any ring R,

gldim k[x1, . . . , xn] = gldim k + n.

In particular, if k is a field then gldim k[x1, . . . , xn] = n.

Example 5.2.12. For a graded module M over S = R/I where R = k[x1, . . . , xn], define βi to be the rank of the ith free S-module in a minimal free resolution of M. These are called i the Betti numbers of M. One can prove that βi = dimk ExtS(M, k). In other words, if I contains anything at all, there exists an S-module M which has infinite projective dimension by Theorem 5.3.5. The question that arises from this is

Question (Serre-Kaplansky). Is the Poincar´e-Bettiseries

M X i PS (t) = βit i≥0

rational? That is, is there some recurrence on the size of the free modules in a minimal resolution of M?

Anick disproved this in 1982 by finding a counterexample in algebraic topology whose M Poincar´e-Bettiseries is irrational. However, there are still interesting cases when PS (t) is rational.

1 If S = k[x1, . . . , xn]/(f1, . . . , fc) for a regular sequence (f1, . . . , fc) – see Section 5.3 – M then PS (t) is rational. S is called a complete intersection.

k 2 (Berglund) If S = R/I with I a monomial ideal, then PS (t) is rational.

3 Let S = k[x, y]/(x2, xy) and consider

 x y −y 0 0  x y −y 0 0 x x 0 ( x y ) ( ) 0 ← k ← S ←−−− S2 ←−−−−−−0 0 x S3 ←−−−−−−−−0 0 0 0 x S5 ← S8 ← · · ·

This is a minimal resolution of k and we can see that the ith map relates the (i − 1)st map in a recursive way. Interestingly, the ranks of the free modules are the Fibonacci numbers (starting at the second 1). Then the Poincar´e-Bettiseries is rational: 1 P k(t) = 1 + 2t + 3t2 + 5t3 + 8t4 + ... = . S 1 − t + t2 √ k n −1 Amazingly, PS (t) is an analytic function with radius of convergence (lim sup βn) . This gives a nice characterization of the growth of the Betti numbers βi.

100 5.3 Regular Local Rings 5 Ring Homology

√ n Definition. The curvature of an R-module M is curv(M) = lim sup βn, where βi are the Betti numbers for M. A powerful characterization theorem using curvature is

Theorem 5.2.13. Let S = k[x1, . . . , xn]/I. Then (1) curv k = 0 ⇐⇒ S is a regular ring.

(2) curv k = 1 ⇐⇒ S is a complete intersection.

5.3 Regular Local Rings

Let R be a commutative noetherian ring, p ⊂ R a prime ideal, Rp the localization at p and ∼ kp = Frac(R/p) = Rp/pRp the residue field at p.

Theorem 5.3.1. Rp is a local ring with unique maximal ideal pRp. Moreover, Rp is a flat R-module.

Corollary 5.3.2. Rp ⊗R − sends projectives to projectives. Conversely, if M is an R-module such that Mp := Rp ⊗R M is projective for all prime ideals p ⊂ R, then M is projective. Theorem 5.3.3. For a commutative noetherian ring R,

gldim(R) = sup{gldim(Rp) | p ⊂ R is a prime ideal}.

Proof. Follows from Theorem 5.2.5.

Definition. For f1, . . . , fc in R a commutative ring, (f1, . . . , fc) is a regular sequence if f1 is a non-zero divisor in R and for each 1 ≤ i ≤ c − 1, fi is a non-zero divisor in R/(f1, . . . , fi−1)R. Let (R, m, k) be a local noetherian ring with maximal ideal m and residue field k = R/m. Then the following types of dimensions are defined for R: ˆ (Krull dimension) dim R = ht(m); ˆ n (Depth) depth(R) = sup{n ≥ 0 | ExtR(R,R) 6= 0};

ˆ d(R) = max{n ≥ 0 | there exists a regular sequence (x1, . . . , xn)}; ˆ 2 1 (Embedding dimension) e dim(R) = dimk(m/m ) = dimk ExtR(k, k); ˆ (Global dimension) gldim(R) = sup{pd(M) | M ∈ R-Mod}; ˆ pd(k), the projective dimension of the residue field. Lemma 5.3.4. If (R, m, k) is a regular local ring, then depth(R) = d(R). Recall that R is said to be regular if dim R = e dim(R). An open question for the first half of the 20th century was

101 5.3 Regular Local Rings 5 Ring Homology

Question. If R is regular and p is a prime ideal of R, is Rp regular? Serre and Auslander-Buchsbaum independently proved the following theorem, answering the question above. Theorem 5.3.5 (Auslander-Buchsbaum-Serre). For (R, m, k) a local ring, the following are equivalent. (1) R is a regular local ring.

(2) pdR(M) < ∞ for all R-modules M, that is, gldim(R) < ∞.

(3) pdR(k) < ∞. (4) The Koszul complex (see Section 5.4) on a minimal generating set of m is acyclic. In this case, dim R = d(R) = e dim(R). Definition. A regular local ring R is Cohen-Macauley if dim R = d(R). Corollary 5.3.6. Every regular ring is Cohen-Macauley. Corollary 5.3.7. An arbitrary noetherian ring R has finite global dimension if and only if Rp is regular for all prime ideals p ⊂ R. Below, we give the proof of (1) ⇐⇒ (2) in Theorem 5.3.5. First, we need:

Lemma 5.3.8. Let x ∈ mrm2 be a nonzero divisor and consider the ring S = R/(x). Then dim S = dim R − 1 d(S) = d(R) − 1 e dim(S) = e dim(R) − 1 gldim(S) = gldim(R) − 1. In particular, R is regular if and only if S is regular. Proposition 5.3.9. R is a regular local ring if and only if gldim(R) < ∞.

Proof. If d(R) > 0, then there exists some x ∈ m r m2 which is a nonzero divisor. Hence by Lemma 5.3.8, R is regular if and only if S = R/(x) is regular which by induction is equivalent to gldim(S) < ∞ and thus also equivalent, again by Lemma 5.3.8, to gldim(R) < ∞. Further, this also shows dim R = d(R) = e dim(R) in the case that d(R) > 0. Now assume d(R) = 0. We will show that R is either a field or gldim(R) = ∞, which proves the proposition. Since d(R) = 0, there exist s ∈ R such that m = AnnR(s), where AnnR(s) = {r ∈ R | rs = 0} is the annihilator of s. Assume A is an R-module with 0 < pd(A) < ∞. By induction on the syzygies in a projective resolution of A, we may assume pd(A) = 1. Then there is a free resolution 0 → Rm → Rn → A → 0 for some m, n ∈ N. Assume Rm ⊆ mn ⊆ Rn. Then sRm ⊆ smRn = 0, but Rm is free so this implies s = 0, i.e. m = R. This is only possible if R is a field. Thus if s 6= 0, then gldim(R) = ∞.

102 5.4 Differential Graded Algebras 5 Ring Homology

Example 5.3.10. Regularity (dim R = e dim(R)) should be regarded as a sort of ‘smooth- ness’ condition on our ring. To be precise, it is a smoothness condition on the algebraic variety described by R. For example, consider the ring R = R[x, y]/(xy). The equation xy = 0 describes the union of the x- and y-axes in the xy-plane. y

(0, 0) x (1, 0)

The ideal (x, y) ⊂ R corresponds to the point (0, 0) at the intersection of the axes, and the ring R/(x, y) = R[x, y]/(x2, y2, xy) has embedding dimension 2, but Krull dimension 1. Therefore it is not regular (this reflects the fact that the union of the two axes is not smooth at the origin). On the other hand, the point (1, 0) corresponds to the ideal (x − 1, y) and the quotient R/(x−1, y) = R[x, y]/((x−1)2, y2, (x−1)y, xy) has Krull dimension and embedding dimension 1, so it is regular.

5.4 Differential Graded Algebras

Let A• be a complex of R-modules, where R is a commutative ring. Write

∂n ∂n−1 ∂2 ∂1 · · · → An −→ An−1 −−−→· · · −→ A1 −→ A0 → 0.

Definition. A• is a differential graded algebra (often abbreviated as DG algebra) pro- vided there is a product on A• such that every pair of homogeneous elements a ∈ An and n b ∈ Am satisfy ∂(ab) = ∂(a)b + (−1) a∂(b). This should look very similar to the product rule in differential calculus, except in this case passing the differential to b costs us a factor of −1. One nice thing about DG algebras is that understanding the maps ∂n on the generators of An means we know the action of the differential on the entire algebra.

Proposition 5.4.1. Z(A•) is a graded subring of A• and B(A•) is a homogeneous ideal of Z(A•). Therefore H(A•) = Z(A•)/B(A•) is a graded ring.

Proof. If a, b ∈ Z(A•) then

∂(ab) = ∂(a)b + (−1)|a|a∂(b) = 0 · b + (−1)|a|a · 0 = 0.

103 5.4 Differential Graded Algebras 5 Ring Homology

So ∂(ab) = 0. The rest of the subring axioms follow from the fact that ∂ is R-linear. To see 0 that B(A•) is an ideal, let a = ∂(a ) + b ∈ Z(A•). Then

ab = ∂(a0)b = ∂(a0b) − (−1)|a0|a0∂(b) = ∂(a0b).

Hence B(A•) is an ideal and therefore H(A•) is a graded ring. The most important example of a DG algebra is the Koszul complex. It is defined in terms of exterior algebras of a free module, so we take a moment to review these objects. Let R be a commutative ring, let F = Rc, a free R-module of rank c, and consider ^ M ^ ∗F = nF n≥0

where VnF is the nth exterior algebra of F , defined by

^n ⊗n F := F / Span{(v1 ⊗ · · · ⊗ vn) | vi = vj for some i 6= j}.

By convention, we set ^ ^ 0F = R and 1F = F.

Vn If {e1, . . . , ec} is a basis for F then we can form a basis for F by

{e1 ∧ · · · ∧ en | 1 ≤ i1 < ··· < in ≤ c}.

In particular, this shows that VkF is free. For any i, j ≥ 0, there is a linear map VjF ⊗VkF → Vj+kF that sends

(ei1 ∧ · · · ∧ eij ) ⊗ (e`1 ∧ · · · ∧ e`k ) 7−→ ei1 ∧ · · · ∧ eij ∧ e`1 ∧ · · · ∧ e`k .

The result isn’t commutative, but it is graded commutative (sometimes called skew commu- tative): if ω ∈ VjF and ω0 ∈ VkF then ω ⊗ ω0 7→ (−1)jk(ω ∧ ω0). V∗ Set deg e1 = 1 (where deg means homological degree). Then F is a graded commutative algebra on which we define

Definition. Let f1, . . . , fc be elements of R. The Koszul complex KR(f1, . . . , fc) is a V∗ c differential graded algebra formed by taking R as a graded algebra, defining ∂(ei) = fi for each i, and extending by the Leibniz rule:

1 ∂(ei1 ∧ · · · ∧ eik ) = ∂(ei1 ) ∧ ei2 ∧ · · · ∧ eik + (−1) ∧ ∂(ei2 ∧ · · · ∧ eik ) k X j−1 = (−1) ei1 ∧ · · · ∧ eij−1 ∧ ∂(eij ) ∧ eij+1 ∧ · · · ∧ eik j=1 k X = f (−1)j−1e ∧ · · · ∧ e ∧ · · · ∧ e . ij i1 cij ik j=1

104 5.4 Differential Graded Algebras 5 Ring Homology

Example 5.4.2. Let R = k[x, y, z, w] for k a field. The Koszul complex for the sequence (x, y, z, w) is shown below, with bases listed below each free module.

 z w 0 0  −y 0 w 0  −y −z −w 0 0 0   0 −y −z 0    −w ! x 0 0 −z −w 0  x 0 0 w      z 0 x 0 y 0 −w 0 x 0 −z −y ( x y z w ) 0 0 x 0 y z 0 0 x y 0 ←− R ←−−−−−− R4 ←−−−−−−−−−−−−−−− R6 ←−−−−−−−−−− R4 ←−−−−x R ←−−−−−− 0

1 e1 e1 ∧ e2 e1 ∧ e2 ∧ e3 e1 ∧ e2 ∧ e3 ∧ e4 e2 e1 ∧ e3 e1 ∧ e2 ∧ e4 e3 e1 ∧ e4 e1 ∧ e3 ∧ e4 e4 e2 ∧ e3 e2 ∧ e3 ∧ e4 e2 ∧ e4 e3 ∧ e4

One can prove that in this example K = KR(x, y, z, w) is acyclic, i.e. all the homologies past H1(K) are trivial. In general, the homology of the Koszul complex measures how many ‘interesting relations’ there are among f1, . . . , fc. In Example 5.4.2, there are only relations in the first homological degree, which correspond to the commutative relations between x, y, z and w. One can similarly define the Koszul complex of an R-module M by

KM (f1, . . . , fc) := KR(f1, . . . , fc) ⊗R M. We have the following characterization theorem.

Theorem 5.4.3. For R a local ring, KR(f1, . . . , fc) is acyclic ⇐⇒ (f1, . . . , fc) is a regular sequence. Example 5.4.4. Let R = k[x, y] and consider the sequence (f, g) = (x2, xy). The Koszul complex for this sequence is simpler than in Example 5.4.2:  −xy  ( x2 xy ) 2 0 ←− R ←−−−− R2 ←−−−−x R ←− 0. −y −y Notice that ∈ ker ∂ but it is not in im ∂ since im ∂ is generated by x . This x 1 2 2 x shows that K = KR(f, g) is not acyclic – the complex is not exact in homological degree 1. Let z = −ye1 + xe2 ∈ Z1(K). Then [z] is a nontrivial class in H1(K), but notice that x[z] = [xz] = 0 because xz ∈ B1(K). This shows that H1(K) is generated by [z]. Example 5.4.5. Let’s examine another situation where the Koszul complex is not acyclic by taking R not to be the whole polynomial ring. For instance, let R = k[x, y]/(x2, xy)– this is actually an example of a Golod ring called the Emmy ring. Ths Koszul complex for the sequence (x, y) looks like

−y ( x y ) ( ) 0 ←− R ←−−− R2 ←−−−x R ←− 0. x 0 Then = [z ] = [xe ] 6= 0 and similarly = [z ] = [ye ] 6= 0. This shows that 0 1 1 y 2 2 H0(K),H1(K) and H2(K) are all nontrivial.

105 5.4 Differential Graded Algebras 5 Ring Homology

A useful object in the study of Koszul complexes is the mapping cone (a term borrowed from algebraic topology, along with many others in homological algebra).

Definition. Given f : A• → B• a chain map of complexes, the mapping cone of f in homological degree n is cone(f)n = An−1 ⊕ Bn. These form a complex cone(f) whose differential ∂cone(f) is given by

 A  cone(f) −∂n−1 0 ∂n = B . −fn−1 ∂n Proposition 5.4.6. (cone(f), ∂cone(f)) is a complex. Proof. It suffices to verify that ∂2 = 0. Note that −∂A 0  −∂A 0   (∂A)2 0  = . −f ∂B −f ∂B f∂A − ∂Bf (∂B)2

A 2 B 2 A B Since A• and B• are complexes, (∂ ) = 0 and (∂ ) = 0, and f∂ − ∂ f = 0 follows from the definition of f as a chain map.

Notice that B• embeds as a subcomplex of cone(f) via the exact sequence

0 → B• → cone(f) → A•(−1) → 0

where A•(−1) is the homological shift of A• by −1. This also associates A• with a quotient of cone(f). Let’s compute the homology of the mapping cone of f : A• → B•. Consider the commu- tative diagram with exact rows

0 Bn An−1 ⊕ Bn An−1 0

B cone(f) A ∂n ∂n −∂n−1

0 Bn−1 An−2 ⊕ Bn−1 An−2 0

By a diagram chase, the connecting map is defined by −f : A• → B•, as seen in the long exact sequence below

−H(f) · · · → Hn(B•) → Hn(cone(f)) → Hn−1(A•) −−−−→ −H(f) −−−−→ Hn−1(B•) → Hn−1(cone(f)) → Hn−2(A•) → · · ·

This is useful for taking a chain map f : A• → B• and making it the connecting homomor- phism in a long exact sequence.

Example 5.4.7. For a commutative ring R, take x ∈ R. If A• is a complex of R-modules x then θx : A• −→ A•, i.e. multiplication by x, is a chain map. Then H(θx) describes the action of x on the homologies Hn(A•) and this action is controlled by H(cone(θx)).

106 5.4 Differential Graded Algebras 5 Ring Homology

Example 5.4.8. Let R = k[x, y, z] and consider the Koszul complex on x:

(x) K(x) : 0 ← R ←− R ← 0.

y Let θy : K(x) −→ K(x) be multiplication by y. Then cone(θy) is

 −x  ( −y x ) −y 0 ← R ←−−−− R ⊕ R ←−−− R ← 0.

Up to a change of basis, this is (isomorphic to) the Koszul complex K(x, y). The homologies are computed to be H0(K(x)) = R/(x) and Hi(K(x)) = 0 for i ≥ 1. The long exact sequence for cone(θy) is

0 → H2(K(x)) → H2(K(x)) → H2(cone(θy))

→ H1(K(x)) → H1(K(x)) → H1(cone(θy))

→ H0(K(x)) → H0(K(x)) → H0(cone(θy)) → 0. ∼ By the isomorphism above, Hi(cone(θy)) = Hi(K(x, y)) for all i, so we can plug this in and the appropriate values for the homology of the first Koszul complex to obtain

0 → 0 → 0 → H2(K(x, y)))

→ 0 → 0 → H1(K(x, y))

→ R/(x) → R/(x) → H0(K(x, y)) → 0.

This shows that H0(K(x, y)) = R/(x, y) since the connecting map is θy (well really it’s −θy but up to relabelling a basis this is the same), and Hi(K(x, y)) = 0 for i ≥ 1. We can go even further: multiplication by z

z θz : K(x, y) −→ K(x, y)

has the following mapping cone

" # y −x 0  x  −z 0 −x y ( −z −y x ) 0 −z −y 0 ← R ←−−−−−− R ⊕ (R ⊕ R) ←−−−−−−−− (R ⊕ R) ⊕ R ←−−−−−z R ← 0.

Notice the partitioning on these matrices corresponds to the differential on K(x, y) and the mapping cone for θz. This complex is clearly isomorphic to K(x, y, z). This procedure generalizes to prove the backward direction of

Theorem 5.4.9. For R a local ring, KR(f1, . . . , fc) is acyclic ⇐⇒ (f1, . . . , fc) is a regular sequence.

In fact, this direction holds for any ring R. The forward direction is the one that requires R to be local. The description above shows that the Koszul complex may be thought of as an iterated mapping cone. It can also be viewed as a tensor product.

107 5.4 Differential Graded Algebras 5 Ring Homology

Definition. Given complexes A• and B•, we define their tensor product in degree n by M (A• ⊗ B•)n := Ai ⊗ Bj. i+j=n

These pieces together form a tensor product of complexes, A• ⊗ B•, which is itself a complex with differential

A•⊗B• A i B ∂n (ai ⊗ bj) = ∂n (ai) ⊗ bj + (−1) ai ⊗ ∂n (bj) for ai ∈ Ai, bj ∈ Bj and i + j = n.

The differential is defined in this specific way in order to make A• ⊗ B• into a DG algebra. In the next example we will see how to tensor Koszul complexes and form larger Koszul complexes.

Example 5.4.10. Consider the Koszul complexes

−y ( x y ) ( ) 0 ←− R ←−−− R2 ←−−−x R ←− 0 −w ( z w ) ( ) 0 ←− R ←−−− R2 ←−−−z R ←− 0 M By definition, (K(x, y) ⊗ K(z, w))n = K(x, y)i ⊗ K(z, w)j. This allows us to compute i+j=n the tensored complex

 x y 0 0  w 0 −y 0  −w x 0 y 0 0     −z 0 0 −y  z 0 x 0 y 0   −y !    0 w x 0  x 0 −z −w 0 0 y 0 −z 0 x −w ( z w x y ) 0 ← R ←−−−−−− R4 ←−−−−−−−−−−−−−−−0 0 0 −z −w −x R6 ←−−−−−−−−−−0 0 −z −w R4 ←−−−−z R ← 0.

This is precisely the Koszul complex K(z, w, x, y) which, after reordering, becomes K(x, y, z, w).

The above shows how to tensor two complexes. We may also wish to apply Hom to two complexes and form a new complex.

Definition. For complexes C• and D• of R-modules, the degree n piece of their Hom complex is defined as M (Hom(C•,D•))n := HomR(Ci,Di+n). i∈Z

These form the Hom complex Hom(C•,D•) whose differential is defined on families (fi) of maps fi : C• → D• by D n C ∂((fi)) = ∂i+nfi − (−1) fi−1∂i .

Hom(C•,D•) consists of families of maps in various degrees. For example, the diagonal arrows below show maps in homological degree 2.

108 5.4 Differential Graded Algebras 5 Ring Homology

··· C2 C1 C0 0

··· D3 D2 D1 D0 0

An element f ∈ Hom(C•,D•) satisfies ∂(f) = 0 (i.e. f is a cycle) if and only if f defines a chain map f : C• → D(−n)•, where D(−n)• is a shift of D• with a sign change:

D(−n) n D D[−n]k = Dk−n and ∂k = (−1) ∂k−n.

Thus the cycles in Hom(C•,D•) are precisely these chain maps. On the other hand, f ∈ D |g| C Hom(C•,D•) is a boundary ⇐⇒ f = ∂ g − (−1) g∂ for some g. Thus Bn(Hom(C•,D•)) consists of the null-homotopic chain maps. This shows that Hn(Hom(C•,D•)) is the homo- topy equivalence classes of degree n chain maps from C• to D•.

109 6 Spectral Sequences

6 Spectral Sequences

6.1 Bicomplexes and Exact Couples

The discussion in Section 5.4 leads naturally into the study of spectral sequences. Given complexes C• and D•, we often want to compute Hn(C• ⊗ D•). This requires one to view C• ⊗ D• as a bicomplex. We need a few definitions first.

Definition. A bigraded module is a family of R modules M = (Mp,q) where p and q each run over the integers.

Definition. For M and N bigraded R-modules, a bigraded map of degree (i, j) is a family of homomorphisms (fp,q : Mp,q → Np+i,q+j), denoted f : M → N. The class of bigraded R-modules form a category. One must be careful in defining submodules and quotients in this category.

0 0 Definition. For M = (Mp,q) a bigraded R-module, M = (Mp,q) is a bigraded submodule 0 2 0 of M if M is bigraded and for all (p, q) ∈ Z , Mp,q is a submodule of Mp,q. The bigraded 0 0 quotient M/M is the bigraded module of quotient modules (Mp,q/Mp,q). Recall that a complex is simply a graded module equipped with a differential. We define the analog for bimodules below.

Definition. A bicomplex is a graded R-module M = (Mp,q) equipped with two differentials ∂h, ∂v : M → M (called the horizontal differential and the vertical differential) satisfying

(i) (∂h)2 = 0 and (∂v)2 = 0, i.e. the rows and columns are differentials.

h v h (ii) ∂ has degree (−1, 0) and ∂ has degree (0, −1), that is, ∂ : Mp,q → Mp−1,q and v ∂ : Mp,q → Mp,q−1.

h v v h (iii) ∂p,q−1∂p,q + ∂p−1,q∂p,q = 0.

Definition. For M = (Mp,q) a bicomplex, its total complex Tot(M) is the complex defined in the nth homological dimension by M Tot(M)n := Mp,q p+q=n with differential ∂Tot(M) defined by

Tot(M) X h X p v ∂n = ∂p,q + (−1) ∂p,q. p+q=n p+q=n

Example 6.1.1. For the bicomplex C•⊗D• from Section 5.4, we can visualize the differential ∂C⊗D in the following diagram.

110 6.1 Bicomplexes and Exact Couples 6 Spectral Sequences

C ∂p ⊗ idD Cp−1 ⊗ Dq Cp ⊗ Dq

D D idC ⊗ ∂q idC ⊗ ∂q

Cp−1 ⊗ Dq−1 Cp ⊗ Dq−1 C ∂p ⊗ idD

Example 6.1.2. For K(x, y) ⊗ K(z, w), we computed their total complex to be 0 0 0

0 R R2 R 0

0 R2 R4 R2 0

0 R R2 R 0

0 0 0

Definition. A bicomplex M = (Mp,q) is concentrated in the first quadrant if Mp,q = 0 whenever p < 0 or q < 0.

Lemma 6.1.3 (Acyclic Assembly). For a bicomplex M = (Mp,q) which is concentrated in the first quadrant and has exact rows (or exact columns), the total complex Tot(M) is acyclic. Pk Proof. Each c ∈ Tot(M)k can be written c = `=0 c` for c` ∈ Mk−`,`. Since M is con- v centrated in the first quadrant, ∂ c0 = 0 so by exactness, there is some b0 ∈ M0,k+1 such v h that ∂ b0 = c0. Then by commutativity, ∂ b0 = c1. Repeating the argument allows us to Pk+1 tot construct b = `=0 b` ∈ Tot(M)k+1 with ∂ b = c.

To generalize this, let C• be any chain complex in an abelian category admitting a filtration −1 0 1 2 · · · ⊆ F C• ⊆ F C• ⊆ F C• ⊆ F C• ⊆ · · · ⊆ C• k k k such that for each k, F C• is a subcomplex of C•. Write F = F C•. Then one way of k k−1 understanding the homology H(C•) is by computing H(F /F ) for each k and comparing these to the homology of the whole complex. Consider the short exact sequence of complexes 0 → F k−1 → F k → F k/F k−1 → 0. This induces a long exact sequence in homology

p−1 α p β p p−1 γ p−1 · · · → Hp+q(F ) −→ Hp+q(F ) −→ Hp+q(F /F ) −→ Hp+q−1(F ) → · · ·

111 6.1 Bicomplexes and Exact Couples 6 Spectral Sequences

p p p−1 Define Dp,q = Hp+q(F ) and Ep,q = Hp+q(F /F ). Then D•,• and E•,• are bicomplexes and there are maps α D•,• D•,•

γ β

E•,•

with bigraded degrees deg(α) = (1, −1), deg(β) = (0, 0) and deg(γ) = (−1, 0). It is a special property that this triangle is exact on each graded piece.

Definition. The data (D•,•,E•,•, α, β, γ) is called an exact couple. Theorem 6.1.4. Suppose (D, E, α, β, γ) is an exact couple such that βγ is a differential on E. Then there exist an exact couple α2 D2 D2

γ2 β2

E2 such that E2 = H(E, βγ). 2 2 Proof. First, let α = α|im α. By exactness of the original couple, (βγ) = βγβγ = β0γ = 0, so γ induces a map ker(βγ)/ im(βγ) → D whose image, again by exactness, lies in im α. Taking D2 = im α and E2 = H(E), we have defined γ2. 2 Next, note that αγ = αγ|ker(βγ) = 0 because αγ = 0 by exactness. Thus ker α ∩ im α = im γ2. For u ∈ im α, define β2(u) = βd where d is any element of α−1(u). If d0 ∈ α−1(u) is another preimage, then α(d − d0) = u − u = 0, so by exactness, d − d0 ∈ im γ. Thus β(d − d0) ∈ im(βγ), so β2(u) is well-defined up to elements of im(βγ). Definition. The couple (D2,E2, α2, β2, γ2) is called the derived couple of (D, E, α, β, γ). In general, the rth derived couple is (Dr,Er, αr, βr, γr), defined recursively. Note that the maps in the rth derived couple have graded degrees deg(αr) = (1, −1), deg(β) = (1 − r, r − 1) and deg(γ) = (−1, 0). k k For each k, write Φ = im(H•(F ) → H•(C•)). We are interested in comparing the groups L k k−1 k k−1 k≥1 H•(F /F ) and Φ H•(C•)/Φ H•(C•). (In other words, how well do homology and associtaed grading commute?) Assume that C• has bounded filtration, i.e. for each n ≥ 0, s t there exist s = s(n) and t = t(n) ∈ Z such that F Cn = 0 and F Cn = Cn. We may view the problem as an exact couple:

p−1 α p Hp+q(F ) Hp+q(F )

γ β

p p−q Hp+q(F /F )

112 6.2 Spectral Sequences 6 Spectral Sequences

We want to show that for large enough N, the map αN in the Nth derived couple of this r p−r+1 p couple has source 0 or target H(C•). Explicitly, Dp,q = im(Hn(F ) → Hn(F )) and

k k r Φ = Φ Hn(C•) = Dk+r−1,n−k−r+1 for r large enough. This gives a short exact sequence

r r r 0 → Dk+r−2,n−k−r+2 → Dk+r−1,n−k−r−1 → Ek,n−k → 0 r ∼ k k−1 and Ek,n−k = Φ /Φ , so it is evident that we will gain insight from computing these higher derived couples.

6.2 Spectral Sequences

Definition. Let (D,E) be an exact couple with derived couples (Dr,Er). The sequence (Er) r is called the spectral sequence for (D,E). We say (E ) converges to a collection (Hn) k r k k−1 of graded abelian groups with filtrations Φ Hn, written Ek,n−k ⇒ Φ Hn/Φ Hn, if ∞ ∞ ∞ \ [ k k−1 En,k−n := Zj/ Bj = Φ Hn/Φ Hn j=2 j=2 where Zj and Bj are the cycles and boundaries in the homology of E.

Our goal is to compute the homology of a total complex, H(Tot(E)), for a bicomplex E•,•. Spectral sequences are the preferred tool for approximating this homology via a filtration

H(Tot(E)) = F0 ⊇ F1 ⊇ F2 ⊇ F3 ⊇ · · · Let us now give an explicit description of the terms Er in the derived couples (Dr,Er) for this complex. We restrict our focus to first quadrant homological bicomplexes, which are bicomplexes that are positive with respect to each differential (meaning its homological di- mensions are all ≥ 0) and dimension decreases along each arrow (as opposed to cohomology). In the diagram below, we seek to compute the sums of the antidiagonals, which represent the pieces of H(Tot(E)).

ˆ + ˆ + ˆ + ˆ − + − + ˆ + ˆ + ˆ + ˆ − + − + ˆ + ˆ + ˆ + ˆ

0 To begin, we define the 0th page of the spectral sequence for Tot(E), denoted E•,•, to be the bicomplex consisting of only the vertical differential. The most useful way to visualize spectral sequences is by drawing a lattice of points (p, q), where each vertex is a graded piece of the bicomplex E•,•, and connecting the points with arrows to depict the differential. A spectral sequence can be thought of as a ‘book’ of such diagrams, and right now we are 0 defining each ‘page’ of the book. E•,• looks like

113 6.2 Spectral Sequences 6 Spectral Sequences

q

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ p

0 E•,•, the 0th page of the spectral sequence

0 E•,• can be thought of as the first approximation to H(Tot(E)). The 1st page of the spectral 1 0 1 sequence is next defined as E•,• := H(Ep,q), whereby the differential becomes ∂ : Ep,q → 1 Ep−1,q. This consists of precisely the horizontal differential on E•,•. q

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ p

1 E•,•, the 1st page of the spectral sequence

Continuing in this fashion, we define the rth page of the spectral sequence for E•,• to be r r−1 r r E•,• := H(Ep,q ) with differential ∂ : Ep,q → Ep−r,q+r−1. After the 0th and 1st pages, the 2 2nd page E•,• no longer has vertical or horizontal differential, but it still has a differential:

114 6.2 Spectral Sequences 6 Spectral Sequences

q

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ p

2 E•,•, the 2nd page of the spectral sequence

Notice that the vertex at (0, 0) is unchanged between the 1st and 2nd pages of the spectral sequence. As we compute more and more pages, there will be a well-defined and stable value ∞ for each vertex. We define the ∞ page of the spectral sequence as the bicomplex E•,• with ∞ r vertices Ep,q equal to the stable value of the Ep,q, if this exists. It turns out that for all first ∞ quadrant homological bicomplexes, Ep,q exists for every (p, q).

r Definition. We say a spectral sequence collapses at the rth page if the differential on E•,• ∞ r is zero. In this case, Ep,q = Ep,q for all (p, q).

Remember that our goal is to approximate Hn(Tot(E)) for any n.

r r Definition. A spectral sequence E•,• converges to Hp,q, written Ep,q =⇒ Hp,q, if there is a filtration of modules that stabilizes,

Hp,q(Tot(E)) = F0 ⊇ F1 ⊇ F2 ⊇ · · · ⊇ Fp+q+1 = Fp+q+2 = ··· = 0

∞ and such that Fi/Fi+1 = Ei,p+q−i.

Example 6.2.1. If H3(Tot(E)) = F0 ⊇ F1 ⊇ F2 ⊇ F3 ⊇ F4 = ··· = 0 where

∞ F0/F1 = E0,3 ∞ F1/F2 = E1,2 ∞ F2/F3 = E2,1 ∞ F3/F4 = E3,0 then this gives us exact sequences

∞ ∞ 0 → E3,0 → F2 → E2,1 → 0 ∞ 0 → F2 → F1 → E1,2 → 0 ∞ 0 → F1 → H3 → E0,3 → 0.

115 6.2 Spectral Sequences 6 Spectral Sequences

If we assume all exact sequences split (for example, in the Grothendieck group), we will have

∞ H3 = F1 + E0,3 ∞ ∞ = F2 + E1,2 + E0,3 ∞ ∞ ∞ ∞ = E3,0 + E2,1 + E1,2 + E0,3.

In particular, if the Fi are free (e.g. if they are vector spaces) then we compute the rank of H3 this way. Example 6.2.2. Suppose that the double complex E consists solely of the two columns p and p−1. Fix n and set q = n−p, so that an element of Hn(T ) is represented by an element (a, b) ∈ Ep−1,q+1 × Ep,1. Show that we have calculated the homology of T = Tot(E) up to extension in the sense that there is a short exact sequence

2 2 0 → Ep−1,q+1 → Hp+q(T ) → Ep,q → 0.

2 Proof. Consider the Ep,q page of the spectral sequence. ˆ ˆ

q + 1 ˆ ˆ E2 q ˆ ˆ p,q

q − 1 ˆ ˆ Hp+q

ˆ ˆ

p − 1 p

Since there are only two columns, all the differentials are 0 on the 2nd page. Therefore the 2 ∞ spectral sequence collapses at this page, i.e. Ep,q = Ep,q. Let T = Tot(E). Then Hp+q(T ) has a filtration 0 = F0 ⊆ F1 ⊆ F2 = Hp+q(T ) ∞ ∞ whose intermediate quotients are given by F1/F0 = F1 = Ep−1,q−1 and F2/F1 = Ep,q. Since each module on E∞ is equal to its counterpart on E2, we can put this together into an exact sequence 2 2 0 → Ep−1,q+1 → Hp+q(T ) → Ep,q → 0.

0 Proposition 6.2.3. For a first quadrant spectral sequence Ep,q (that is, Ep,q = 0 unless 2 p, q ≥ 0), E0,0 = H0(T ) and there is an exact sequence

2 d 2 2 H2(T ) → E2,0 −→ E0,1 → H1(T ) → E1,0 → 0.

116 6.2 Spectral Sequences 6 Spectral Sequences

Proof. Consider the filtrations

0 0 0 H2 = F0 ⊇ F1 ⊇ F2 ⊇ 0

H1 = F0 ⊇ F1 ⊇ F2 ⊇ 0

0 0 ∞ 0 0 ∞ 0 0 ∞ ∞ ∞ where F0/F1 = E2,0, F1/F2 = E1,1, F2/F3 = E0,2, F1/F0 = E1,0 and F1/F2 = F1 = E0,1. On 3 the Ep,q page, the arrows are hard to compute but the vertices stabilize. In particular we have

∞ 3 E2,0 = E2,0 = ker d/(0) = ker d ∞ 3 2 E0,1 = E0,1 = E0,1/ im d = coker d.

2 ∞ 2 We also have stabilization on the Ep,q page at the (1, 0) vertex: E1,0 = E1,0. This shows that ∞ ∞ 2 H1/E0,1 = E1,0 = E1,0 so we can start building the exact sequence from the right:

∞ 2 0 → E0,1 → H1 → E1,0 → 0.

∞ We can splice in the short exact sequence induced by d, since coker d = E0,1 from above, which gives us the desired sequence:

2 d 2 2 H2 E2,0 E0,1 H1 E1,0 0

∞ ∞ E2,0 E0,1

0 0 0 0

Proposition 6.2.4 (Two rows). Suppose that a spectral sequence converging to H∗ has 2 Ep,q = 0 unless q = 0, 1. Then there is a long exact sequence

2 d 2 2 d 2 ··· Hp+1 → Ep+1,0 −→ Ep−1,1 → Hp → Ep,0 −→ Ep−2,1 → Hp−1 → · · · .

r Proof. Since Ep,q =⇒ Hp+q, there is a filtration

0 = ··· = Fr ⊆ · · · ⊆ F3 ⊆ F2 ⊆ F1 ⊆ F0 = Hp+q

∞ 2 such that Fi/Fi+1 = Ei,p+q−i. The E•,• page of the spectral sequence is just the bottom two rows: q

ˆ ˆ ˆ ˆ ˆ ˆ

∂1 ∂2 ∂3 ∂4 ˆ ˆ ˆ ˆ ˆ ˆ p

117 6.2 Spectral Sequences 6 Spectral Sequences

so we see that the vertices E0,0 and E1,0 have already stabilized by the 2nd page. Even better, on the 3rd page all differentials are 0 (they run past the 2nd row) so the spectral sequence converges at r = 3 and we can fill in the modules using the differential rule for passing from E2 to E3:

3 E•,•

2 2 2 2 E0,1/ im ∂1 E1,1/ im ∂2 E2,1/ im ∂3 E3,1/ im ∂4

···

2 2 E0,0 E1,0 ker ∂1 ker ∂2

(The dashed arrows show the anti-diagonals along which we will compute homology.) We start building our long exact sequence with

2 2 E0,1 → H1 → E1,0 → 0 by the definition of H1. Then we also have

2 0 → E1,1/ im ∂2 → H2 → ker ∂1 → 0

2 so to work this into our sequence, we add in the kernel of E0,1 → H1, which is im ∂1, and tack on the ∂1 arrow:

2 ∂1 2 2 E2,0 E0,1 H1 E1,0 0

im ∂1

0 0

Now we are able to splice in the short exact sequence for H2 from above: 0 0

ker ∂1

2 ∂1 2 2 H2 E2,0 E0,1 H1 E1,0 0

im ∂1

0 0

Continuing in this way, we have

118 6.3 Applications of Spectral Sequences 6 Spectral Sequences

0 0 0 0

ker ∂2 ker ∂1

2 ∂2 2 2 ∂1 2 2 ··· H3 E3,0 E1,1 H2 E2,0 E0,1 H1 E1,0 0

im ∂2 im ∂1

0 0 0 0

The pattern continues in this way, showing that we have an exact sequence

2 ∂p 2 2 ∂p−1 2 · · · → Hp+1 → Ep+1,0 −→ Ep−1,1 → Hp → Ep,0 −−→ Ep−2,1 → Hp−1 → · · ·

In general, we have:

Theorem 6.2.5. If E•,• is a bounded bicomplex, then the spectral sequence (Hp+q(Tot(E))) converges to H(E).

6.3 Applications of Spectral Sequences

Example 6.3.1. Let R be a ring, M and N a pair of R-modules, P• → M and Q• → N projective resolutions and consider the bicomplex P• ⊗R Q•. There are two natural filtrations of Tot(P• ⊗R Q•) to consider, the ‘vertical’ and the ‘horizontal’ filtration:

k k v k M h k M F := (Pi ⊗R Q•) and F := (P• ⊗R Qj). i=0 j=0

Computing the spectral sequences for each filtration yields: ( v 1 M ⊗R Qp, q = 0 v 2 R E = E = Hp(M ⊗R Qp) = Tor (M,N) p,q 0, q > 0 p,q p ( h 1 Pp ⊗R N, q = 0 h 2 R E = E = Hp(P• ⊗R N) = Tor (M,N). p,q 0, q > 0 p,q p

Therefore each spectral sequence converges and we have

R ∼ ∼ ∼ Torp (M,N) = Hp(P• ⊗R N) = Hp(M ⊗R Q•) = Hp(Tot(P• ⊗R Q•))

for any projective resolutions P• → M and Q• → N. We next give a generalization, using spectral sequences, of the universal coefficient theo- rem (4.5.1) that does not require R to be a PID.

119 6.3 Applications of Spectral Sequences 6 Spectral Sequences

Theorem 6.3.2 (Universal Coefficients). Let C• be a flat chain complex of R-modules and fix an R-module M. Then for all n ∈ Z, there is a natural isomorphism ∼ R Hn(M ⊗R C•) = M ⊗ Hn(C•) ⊕ Tor1 (M,Hn−1(C•)).

Proof. Let P• → M be a projective resolution and consider the bicomplex P• ⊗R C•. In the v 1 R spectral sequence for the vertical filtration of Tot(P• ⊗R C•), we have Ep,q = Torq (M,Cp). Notice that since the Cp are all flat, Torq(M,Cp) = 0 for q > 0. Thus we are only left with a horizontal differential in the 0th row, so the spectral sequence collapses on the 2nd page, yielding ∼ H•(Tot(P• ⊗R C•)) = H•(M ⊗R C•). On the other hand, the first page of the spectral sequence for the horizontal filtration is h 1 Ep,q = Pq ⊗R Hp(C•) and by flatness again, there is only a vertical differential. Thus

h 2 R Ep,q = Torq (M,Hp(C•)) ⇒ Hp+q(M ⊗ C•). This proves the theorem.

Theorem 6.3.3 (K¨unnethFormula). If C• and D• are flat complexes of R-modules, then for all n ∈ Z there is an isomorphism ∼ M  R  Hn(Tot(C• ⊗R D•)) = Hp(C•) ⊗R Hq(D•) ⊕ Tor1 (Hp−1(C•),Hq(D•)) . p+q=n

Proof. The first two pages of the standard spectral sequence for Tot(C• ⊗R D•) are

1 2 Ep,q = Cp ⊗R Hq(D•) and Ep,q = Hp(Cp ⊗R Hq(D•)).

∞ R It follows that Ep,q = Hp(C•)⊗Hq(D•)⊕Tor1 (Hp−1(C•),Hq(D•)) by the universal coefficient theorem, and the spectral sequence converges to

∼ M ∞ Hn(Tot(C• ⊗R D•)) = Ep,q. p+q=n

The universal coefficient theorem and the K¨unnethformula are important results in topology, and both are actually special cases of a more general phenomenon.

Theorem 6.3.4 (Grothendieck Spectral Sequence). Let A −→BG −→CF be additive, right exact functors between abelian categories with enough projectives such that the left derived functors of F satisfy LiF ◦ G(P ) = 0 for all i > 0 and all projectives P ∈ A. Then for all M ∈ A, there is a spectral sequence

LpF ◦ LqG(M) ⇒ Lp+q(F ◦ G)(M).

Proof. Take a projective resolution P• → M in A. Applying G and taking compatible projective resolutions in B (e.g. using the mapping cone; see Section 5.4), we get a spectral sequence whose 0th page is

120 6.3 Applications of Spectral Sequences 6 Spectral Sequences

0 Ep,q C0,1 C1,1 C2,1

C0,0 C1,0 C2,0

GP0 GP1 GP2 ···

0 0 0

i Note that the homology in the vertical direction on this page is (L F ◦ G)(Pi) = (FG)Pi in degree 0. Thus we start by taking the spectral sequence for the vertical filtration of E0. Set Zi = ker(GPi → GPi−1) and Bi = im(GPi+1 → GPi). Then taking projective resolutions Zi,• → Zi and Bi,• → Bi which are compatible with each other and with Ci,• → GPi, we obtain the following on the second page of the sequence:

2 Ep,q 0 0 0

0 0 0

(FG)M L1(FG)M L2(FG)M

0 0 0

∼ i That is, the sequence collapsese on the second page, so Hi(Tot(C•,•)) = L (FG)M. On the other hand, if we compute the spectral sequence with respect to the horizontal filtration, the first page becomes

1 Ep,q FH0,1 FH1,1 FH2,1

FH0,0 FH1,0 FH2,0

(L0G)M (L1G)M (L2G)M

0 0 0

i for some projective resolution Hi,• → (L G)M. Taking the homology vertically and passing to the 2nd page, we have

121 6.3 Applications of Spectral Sequences 6 Spectral Sequences

2 2 2 1 2 2 Ep,q L F (GM) L F (L GM) L F (L GM)

L1F (GM) L1F (L1GM) L1F (L2GM)

(FG)M F (L1GM) F (L2GM)

0 0 0

∼ i • Therefore Hi(Tot(C•,•)) = L F (L GM), and comparing the two computations gives the result. Example 6.3.5. Let ϕ : Q → R be a homomorphism of commutative rings, let M be an R-module and let N be a Q-module. There is a spectral sequence (called a Cartan-Eilenberg spectral sequence) such that

2 R Q Q Ep,q = Torp (M, Torq (R,N)) =⇒ Torp+q(M,N). Take V a free resolution of M over R and W a free resolution of N over Q; these are denoted V −→' M and W −→' N, respectively. (Note that we could just as easily work with projective resolutions, but over a module category it does not matter.) Then V −→' M is a quasi-isomorphism and tensor respects quasi-isomorphisms, so V ⊗Q W → M ⊗Q W is also a quasi-isomorphism. Since this is a complex, we can compute its homology

Q Hn(M ⊗Q W ) = Torn (M,N). ∼ ∼ Note that V ⊗Q W = (V ⊗R R) ⊗Q W = V ⊗R (R ⊗Q W ) by associativity of tensor products. This allows us to define a bicomplex:

Ep,q := Vp ⊗R (R ⊗Q Wq).

The 0th page of the spectral sequence for Ep,q looks like

0 Ep,q V0 ⊗R (R ⊗Q W2) V1 ⊗R (R ⊗Q W2)

V0 ⊗R (R ⊗Q W1) V1 ⊗R (R ⊗Q W1)

V0 ⊗R (R ⊗Q W0) V1 ⊗R (R ⊗Q W0)

0 0

0 To compute the 1st page of this spectral sequence, we compute the homology of Ep,q. How- ever, since homology and tensor commute (R is projective, so R ⊗R − is an exact functor) we have

122 6.3 Applications of Spectral Sequences 6 Spectral Sequences

1 Ep,q 0 V0 ⊗R H2(R ⊗Q W ) V1 ⊗R H2(R ⊗Q W )

0 V0 ⊗R H1(R ⊗Q W ) V1 ⊗R H1(R ⊗Q W )

0 V0 ⊗R H0(R ⊗Q W ) V1 ⊗R H0(R ⊗Q W )

Q However these are really just Vp ⊗ Torq (R,N) by definition of Tor in terms of Hq. This allows us to write the 1st page as

1 Ep,q 0 V0 ⊗R Tor2(R,N) V1 ⊗R Tor2(R,N)

0 V0 ⊗R Tor1(R,N) V1 ⊗R Tor1(R,N)

0 V0 ⊗R Tor0(R,N) V1 ⊗R Tor0(R,N)

1 Next, computing the 2nd page involves taking the homology of the rows of Ep,q, each of which has the form 0 ←− V0 ⊗R Torq ←− V1 ⊗R Torq ←− · · · R but this is just Torp (M, Torq), so we have

2 R Q R Q R Q Ep,q Tor0 (M, Tor2 (R,N)) Tor1 (M, Tor2 (R,N)) Tor2 (M, Tor2 (R,N))

R Q R Q R Q Tor0 (M, Tor1 (R,N)) Tor1 (M, Tor1 (R,N)) Tor2 (M, Tor1 (R,N))

R Q R Q R Q Tor0 (M, Tor0 (R,N)) Tor1 (M, Tor0 (R,N)) Tor2 (M, Tor0 (R,N))

Unfortunately, further pages are rather difficult to compute since there’s no good interpre- 2 tation (e.g. in terms of Tor) of the homology of Ep,q. Example 6.3.6. Let Q be a ring and x ∈ Q a non-zero divisor. A common application of the above spectral sequence is to R = Q/(x) with R-modules M and N. We compute some

123 6.3 Applications of Spectral Sequences 6 Spectral Sequences

Tor modules to simplify the spectral sequence from Example 6.3.5:

 x  Tori(R,N) = Hi (0 → Q −→ Q → 0) ⊗Q N

 x  = Hi 0 → N −→ N → 0 ( N if i = 0, 1 = 0 otherwise.

2 This shows that all of the rows above the first two in the Ep,q page of the spectral sequence vanish, leaving

2 Ep,q

R R R R Tor0 (M,N) Tor1 (M,N) Tor2 (M,N) Tor3 (M,N)

R R R R Tor0 (M,N) Tor1 (M,N) Tor2 (M,N) Tor3 (M,N)

2 To compute some terms in the Ep,q page of the spectral sequence, one builds the long exact sequence

Q R R Q R ··· Tor2 (M,N) Tor2 (M,N) Tor0 (M,N) Tor1 (M,N) Tor1 (M,N) 0

3 3 E2,0 E0,1

0 0 0 0

This may be surprising, as the connecting map in this long exact sequence drops the homo- logical degree by two. In general, this sequence will continue with the following pattern:

R Q R · · · −→ Torn−1(M,N) −→ Torn (M,N) −→ Torn (M,N) R Q R −→ Torn−2(M,N) −→ Torn−1(M,N) −→ Torn−1(M,N) −→ · · ·

We conclude the discussion of spectral sequences with another application of the spectral sequence for ϕ : Q → R. This context is called a change of rings. In Example 6.3.6, we 2 were able to gain some information from the Ep,q page since all but the bottom two rows vanished. When Q is not a PID, that is, when R is not simply Q/(x) for a non-zero divisor x, things are more complicated.

124 6.3 Applications of Spectral Sequences 6 Spectral Sequences

Example 6.3.7. (Abramov) Suppose ϕ : Q → R is a surjective homomorphism of local rings Q and R, with respective maximal ideals mQ and mR and residue fields k = Q/n = R/m (this is equality is easy using the properties of local rings). If M is an R-module, there is a spectral sequence

2 h TorQ(R,k) Q i R Ep,q = Torp (Tor (M, k), k) =⇒ Torp+q(M, k). q

We will investigate this when Q = k[x1, . . . , xn] and R = Q/I for an ideal I ⊂ Q (R is isomorphic to such a quotient anyways since ϕ is surjective). To simplify the expression above, we want to express TorQ(R, k) as the homology of some minimal free resolution of k over Q. Luckily, we described such a MFR in Section 5.4: the Koszul complex KQ(x1, . . . , xn). We have Q Tor∗ (R, k) = H∗(R ⊗Q KQ(x1, . . . , xn)) but since tensoring with R gives a MFR of k over R, we have

H∗(R ⊗Q KQ(x1, . . . , xn)) = H∗(KR(x1, . . . , xn))

R which we denote by H∗(K ). This is a finite dimensional graded algebra. We can do the same thing for M: Q M Tor∗ (M, k) = H∗(K ) R but this is no longer an algebra – it is still a finite dimensional graded module over H∗(K ) though. In any case this allows us to write the spectral sequence as

2 h H(KR) M i R Ep,q = Torq (H(K ), k) =⇒ Torp+q(M, k). q This is not immediately helpful numerically, but a tool called the bar resolution can help us 2 make sense of Ep,q. The basic setup is to let A −→' R be a resolution of R over Q such that A is a DG algebra, Q and U −→' M be a resolution of M over Q such that U is a DG module over A. Also define Q ' B = A ⊗Q k and V = U ⊗Q k, so that V −→ B. Notice that when Q = k[x1, . . . , xn], Q

Q R H(B) = H(A ⊗Q k) = Tor (R, k) = H(K ) and H(V ) = TorQ(M, k) = H(KM ).

Then we define the bar resolution in the pth degree by

k ' Sp (H(B),H(V )) −→ H(V ). H(B)

1 This in turn allows us to describe the Ep,q page of the above spectral sequence:

E1 = Sk(H(B),H(V )) ⊗ k . p,q p H(B) q

125 6.3 Applications of Spectral Sequences 6 Spectral Sequences

k Note that we may instead write Sp in terms of a tensor product of graded vector spaces, k Sp (H(B),H(V )) = H(B) ⊗k H+(B) ⊗k · · · ⊗k H+(B) ⊗kH(V ). | {z } p To further describe this, we need some properties of Hilbert series, which we combine into a lemma: Lemma 6.3.8. If X and Y are graded k-vector spaces with Hilbert series Hilb(X) and Hilb(Y ) then Hilb(X ⊗k Y ) = Hilb(X) Hilb(Y ). What’s more, if T (V ) is the tensor algebra of V , T (V ) = k ⊕ V ⊕ (V ⊗ V ) ⊕ (V ⊗ V ⊗ V ) ⊕ ... 1 then Hilb(T (V )) = (this should look just like a geometric series). 1 − Hilb(V ) We want to apply this to the bar resolution; since tensor is commutative in our context, k we can tensor with k on either side of the vector space expansion of Sp to obtain k ∼ Sp (H(B),H(V )) ⊗H(B) k = H+(B) ⊗k · · · ⊗k H+(B) ⊗k H(V ). k  p Then by Lemma 6.3.8, Hilb(Sp (H(B),H(V )) ⊗H(B) k) = Hilb(H+(B)) Hilb(H(V )). Re- 2 Q call that we are trying to show the spectral sequence Ep,q converges to Torp+q(M, k). The Poincar´eseries for M over R is ∞ ∞ ! R X R n X X ∞ n PM (t) = dimk Torn (M, k)t = dim Ep,q t . n=0 n=0 p+q=n The beauty of spectral sequences (at least when we are dealing with graded vector spaces) i is that we know the dimension at each vertex Ep,q must stay the same or decrease. This implies ∞ ! ∞ ! X X ∞ n X X 1 n dim Ep,q t 4 dim Ep,q t n=0 p+q=n n=0 p+q=n where “4” denotes that each coefficient of the series on the left is less than or equal to the corresponding coefficient of the series on the right. Using the fact that Hilb H(B) = n Q Q Q (1 + t) = Pk (t) and Hilb H(V ) = Hilb H(U ⊗Q k) Hilb Tor (M, k) = PM (t), we write ∞ ! ! X X 1 n X X 1 q p dim Ep,q t = dim Ep,qt t n=0 p+q=n p q X = (Hilb(H(B)) − 1)p Hilb(H(V ))tp p X  Q p Q  p = (Pk (t) − 1) PM (t) t p Q X Q p p = PM (t) (Pk (t) − 1) t p

Q 1 = PM (t) Q by geometric series 1 − t(Pk (t) − 1) This bound is true for all R-modules M. We summarize our findings in the theorem below.

126 6.3 Applications of Spectral Sequences 6 Spectral Sequences

Theorem 6.3.9. Let Q = k[x1, . . . , xn] be a commutative local ring, where k = Q/mQ, and let R = Q/I for I a homogeneous ideal of Q. Then

P Q(t) (1 + t)n P R(t) k = . k 4 Q 1 − t(Hilb(H(KR)) − 1) 1 − t(PR (t) − 1) Definition. A commutative local ring R is called a Golod ring if equality holds:

(1 + t)n P R(t) = . k 1 − t(Hilb(H(KR)) − 1)

It is known that R is Golod ⇐⇒ the homology of the Koszul complex H(KR) admits a trivial Massey operation. A rather interesting open question is to prove the following conjecture.

Conjecture 6.3.10. R is Golod ⇐⇒ H(KR) is a trivial k-algebra.

127 7 Group Cohomology

7 Group Cohomology

7.1 G-Modules

Let G be a group and consider an abelian group A (or a vector space over a field k) on which G acts on the left. A is called a G-module, which can be thought of in several ways: ˆ As a group action, i.e. a function G × A −→ A (g, a) 7−→ g · a satisfying g · (h · a) = (gh) · a and 1 · a = a for all a ∈ A and g, h ∈ G. ˆ As a group homomorphism ρ : G → AutZ(A). This is the representation theoretic viewpoint.

ˆ As a module over the group algebra ZG (or kG for k-vector spaces). This is the preferred context for group cohomology. In general, one defines the group ring ( ) X RG = rgg : g ∈ G, rg ∈ R g∈G

for any ring R. The group algebras ZG and kG are special cases of this construction.

Group cohomology is the study of Tor and Ext over these group algebras. Let G-Mod be the category of left G-modules, which is equivalent to the category of left ZG-modules. There is an exact functor Ab −→ G-Mod A 7−→ A with the trivial G-action which is rather uninteresting. However, this admits a left and right adjoint: (−)G : G-Mod −→ Ab A 7−→ AG := {a ∈ A | g · a = a}

and (−)G : G-Mod −→ Ab

A 7−→ AG := A/hg − 1iA. The functor (−)G sends A to the set of G-invariants of A, that is, AG is the largest subspace of A which is fixed by G. Note that elements in AG have singleton orbits. On the other hand, the image of (−)G are the quotients AG = A/hg − 1iA which contain the G-coinvariants of A. Here the elements are actual orbits. One can see that these two functors are related in some way; the connection is made clear in the next proposition. G ∼ ∼ Proposition 7.1.1. For every G-module A, A = HomZG(Z,A) and AG = Z ⊗ZG A. In G other words, (−) and HomZG(Z, −) are naturally isomorphic functors, as are (−)G and Z ⊗ZG −.

128 7.1 G-Modules 7 Group Cohomology

Proof. Take ϕ ∈ HomZG(Z,A). Then ϕ is determined by where it sends 1. For any g ∈ G, note that the definition of ZG-linearity implies g · ϕ(1) = ϕ(g · 1) = ϕ(1).

In particular this shows that 1 is G-invariant, so ϕ(1) ∈ AG. Conversely, given a ∈ AG we define a map

ϕa : Z −→ A 1 7−→ a

which is well-defined using the abelian group (Z-module) structure on A. From this we construct a map

G HomZG(Z,A) −→ A ϕ 7−→ ϕ(1)

G which is an isomorphism of abelian groups. Hence (−) and HomZG(Z, −) are naturally isomorphic. Now consider

Ψ: Z × A −→ AG (m, a) 7−→ [ma]

where [ma] denotes the image of ma in the quotient A/hg − 1iA. For this to be biadditive, we must have Ψ(mg, a) = Ψ(m, ga), but

Ψ(m, ga) = [m(ga)] = m[ga] = m[a] = [ma] = Ψ(m, a) = Ψ(mg, a).

By the universal property of tensors, we get a map ψ : Z ⊗ZG A → AG which is clearly onto. To define an inverse, set

ϕ : A −→ Z ⊗ZG A a 7−→ 1 ⊗ a.

Then ϕ(ga−a) = 1⊗(ga−a) = (1⊗ga)−(1⊗a) = (1·g⊗a)−(1⊗a) = (1⊗a)−(1⊗a) = 0.

So ϕ is well-defined. By definition of AG as a quotient, we also get a map ϕ : AG → Z⊗ZG A.

It is easy to see that ψϕ = idAG and ϕψ = idZ⊗A, so they are inverses. We have thus proven that (−)G and Z ⊗ZG − are naturally isomorphic. Both invariants and coinvariants are a special case of a more general construction. Definition. Let ϕ : G → H be a group homomorphism. Then the inflation functor is defined by

G InfH : H-Mod −→ H-Mod

A 7−→ GA

where GA is the G-module given by g · a = ϕ(g)a.

129 7.2 Cohomology of Groups 7 Group Cohomology

Lemma 7.1.2. Inflation is an exact functor.

Proof. This is obvious, since exactness in G-ModandH−Mod is defined on the level of abelian groups. Therefore Inf admits both a left and a right adjoint:

Definition. For a group homomorphism ϕ : G → H, the induction functor is defined by

H IndG : G-Mod −→ H-Mod

N 7−→ ZH ⊗ZG N, and the coinduction functor is defined by

H CoindG : G-Mod −→ H-Mod N 7−→ HomG(ZH,N).

Example 7.1.3. There is a canonical map 1 → H for every group H, inducing Z → ZH. In this case, the inflation functor is just the forgetful functor sending M ∈ G-Mod H H to the underlying abelian group. Then Ind1 = ZH ⊗Z − and Coind1 = HomZ(ZH, −). Since coinduction is the right adjoint of inflation, we know from Theorem 3.2.10 that Coind preserves injectives. In particular, this gives another proof (see Proposition 2.3.9) that the ∗ dual group S = HomZ(ZH, Q/Z) is injective. Example 7.1.4. Let k be a field. Then any homomorphism k → A turns A into a k-algebra, A and Coindk = Homk(A, −). P Example 7.1.5. Let α : ZG → Z be the augmentation homomorphism g∈G ngg 7→ P G g∈G ng, corresponding to the trivial map G → 1. Then for any abelian group A, Inf1 (A) = A with the trivial G-action, while induction and coinduction recover the coinvariants and invariants of A:

1 ∼ IndG(A) = Z ⊗ZG A = AG 1 ∼ G CoindG(A) = HomZG(Z,A) = A .

7.2 Cohomology of Groups

Definition. For an abelian group A which is a left G-module, we define group homology and group cohomology by

ZG n n Hn(G; A) := Torn (Z,A) and H (G; A) := ExtZG(Z,A).

In the language of derived functors, H∗(G; A) are the left derived functors L∗(−)G(A) of the G-coinvariant functor; likewise H∗(G; A) are the right derived functors R∗(−)G(A) of 0 G the G-invariant functor. Note that H0(G; A) = AG and H (G; A) = A . One perspective of group cohomology is that it measures how far G is from being a finite n n group. In particular, we use H (G; Z) = ExtZG(Z, Z) to measure this property of G.

130 7.2 Cohomology of Groups 7 Group Cohomology

Example 7.2.1. For G = {1} and any G-module A, H0(G; A) = AG = A and likewise 0 G n H (G; A) = A = A. Further, the functor is exact in this case so Hn(G; A) = H (G; A) = 0 for all n ≥ 1.

The definition of Hn(G; A) in terms of Ext will require us to construct a resolution of Z as a ZG-module; a common resolution fulfilling this role is called the bar resolution, which was introduced in Section 6.3. Because we want such a resolution to work for any arbitrary group G, the bar resolution will be highly non-minimal. Example 7.2.2. (Infinite cyclic groups) Let G = hti be an infinite cyclic group. Then we have ZG = Z[t, t−1] which is a domain, and so the following is a resolution of Z over ZG:

t−1 0 → ZG −−→ ZG → Z → 0. t−1 This allows us to compute cohomology using the deleted resolution 0 → ZG −−→ ZG → 0. n In particular, we see that for n ≥ 2, Hn(G; A) = H (G; A) = 0. For n = 0, 1 we tensor the t−1 deleted resolution with A: (0 → ZG −−→ ZG → 0) ⊗ZG A which produces

0 → A −−→t−1 A → 0.

If we denote the middle arrow, which is multiplication by t − 1, by f then H0(G; A) = G coker f = A/ im f = AG and similarly H1(G; A) = ker f = {a ∈ A | (t − 1)a = 0} = A .

Now to compute the Tor modules, apply HomZG(−,A) to the same deleted resolution to produce the following: 0 → A −−→t−1 A → 0. 0 G 1 Once again, this shows that H (G; A) = ker f = A and H (G; A) = coker f = AG. Example 7.2.3. (Finite cyclic groups and the exact hexagon) Let G = hti where this time tn = 1 for some n. We construct a resolution of Z over ZG by

N t−1 N t−1 ··· −→ ZG −−→ ZG −→ ZG −−→ ZG → Z → 0 X where N = 1 + t + t2 + ... + tn−1 = g is called the norm element of G. In each step, g∈G r we extend Fk → Fk−1 by adding the map Fk+1 −→ Fk where r is a generator of Fk. We see that this procedure produces an infinite, 2-periodic resolution of Z as a left ZG-module. Therefore ∼ ∼ H2i−1(G; A) = H2j−1(G; A) and H2i(G; A) = H2j(G; A) for all integers i, j ≥ 1. Similarly, for cohomology we have

H2i−1(G; A) ∼= H2j−1(G; A) and H2i(G; A) ∼= H2j(G; A) for all i, j ≥ 1. In other words, the periodicity of the resolution of Z over ZG implies that we only have two (co)homologies past the 0th homological degree: an even and an odd (co)homology. Now given an exact sequence 0 → A0 → A → A00 → 0 of ZG-modules, the long exact sequence in cohomology can be written as an exact hexagon:

131 7.2 Cohomology of Groups 7 Group Cohomology

Hodd(G; A00) Hodd(G; A)

Heven(G; A0) Hodd(G; A0)

Heven(G; A) Heven(G; A00)

This is of particular importance in class field theory, where G = Gal(K/F ) is the Galois group of some cyclic extension K/F and A, A0,A00 are fractional ideal groups in the rings of integers of F and K. Let’s compute the homology and cohomology of finite groups. X Proposition 7.2.4. Let G be a finite group with |G| = m and set N = g to be the norm g∈G 1  1 
of G. Also let e = m N which is an element of Z m G ⊆ QG. Then (1) e is a central idempotent in QG.

(2) If A is a QG-module (or any G-module on which the action of m is invertible) then 0 ∼ H (G; A) = H0(G; A) = eA.

n (3) Given A as in (2), H (G; A) = Hn(G; A) = 0 for all n 6= 0. ! X X 2 1 Proof. (1) Since G acts on itself transitively, mN = N = N g = N . Set e = m N. g∈G g∈G 2 1 2 1 1 Then e = m2 N = m2 mN = m N = e so e is idempotent. We also have eg = e = ge for all g ∈ G so e is central as well. The above work shows that 1 − e is also a central idempotent, since e is central and

(1 − e)2 = (1 − e)(1 − e) = (1 − e) − (e − e2) = (1 − e) − (e − e) = 1 − e.

 1  Note that e+(1−e) = 1 so R = Z m G decomposes into the direct sum R = eR⊕(1−e)R. ∼  1   1  We want to show that eR = Z m since then Theorem 2.1.1 will imply that Z m is a  1  projective R-module. To prove this claim, we define a map eR → Z m in two parts and show it is an isomorphism. One direction is easy:

 1  eR ←− Z m ez 7−→ z.

X  1  P P Now let er ∈ eR. Write r = rgg where each rg ∈ Z m . Then er = e ( rgg) = rge = g∈G P  1  e ( rg) so we can define the forward map eR → Z m by er 7→ n where n is the unique  1  0 P element in Z m satisfying er = en. This is well-defined, since er = er =⇒ e ( rg) =

132 7.3 Some Results for the First Group Cohomology 7 Group Cohomology

P 0
0  1  e rg =⇒ rg = rg for all g ∈ G since R is a free Z m -module. The two maps we have defined are clearly inverses and preserve R-module structure on both sides, so we conclude ∼  1  that eR = Z m . G G (2) We must show that eA = A = AG. Clearly eA ⊆ A since G acts on itself transitively, so the sum in the definition of e doesn’t change under the G-action. On the other hand, we know that for any a ∈ AG,

1 X 1 X a = a = g · a = ea ∈ eA. m m g∈G g∈G

This proves the first equality. The equality for coinvariants is proven similarly. R (3) We proved above that eR is projective, so Torn (eR, A) = 0 for all n 6= 1. What’s  1  more, it’s easy to prove that R = Z m G is a flat ZG-module, so for n 6= 0, we have

ZG Hn(G; A) = Torn (Z,A) R = Torn (Z ⊗ZG R,A) (tensoring with R leaves A unchanged) R = Torn (RG,A) by Proposition 7.1.1 R = Torn (eR, A) by (2) = 0.

We can use a similar trick with Ext∗ to prove Hn(G; A) = 0 for all n 6= 0.

Example 7.2.5. Let Z/2Z act on C× = C r {0} by complex conjugation. Then we get a resolution × α × β × α × β C −→ C −→ C −→ C −→· · · z¯ where α(z) = z and β(z) = zz¯. Notice that

× 1 1 ker α = R , im α = S , ker β = S and im β = R>0.

Therefore Z/2Z has the following cohomology groups with coefficients in A = C×:

 × R , n = 0 n ×  H (Z/2Z; C ) = 0, n > 0 is odd  Z/2Z, n > 0 is even.

7.3 Some Results for the First Group Cohomology

0 In the last section we proved some easy results for H0(G; A) and H (G; A). Now we turn 1 our attention to H1 and H . Recall that for a group G, the commutator of G is denoted [G, G] (or sometimes G0).

ab Proposition 7.3.1. For any group G, H1(G; Z) = G/[G, G] = G .

133 7.3 Some Results for the First Group Cohomology 7 Group Cohomology

Proof. Let I be the ideal of ZG generated by elements g − 1, g 6= 1. Recall that the coin- variants of a G-module A are defined by AG = A/IA. Consider the short exact sequence

0 → I → ZG → Z → 0.

Applying H1 induces an exact sequence

H1(G; ZG) → H1(G; Z) → IG → (ZG)G → Z → 0.

Now since ZG is projective over itself, H1(G; ZG) = 0. Moreover, the right part of the ∼ ∼ sequence describes the isomorphism (ZG)G = Z, so we see that H1(G; Z) = IG. It remains ∼ to show that IG = G/[G, G], but this is done in Weibel. Definition. The set of derivations from G to A is

Der(G, A) := {D : G → A | D(gh) = gD(h) + D(g)}.

The subset of principal derivations is defined by

PDer(G, A) := {D : G → A | there exists a ∈ A such that D(g) = ga − a ∀g ∈ G}.

Der(G, A) is an abelian group under pointwise addition, and PDer(G, A) forms a subgroup under the same operation. In fact, it can be shown that PDer(G, A) ∼= A/AG. Even further, Der(G, −) is a functor:

G-Mod −→ Ab A −→ Der(G, A) a 7−→ (g 7→ ga − a).

Der(G, −) is naturally isomorphic to the functor HomG(I, −), where I is the ideal generated by {g − 1 | g ∈ G r {1}} defined above. Theorem 7.3.2. H1(G; A) ∼= Der(G, A)/ PDer(G, A).

Proof. Consider the exact sequence 0 → I → ZG → Z → 0 again. Applying HomG(−,A) induces a commutative diagram with exact rows (and equality vertically): Hom ( ,A) Hom ( G, A) Hom (I,A) 1 0 G Z G Z G ExtZG(Z,A) 0

0 AG A Der(G, A) H1(G; A)

An element a ∈ A gets sent to ϕ : g − 1 7→ (g − 1)a in Der(G, A) = HomG(I,A). From this i is easy to see ϕ is a principal derivation, so the image of A in Der(G, A) is PDer(G, A). Hence by exactness of the bottom row, H1(G; A) ∼= Der(G, A)/ PDer(G, A). As a special case of this theorem, we can state the well-known Hilbert’s Theorem 90 from his volume on number fields.

134 7.3 Some Results for the First Group Cohomology 7 Group Cohomology

Theorem 7.3.3 (Hilbert’s Theorem 90). If G = Gal(L/K) is the Galois group for L/K a finite, Galois extension of fields, then H1(G, L×) = 0 where L× is the invertible elements of L. 1 Here is another perspective on H (G; A). Let B0 = ZG and for n ≥ 1, let Bn be the free ZG-module on the set of all symbols [g1| · · · |gn] where gi ∈ G r {1}. Define the differential Pn i d : Bn → Bn−1 to be d = i=0(−1) di, where the di are defined in the following fashion:

d0([g1| · · · |gn]) = g1[g2| · · · |gn]; ( [g1| · · · |gigi+1| · · · |gn] if gigi+1 6= 1 di([g1| · · · |gn]) = for i = 1, . . . , n − 1; 0 if gigi+1 = 1

and dn([g1| · · · |gn]) = [g1| · · · |gn−1].

We adopt the convention that [g1| · · · |gn] = 0 if gi = 1 for any i. Let ε : ZG → Z be the augmentation map. Definition. The normalized bar resolution for a group G is the free resolution

d d d ε ··· −→ B2 −→ B1 −→ B0 −→ Z → 0.

Truncating the bar resolution at B0 and applying HomG(−,A) gives a cochain complex HomG(B•,A) with differential D : HomG(Bn,A) → HomG(Bn+1,A) given by Dϕ = ϕd. n It’s easy to check that the elements of HomG(Bn,A) are set maps ϕ : G → A such that ϕ(g1, . . . , gn) vanishes whenever gi = 1 for some i. These elements are called n-cochains, and the differential on these is given by

n−1 X (Dϕ)(g0, . . . , gn) = g0ϕ(g1, . . . , gn) + ϕ(g0, . . . , gigi+1, . . . , gn) + ϕ(g0, . . . , gn−1). i=1 The n-cochains satisfying Dϕ = 0 are called n-cocycles and they form a group denoted n by Z (G; A). The n-cochains of the form Dϕ for some ϕ ∈ HomG(Bn−1,A) are the n- coboundaries, which form a group Bn(G; A). Proposition 7.3.4. For all n ≥ 0 and any G-module A, Hn(G; A) ∼= Zn(G; A)/Bn(G; A). Example 7.3.5. For n = 1, we have {ψ : G → A | ψ(g g ) = ψ(g ) + g ψ(g )} H1(G; A) = 1 2 1 1 2 , {ψ : G → A | there exists a ∈ A such that ψ(g) = ga − a}

exactly as Theorem 7.3.2 tells us. Maps ψ : G → A such that ψ(g1g2) = ψ(g1) + g1ψ(g2) are sometimes called crossed homomorphisms (or derivations, as we saw). Example 7.3.6. For n = 2, there is also a description of cohomology: {ψ : G2 → A | g ψ(g , g ) + ψ(g , g g ) = ψ(g g , g ) + ψ(g , g )} H2(G; A) = 1 2 3 1 2 3 1 2 3 1 2 . {ψ | ψ(g1, g2) = g1ϕ(g2) − ϕ(g1g2) + ϕ(g1) for ϕ : G → A} These elements turn out to have very natural group-theoretic interpretations, as we will see in the next section.

135 7.4 Group Extensions 7 Group Cohomology

7.4 Group Extensions

Definition. Let G and E be groups, and let N be a normal subgroup of E.A group extension of G by N is a short exact sequence of groups

1 → N → E → G → 1.

Definition. An extension 0 → A → E → G → 1 where A is an abelian group is called a central extension of G by A.

Given a central extension 0 → A → E −→π G → 1, we get a G-module structure on A: if σ : G → E is a function (not necessarily a homomorphism) such that σ(1G) = 1E and −1 πσ = idG, then G acts on A by g · a = σ(g)aσ(g) . Proposition 7.4.1. A is a G-module under this action.

Proof. For g, h ∈ G and a ∈ A, h · (g · a) = σ(h)σ(g)aσ(g)−1σ(h)−1 and (hg) · a = σ(hg)aσ(hg)−1 but these are not equal at first glance since we did not assume σ to be a −1 homomorphism. However, if we apply π to σ(hg) σ(h)σ(g) and remember that πσ = idG,

πσ(hg)−1σ(h)σ(g)
= (πσ)(hg)−1(πσ)(h)(πσ)(g) = (hg)−1hg = 1

which is sufficient to imply equality above. Hence A is a G-module.

In general the condition that there exists a homomorphism σ such that πσ = idG defines a split extension in a similar sense as split sequences in Section 1.2. Split extensions of groups are important for the following reason.

Definition. We say that two extensions 1 → A → E1 → G → 1 and 1 → A → E2 → G → 1 are equivalent if there is an isomorphism ϕ : E1 → E2 making the following diagram commute:

1 A E1 G 1

ϕ

1 A E2 G 1

Proposition 7.4.2. A central extension 0 → A → E → G → 1 splits if and only if E is isomorphic to the semidirect product A o G. In general, group extensions arise from a generalization of semidirect products.

Definition. Given a group G and an abelian group A on which G acts, a function [·, ·]: G × G → A satisfying

ˆ [g, 1] = 0 and [1, g] = 0 for all g ∈ G

ˆ [g, h] − [fg, h] + [f, gh] − [f, g] = 0 for all f, g, h ∈ G

136 7.4 Group Extensions 7 Group Cohomology is called a normalized 2-cocycle.

Normalized 2-cocycles arise from the bar resolution. The first condition is the normalized condition, while the second condition makes the map [·, ·] a cocycle map; this should look something like a cross-ratio from geometry. Given an extension 0 → A → E −→π G → 1, we may choose a set map σ : G → E such that σ(1) = 1E and πσ = idG. Such a map is called a based section. Then, given g, h ∈ G, we have that σ(gh) and σ(g)σ(h) are elements of E which map to gh under π, so their difference lies in A. Set [g, h] = σ(g)σ(h)σ(gh)−1; this lies in the subgroup of E corresponding to the image of A.

Definition. The function [·, ·]: G × G → A is called the factor set determined by the extension E and based section σ.

Lemma 7.4.3. If two extensions 0 → A → E1 → G → 1 and 0 → A → E2 → G → 1 with based sections σi : G → Ei yield the same factor set, then the extensions are equivalent. Notice that a factor set [·, ·] is the zero map if and only if σ is a homomorphism, that is, a section. In this case the extension is split. We saw above that every central extension 0 → A → E → G → 1 defines a G-module structure on A. The converse is true as well; namely, if A is a G-module then there is a central extension 0 → A → E → G → 1. To construct such an extension, let E = A × G as sets and define a binary operation on E by

(a, g) · (b, h) = (a + g · b + [g, h], gh).

It is routine to check that this operation gives E a group structure (the only sticking point is the associative axiom, but the 2-cocycle condition defined above is precisely what is needed). In general we are interested in the following question.

Question. How many ways are there to put a group structure on E = G × A?

The answer, as it turns out, is best described in terms of group cohomology. From the bar resolution, we have that d[g|h] = g[h] − [gh] + [g] and d[f|g|h] = f[g|h] − [fg|h] + [f|gh] − [f|g], so Z2(G; A) is the set of all ϕ : G × G → A such that ϕ(1, g) = ϕ(g, 1) = 0 and

(Dϕ)(f, g, h) = fϕ(g, h) − ϕ(fg, h) + ϕ(f, gh) − ϕ(f, g) = 0 for all f, g, h ∈ G. Meanwhile, B2(G; A) consists of all maps ϕ : G × G → A such that ϕ(1, g) = ϕ(g, 1) = 0 and

ϕ(f, g) = β(d[f|g]) − β(fg) + β(f) for some β : G → A. We then have the following lemma.

Lemma 7.4.4. Let A be a G-module. A set function [·, ·]: G × G → A is a factor set if and only if it is an element of Z2(G; A).

137 7.4 Group Extensions 7 Group Cohomology

Proof. Given a factor set [·, ·] associated to an extension E and based section σ, it is a routine computation to check that [·, ·] is a 2-cocycle. On the other hand, given a 2-cocycle [·, ·], let E be the set A × G with multiplication given by

(a, g) · (b, h) = (a + gb + [g, h], gh).

The element (0, 1) is an identity of E, the operation associates and

(a, g) · (−g−1a − g−1[g, g−1], g−1) = (0, 1)

so E in fact is a group. Further, A×1 is a subgroup of E isomorphic to A and E/(A×1) ∼= G. Hence 0 → A → E → G → 1 is an extension with factor set corresponding to the based section G,→ 0 × G. Theorem 7.4.5. Elements of H2(G; A) are in one-to-one correspondence with equivalence classes of extensions of the form

0 → A → E → G → 1.

Proof. Let 0 → A → E −→π G → 1 be an extension of groups and let σ, σ0 be two based sections of π, with corresponding factor sets [·, ·] and [·, ·]0. We note that σ(g) and σ0(g) lie in the same coset of A in E for all g ∈ G, so there is a β(g) ∈ A such that σ0(g) = β(g)σ(g). Then

[g, h]0 = σ0(g)σ0(h)σ0(gh)−1 = β(g)σ(g)β(h)σ(h)σ(gh)−1β(gh)−1 = β(g)σ(g)β(h)σ(g)−1σ(g)σ(h)σ(gh)−1β(gh)−1 = β(g) + σ(g)β(h)σ(g)−1 + σ(g)σ(h)σ(gh)−1 − β(gh) = β(g) + gβ(h) + [g, h] − β(gh).

Hence [g, h]0 −[g, h] = gβ(h)−β(gh)+β(g) = (Dβ)(g, h). Thus the factor set of an extension is unique up to boundary, so we have a well-defined map Φ from the set of equivalence classes of extensions of G by A into H2(G; A), taking each extension to the cohomology class of the factor set corresponding to any of its based sections. Now notice that the formula σ0(g) = β(g)σ(g) gives a bijective correspondence between based sections σ and the set of all maps β : G → A with β(1) = 1. Hence, given two extensions yielding the same cohomology class, we see that a choice of based sections will give the same factor sets, lying in the same cohomology class as the initial factor sets. By Lemma 7.4.3, the extensions are equivalent so Φ is injective. On the other hand, by Lemma 7.4.4, every 2-cocycle is a factor set so Φ is surjective. This completes the proof. Theorem 7.4.6. Let G be a finite group of order m. If A is a G-module on which multipli- cation by m is an isomorphism, then Hi(G; A) = 0 for all i ≥ 1. The main result we can prove using this machinery is: Theorem 7.4.7 (Schur-Zassenhaus). If m and n are relatively prime, then any extension 1 → A → E → G → 1 of a group G of order m by a group A of order n is split.

138 7.5 Central Simple Algebras 7 Group Cohomology

Proof. First assume A is abelian, as then the extensions of G by A are classified by H2(G; A). By Theorem 7.4.6 this is zero, so the only equivalence class of extensions is the split one. In the general case, we will induct on n. Note that to show E = A o G, it suffices to show that E contains a subgroup of order m, for such a subgroup will be isomorphic to G under the map E → G. Let p | n be prime and let P be a Sylow p-subgroup of A, and thus also of E. Let Z = Z(P ) and N = NE(Z(P )). Then m divides |N|. For a prime q | m, choose a Sylow q-subgroup Q ≤ E. It follows that QA = NQA(P )A, so |Q| divides |NQA(P )| and thus |NE(P )|. A short argument shows that NE(P ) is contained in N. Hence E = AN, so N/AN ∼= N(A ∩ N) which gives us an extension

1 → A ∩ N → N → G → 1.

If N 6= E then A ∩ N 6= A and (|A ∩ N|, m) = 1 so the sequence splits by induction. Hence there is a subgroup of N isomorphic to G, which is also a subgroup of E isomorphic ∼ ∼ to G. On the other hand, if N = E, then Z C E and (E/Z)/(A/Z) = E/A = G and 0 → A/Z → E/Z → G → 1 is split, again by induction. Let G0 be the subgroup of E/Z isomorphic to G and set E0 = {x ∈ E | xZ ∈ G0}. Then 0 → Z → E0 → G → 1 is exact and Z is abelian, so the extension splits. Hence there is a subgroup of E0 isomorphic to G, which is also a subgroup of E. This completes the proof of the Schur-Zassenhaus theorem.

7.5 Central Simple Algebras

¯ ∼ ¯ ¯ Definition. An algebra A over a field k is central simple if A⊗k k = Mn×n(k) as k-algebras for some n ≥ 1, where k¯ denotes the algebraic closure of k.

Lemma 7.5.1. A k-algebra A is central simple if and only if the following conditions hold:

(1) Z(A) = k.

(2) A is finite dimensional over k.

(3) A is simple as a k-algebra.

Example 7.5.2. The quaternions are an R-algebra defined by

2 2 2 H = Rhi, j, k | i = j = k = −1, ij = k, jk = i, ki = ji. ∼ Over R = C, H ⊗R C = M2×2(C) via the assignment i 0  0 −1  0 −i i 7→ , j 7→ , k 7→ . 0 −i 1 0 −i 0

Example 7.5.3. In general, every finite dimensional division algebra over k is central simple. Let D be a finite dimensional division algebra. Then Mn×n(D) is also a central simple algebra. In fact, Wedderburn theory shows that every central simple algebra is isomorphic to Mn×n(D) for some division algebra D.

139 7.5 Central Simple Algebras 7 Group Cohomology

Example 7.5.4. If A and B are simple algebras over k with A central simple, then A ⊗k B is a central simple algebra over Z(B). In particular, if B is also central simple then A ⊗k B is central simple over k.

Definition. We say two central simple algebras A and B are Morita equivalent if B ∼= Mn×n(A) for some n ≥ 1. Lemma 7.5.5. For two central simple algebras A and B over k, the following are equivalent:

(1) A and B are Morita equivalent. ∼ (2) A ⊗ Mn×n(k) = B ⊗ Mm×m(k) as k-algebras for some m, n. ∼ (3) If A = Mm×m(D1) and B = Mn×n(D2), then D1 = D2 as k-algebras. Theorem 7.5.6. The Morita equivalence classes of central simple algebras over a field k are in bijection with H2(Gal(k/k¯ ); k¯×).

Definition. The set of central simple algebras over k up to Morita equivalence is called the Brauer group of k, denoted Br(k).

By Theorem 7.5.6, the group structure on H2(Gal(k/k¯ ); k¯×) induces a group structure on Br(k). The next result shows that there is a natural group structure on the Brauer group, and in fact these coincide.

Theorem 7.5.7. The Brauer group Br(k) is an abelian group under tensor product, with inverses [A]−1 = [Aop], where Aop denotes the opposite ring, and identity [k].

Definition. A splitting field for a central simple algebra A is a field extension L/k such ∼ that A ⊗k L = Mn×n(L) for some n. For all extensions L/k, there is an induced map on Brauer groups

fL/k : Br(k) −→ Br(L)

[A] 7−→ [A ⊗k L].

It is easy to check that this is a group homomorphism.

Definition. The relative Brauer group of an extension L/k is

Br(L/k) := ker fL/k = {[A] ∈ Br(k) | L is a splitting field for A}.

Theorem 7.5.8. For any field k, the Brauer group of k is a direct limit [ Br(k) = Br(L/k) L/k where the limit is taken over all finite Galois extensions L/k.

140 7.5 Central Simple Algebras 7 Group Cohomology

We now explore Theorem 7.5.6 a bit. The absolute Galois group Gal(k/k¯ ) is an inverse limit: Gal(k/k¯ ) = lim Gal(L/k) ←− over all finite Galois extensions L/k. Since H2 commutes with limits, we can understand H2(Gal(k/k¯ ); k¯×) by computing H2(Gal(L/k); L×) for finite Galois extensions instead. For such an extension L/k with group G = Gal(L/k), let A = Lheσiσ∈Gal(L/k) be the k-algebra generated by formal symbols eσ for each σ ∈ Gal(L/k) which satisfy

eσeτ = γ(σ, τ)eστ for some 2-cocycle γ : G × G → L. Then γ 7→ A defines a map H2(G; L×) → Br(L/k) which becomes H2(G; k¯×) → Br(k) after passing to the limit. Going the other direction, one needs to understand the G-module structures on A ⊗k L and Mn×n(L). Both of these actions in some sense “live” in the action of G n P GLn(L), by which we mean a central simple algebra A over L/k determines a cohomology class 1 2 × βA ∈ H (G; P GLn(L)). To construct a class in H (G; L ), consider the short exact sequence of groups × 1 → L → GLn(L) → P GLn(L) → 1. This determines a long exact sequence in group cohomology, which by Hilbert’s Theorem 90 (7.3.3) starts with:

1 × 1 1 2 × 0 = H (G; L ) → H (G; GLn(L)) → H (G; P GLn(L)) → H (G; L ) → · · ·

Definition. A projective representation of G is a homomorphism ρ : G → P GL(V ) for some L-vector space V .

Explicitly, a projective representation ρ assigns to each g ∈ G a linear subspace Lϕg ∈ × × × End(V ) such that (L ϕg)(L ϕh) = L ϕgh for all g, h ∈ G. Therefore ϕgϕh = τ(g, h)ϕgh for some τ(g, h) ∈ L×. Since matrix multiplication is associative, we have

(ϕgϕh)ϕa = ϕg(ϕhϕa) for all g, h, a ∈ G

=⇒ τ(g, h)ϕghϕa = ϕgτ(h, a)ϕha

=⇒ τ(g, h)τ(gh, a)ϕgha = τ(h, a)τ(g, ha)ϕgha.

Since ϕgha 6= 0, we can divide it out to obtain τ(g, h)τ(gh, a) = τ(h, a)τ(g, ha). That is, τ is a 2-cocycle for G with coefficients in L×. One can check that a different choice of the coset representatives for g, h, a ∈ G give a 2-cocycle of the form τ 0 = τdψ for some ψ : G → A. This proves:

Proposition 7.5.9. Every projective representation of G determines a unique cohomology class in H2(G; L×) which is trivial if and only if the representation lifts to a representation G → GL(V ).

Let Ge be the pullback in the diagram of groups

141 7.6 Classifying Space 7 Group Cohomology

Ge GL(V )

ρ G P GL(V )

Explicitly, Ge = {(g, a) ∈ G × GL(V ) | a ∈ ρ(g)}. This defines a short exact sequence of groups 1 → L× → Ge → G → 1 but by Theorem 7.4.5, such extensions are in bijection with elements of H2(G; L×), with the trival class corresponding to split extensions. In this case, a split extension is a lift of ρ to G → GL(V ). This gives the map Br(k) → H2(Gal(k/k¯ ); k¯×) in the forward direction of Theorem 7.5.6, modulo checking compatibility with the equivalence A ∼ Mn×n(A).

7.6 Classifying Space

Given how topological the definitions of homology and cohomology seem to be in Chapter 4, one may wonder if there is a corresponding topological interpretation of group homology and cohomology. It turns out that for a group G, there is topological space BG such that

Hn(G; A) ∼= Hn(BG; A)

where the latter denotes the singular cohomology of the space BG with coefficients in A. To construct BG as a CW-complex, start with the 1-skeleton defined by specifying a directed edge g for each element g ∈ G and gluing according to the group law on G:

gh h

g

Then extend over the n-skeleton for n ≥ 2 by gluing every n-fold product g1g2 ··· gn in G. Let π1(BG) denote the fundamental group of this space.

Proposition 7.6.1. For all groups G, π1(BG) = G. This constructs BG, called the classifying space of G. One can construct a universal cover EG → BG where EG is a CW-complex with n-skeleton consisting of faces for ev- ery (n + 1)-tuple of elements in G where (g0, . . . , gn) is glued to each of the (n − 1)-faces ∼ (g0,..., gˆi, . . . , gn). Then BG = EG/G. Note that EG is contractible.

Proposition 7.6.2. Suppose X is a topological space with contractible universal cover Xe ∼ and π1(X) = G. Then X is homotopy equivalent to BG and for all abelian groups A, H•(X; A) ∼= H•(G; A) where A is viewed as a trivial G-module.

142 7.6 Classifying Space 7 Group Cohomology

Let A be a G-module and consider the homomorphism ρ : G → Aut(A) with kernel K = ker ρ. Then K determines a covering space Xρ → BG such that π1(Xρ) = K. Moreover, n G ∼ n G ∼ for each n ≥ 0 we have H (G; A ) = H (K; A) where A = HomZG(Z,A) is the module of G ∼ [G:K] G-invariants of A. For example, let A = C as a trivial K-module, so that C = C as a complex vector space (explicitly, CG is equal to the regular representation of G on the cosets of K). Then

n ∼ n H (K; C) = H (G; CH) where H = G/K   ∼ n M = H G; Vχ where Vχ are the irreducible representations of G χ∈Gb ∼ M n = H (G; Vχ). χ∈Gb

Further, when H = G/K is a finite abelian group, each Vχ is just a 1-dimensional C-vector space with some H-action given by a homomorphism τ : H → C×. Then Hn(K; C) is easily n understood in terms of the H (G; Cτ ) for different irreducible representations of H.

143 8 Sheaf Theory

8 Sheaf Theory

In this chapter, we describe the theory of sheaves on topological spaces. The basic intuition is that for a topological space X, one wishes to study a given property defined on open sets U ⊆ X that behaves well under union and intersection of open sets.

8.1 Sheaves and Sections

Definition. Let p : E → X be a morphism in a category C.A section of p is a morphism σ : X → E such that p ◦ σ = idX . Definition. When p : E → X is a continuous map of topological spaces, a local section of p over an open set U ⊆ X is a map σ : U → E such that p ◦ σ = idX |U . The set of all sections of p over U is denoted Γp(U).A global section of p is a section over the open set U = X.

The following observations, which are easy to prove, are a natural starting point for the study of sheaves.

Lemma 8.1.1. Let p : E → X be a continuous map of topological spaces. Then

(a) If V ⊆ U ⊆ X are open sets, then there is a restriction map

Γp(U) −→ Γp(V )

σ 7−→ σ|V .

That is, Γp(−) is a contravariant functor on the category TopX of open sets on X. S (b) Let {Ui} be a collection of open sets in X, with U = Ui, and suppose there exist

local sections σi ∈ Γp(Ui) such that σi|Ui∩Uj = σj|Ui∩Uj for all i, j. Then there exists a

unique section σ ∈ Γp(U) for which σ|Ui = σi holds for all i. This generalizes in the following way.

Definition. For a topological space X, a presheaf on X is a contravariant functor F :

TopX → C, where C is any category. For each inclusion of open sets V,→ U, the induced morphism FUV : F(U) → F(V ) is called restriction, written FUV (σ) = σ|V .

Definition. Let C be a set category. A presheaf F : TopX → C is called a sheaf on X provided it satisfies the following ‘descent conditions’:

S 0 (1) For any open covering U = Ui, if there exist sections σ, σ ∈ F(U) such that σ|Ui = 0 0 σ |Ui for all i, then σ = σ . S (2) For any open covering U = Ui admitting sections σi ∈ F(Ui) such that σi|Ui∩Uj =

σj|Ui∩Uj for all i, j, there exists a unique section σ ∈ F(U) such that σ|Ui = σi for all i.

144 8.1 Sheaves and Sections 8 Sheaf Theory

Notice that the uniqueness in (2) is guaranteed by (1) so it is not strictly necessary to include it in the definition. Definition. A morphism of sheaves is simply a natural transformation of functors F → G. Likewise, an isomorphism of sheaves is a natural isomorphism of the underlying functors.

Example 8.1.2. For a (pre)sheaf F on X and an open set U ⊆ X, the functor F|U : V 7→ F(V ) is a (pre)sheaf. The operation F 7→ F|U is called restriction of a sheaf. Remark. The descent conditions in the definition of a sheaf are stated for set categories, but they can be reinterpreted in an arbitrary category C by saying that for every open set S U ∈ TopX and every open covering U = Ui, the object F(U) is a limit of the following diagram (also called an equalizer): Y Y F(U) F(Ui) F(Ui ∩ Uj) i i,j

Example 8.1.3. The classic example of a sheaf on a topological space X assigns to each open U ⊆ X the set F(U) of all continuous (or differentiable, smooth, holomorphic, etc.) functions on U, with restriction maps given by restriction of functions.

Example 8.1.4. Let S be a set and define a functor F : TopX → Set by F(U) = S for all U ⊆ X, with the restriction maps given by σ|V = σ ∈ S for any inclusion V,→ U. Then F is a presheaf, called the constant presheaf on S, but if X has two disjoint open sets, then F is not a sheaf. Example 8.1.5. The right notion of a constant sheaf is constructed as follows. Let S be a set considered with the discrete topology. For all open sets U ⊆ X, let FS(U) = C(U, S) be the set of all continuous (i.e. locally constant) functions U → S. Then FS defines a sheaf ∼ on X. Notice that if U is a connected open set, then FS(U) = S, hence the name constant sheaf.

Example 8.1.6. For a continuous map p : E → X, the assignment U 7→ Γp(U) defines a sheaf on X by Lemma 8.1.1. Conversely, every sheaf F on X determines a topological space EF , called the ´espace ´etale (or ´etalespace) of F, together with a map pF : EF → X, called the ´etalecover associated to F. This is constructed below. Definition. For a sheaf F on X and a point x ∈ X, the stalk of F at x is the direct limit

Fx := lim F(U) −→ taken over all open sets U ⊆ X containing x.

Example 8.1.7. For the constant sheaf FS, the stalks are all the same: FS,x = S. Example 8.1.8. Suppose X is a T1 topological space and A is an abelian group. The skyscraper sheaf at x ∈ X with coefficients in A is the sheaf Ax defined by ( A, x ∈ U Ax(U) = 0, x 6∈ U.

For a point y ∈ X, the stalk (Ax)y is A if y = x and 0 otherwise. Therefore we can think of a skyscraper sheaf as a sort of constant sheaf concentrated at the point x.

145 8.1 Sheaves and Sections 8 Sheaf Theory

Example 8.1.9. Let X = C be the space of complex numbers and for each open U ⊂ C, let O(U) denote the set of all holomorphic functions U → C. Then O is a sheaf of rings with stalks given by Oz = C[t](t−z), the ring of Taylor series at z with positive radius of convergence.

Proposition 8.1.10. Let p : E → X be a local homeomorphism. Then

(1) The functor Γp(−): TopX → Set,U 7→ Γp(U), is a sheaf on X with restriction maps Γp(U) → Γp(V ) for V ⊆ U given by restriction of sections s 7→ s|V .

(2) Sets of the form s(U), where U ⊆ X is open and s ∈ Γp(U), form a basis for the topology on E.

(3) The stalk of Γp(−) at x ∈ X is the fibre Ex. Proof. (1) follows from Lemma 8.1.1 and (2) is routine. Here’s a proof of (3): For each neighborhood U ⊆ X of x, define a map

νU :Γp(U) −→ Ex s 7−→ s(x).

This is compatible with restriction, i.e. if V ⊆ U is a smaller neighborhood of x, the diagram

Γp(U) νU

Ex

νV Γp(V ) commutes. Therefore the νU induce a morphism ν :Γp(−)x = lim Γp(U) → Ex. For e ∈ Ex, −→ take a neighborhood Ve ⊆ E such that p|Ve : Ve → π(Ve) = U is a homeomorphism. Then the map s = p|−1 is a section of p over U and s(x) = s(p(e)) = e, so ν is onto. Finally, if Ve s and t are sections of p over some neighborhoods U1 and U2 of x, respectively, such that ∼ s(x) = t(x) in Ex, then we may choose neighborhoods V1,V2 ⊆ E such that p|V1 : V1 −→ U1 and p|V2 : V2 → U2 are homeomorphisms. Set V = V1 ∩ V2 and U = π(V ). Then p is a homeomorphism on V , so p ◦ s|U = p ◦ t|U implies s|U = t|U so s = t in Γp(−)x. Thus ν is a bijection. S Definition. The ´espace´etale of a sheaf F on X is the space EF = x∈X Fx with the topology induced by the open sets

VU,f := {(x, f|Fx ) | x ∈ U, f ∈ F(U)} where U runs over all open sets in X. The ´etalecover pF : EF → X is naturally defined as (x, f|Fx ) 7→ x.

146 8.1 Sheaves and Sections 8 Sheaf Theory

Example 8.1.11. For the sheaf of holomorphic functions O on C, the open sets VU,f (for holomorphic functions f : U → C) are given by

VU,f = {(z0, p(z)) | p(z) is the Taylor series expansion of f(z) at z0}.

Theorem 8.1.12. There is an equivalence of categories between maps p : E → X and sheaves F on X given by (p : E → X) 7→ Γp and F 7→ (pF : EF → X).

Proof. We have seen, courtesy of Proposition 8.1.10, that Γp is a sheaf on X. On the other hand, it follows from the definition of the topology on EF that the map pF : EF → X is continuous. Therefore it remains to show that the assignments p 7→ Γp and F 7→ pF define an equivalence of categories. That is, we must show that for a sheaf F on X, there is an ∼ ∼ isomorphism ΓpF = F, and for a map p : E → X, there is a homeomorphism EΓp = E making the diagram ∼ EΓp E

p

X

commute. First let F be a sheaf, let U ⊆ X be open, take σ ∈ F(U) and define s ∈ ΓpF (U) −1 by s(x) = (x, σ|x). Consider s (VU,f ) ⊆ U for some f ∈ F(U). Then

−1 x ∈ s (VU,f ) ⇐⇒ s(x) = f(x)

⇐⇒ s|W = f|W for some neighborhood W of x.

−1 For such an x we have W ⊆ s (VU,f ), so the morphism F → ΓpF sending σ 7→ s is well- 0 0 0 defined and continuous. Suppose s ∈ ΓpF (U) such that s(x) = s (x) for all x ∈ U . Then S 0 there exists an open cover U = Ui with s|Ui = s |Ui for all Ui, so by the sheaf axioms, 0 s = s . Therefore F → ΓpF is one-to-one.

On the other hand, suppose s : U → EΓp is a section of pF . Then

−1 0 0 0 0 0 s (VU 0,s0 ) = {x ∈ U | s(x ) = s (u )}.

−1 0 0 On the open set s (VU 0,s0 ) ⊆ U, s comes from s uniquely. Therefore if U is a neighborhood −1 0 of x, then s (VU 0,s0 ), where s (x) = s(x), is a neighborhood of x. This means there is an S open cover U = Ui with σi ∈ F(Ui) such that σi(x) = s(x) for all x ∈ U. Moreover,

since F → ΓpF is one-to-one, we have σi|Ui∩Uj = σj|Ui∩Uj for all i, j. Therefore by the sheaf

axioms, there is a unique section σ ∈ F(U) with σ|Ui = σi for all i, which then satisfies

σ(x) = s(x) for all x ∈ U. Thus F → ΓpF is onto, hence an isomorphism.

Now let p : E → X be a continuous map. Define E → EΓp as follows. For y ∈ E, set x = p(y) ∈ X. Then for all open sets U ⊆ X containing X, y ∈ p−1(U), so y in fact lies in the stalk Γp,x = lim Γp(U). Thus the assignment E → EΓ , y 7→ y is well-defined. It is −→ p clearly one-to-one and onto, and continuity follows easily from the definition of the topology

on EΓp .

147 8.1 Sheaves and Sections 8 Sheaf Theory

Definition. A sheaf F on X is locally constant if for all x ∈ X, there is a neighborhood U ⊆ X of x such that F|U is isomorphic to a constant sheaf on U.

Example 8.1.13. If p : E → X is a topological covering, then Γp is a locally constant sheaf on X. Indeed, for each x ∈ X, choose a connected, locally trivial neighborhood U of X such that p−1(U) ∼= U × p−1(x). Then for a local section σ : U → E, p maps σ(U) homeomorphically onto U, so σ(U) must be a connected component of p−1(U). It follows ∼ −1 that Γp(U) = p (x) and therefore Γp|U is isomorphic to the constant sheaf Fp−1(x). Conversely, for every locally constant sheaf F, the ´etale cover EF → X is a topological cover. To see this, note that for each locally constant neighborhood U ⊆ X, there is a fixed −1 ∼ discrete set S such that Fx = S for all x ∈ U. Thus p (U) = U × S, so EF is a cover of X. Thus we have proven:

Corollary 8.1.14. There is an equivalence of categories between covering spaces of X and locally constant sheaves on X.

Remark. For a presheaf F on X, there is still a notion of ´espace ´etale pF : EF → X, but

in general the natural maps F(U) → ΓpF (U) need not be isomorphisms. Theorem 8.1.15 (Sheafification). For every presheaf (of sets) F on X, there exists a presheaf F sh together with a morphism of presheaves θ : F → F sh such that for every sh x ∈ X, θx : Fx → Fx is a bijection. Moreover, for any morphism ϕ : F → G where G is a sheaf, there is a unique morphism of sheaves ψ : F sh → G such that the diagram

θ F F sh

ϕ ψ

G

commutes.

Corollary 8.1.16. If F is a sheaf, then F sh = F .

sh Proof. This will from Proposition 8.2.8(2) and the fact that Fx → Fx is a bijection for all x ∈ X. π Proof of 8.1.15. Let E = EF −→ X be the ´etalespace of F and define F = Γπ(−), the sheaf of sections of π. Then the morphism θ : F → F sh is defined by

sh θU : F (U) −→ F (U)

s 7−→ (˜s : U → E, x 7→ s|x).

Suppose V ⊆ U is an inclusion of open sets. It is immediate that the diagram

148 8.1 Sheaves and Sections 8 Sheaf Theory

θU F (U) F sh(U)

F (V ) F sh(V ) θV

sh commutes, so the θU indeed give a morphism θ : F → F . On stalks, we have that sh Fx = Γ(−, π)x = Ex = Fx by Proposition 8.1.10(3), and it is clear that θx : Fx → Fx is just the identity morphism sx 7→ sx. To prove the universal property, note that Corollary 8.1.16 already follows from the paragraph above, so for any sheaf G, θ : G → Gsh is the identity on G. If ϕ : F → G is a morphism (of presheaves), then by Theorem 8.1.12 there is a continuous map Eϕ : EF → EG sh sh and thus a morphism ψ : F = ΓπF → ΓπG = G = G. Now for all open U ⊆ X and sections s ∈ F (U), ψU ◦ θU (s) = ψU (˜s) = Eϕ ◦ se = ϕU (s) by construction, so the diagram

θU F (U) F sh(U)

ϕU ψU

G(U)

commutes, and once again this is compatible with restriction of sections along V ⊆ U. The sh fact that such a ψ is unique follows from the bijection Fx → Fx and Proposition 8.2.8. Definition. For a presheaf F on X, we call F sh the sheafification of F , or the sheaf associated to F .

Remark. Abstractly, we have an adjoint pair of functors

sh Presh(C) Sh(C) forget

which means sheafification is right exact and the forgetful functor is left exact. In a more general setting, such as when we consider the categories of presheaves and sheaves of abelian groups, rings, algebras, etc., we can take this as our definition of the sheafification functor: it is the left adjoint to the (left exact) forgetful functor ShX ,→ PreshX . Alternatively, we can equip an associated sheaf F sh with the structure of a sheaf of abelian groups (or rings, algebras, etc.) as follows. Let E −→π X be the ´etalecover for the presheaf F and consider the fibre product

E ×X E := {(e1, e2) ∈ E × E | π(e1) = π(e2)}.

149 8.2 The Category of Sheaves 8 Sheaf Theory

Then E ×X E → E, (e1, e2) 7→ e1 + e2 is a well-defined, continuous map of topological spaces since the fibres of E (i.e. the stalks of F ) are abelian groups. The induced map sh sh ∼ sh F (U) × F (U) = Γπ(U) × Γπ(U) = Γπ×X π(U) −→ Γπ(U) = F (U) gives an additive structure on each F sh(U), making F sh into a sheaf of abelian groups.

8.2 The Category of Sheaves

In this section, we study the category ShX of sheaves on a topological space X. We start by introducing several constructions in the category PreshX of presheaves on X. Many categorical constructions in PreshX can be constructed locally, i.e. over open sets U ⊆ X. For example: Definition. For two presheaves F and G on X, the presheaf product F × G is defined by (F × G)(U) = F(U) × G(U). Likewise, the presheaf coproduct F ` G is defined by (F ` G)(U) = F(U) ` G(U). Definition. Let ϕ : F → G be a morphism of presheaves. ˆ The presheaf image im ϕ is defined by (im ϕ)(U) = im(F(U) → G(U)). ˆ The presheaf kernel ker ϕ is defined by (ker ϕ)(U) = ker(F(U) → G(U)). ˆ The presheaf cokernel coker ϕ is defined by (coker ϕ)(U) = coker(F(U) → G(U)). Lemma 8.2.1. If ϕ : F → G is a morphism of sheaves, then ker ϕ is a sheaf. S Proof. Suppose U = i Ui is an open cover in X and s, t ∈ (ker ϕ)(U) such that s|Ui = t|Ui for all Ui. Then since (ker ϕ)(U) ⊆ F(U), the first sheaf axiom for F implies directly that

s = t. Now suppose there are sections si ∈ (ker ϕ)(Ui) such that si|Ui∩Uj = sj|Ui∩Uj for all overlapping Ui,Uj. Since (ker ϕ)(Ui) ⊆ F(Ui), the second sheaf axiom for F implies there

exists a section s ∈ F(U) such that s|Ui = si for all i. We must show s ∈ (ker ϕ)(U) ⊆ F(U).

Note that ϕU (s)|Ui = ϕUi (s|Ui ) = ϕUi (si) = 0 for each Ui, so by the first sheaf axiom for G, we must have ϕU (s) = 0.

Theorem 8.2.2. If C is an abelian category, then the category PreshX (C) of presheaves TopX → C is an abelian category. Example 8.2.3. For a morphism ϕ of sheaves, coker ϕ is not a sheaf in general. For example, take X = C and let F = O be the sheaf of holomorphic functions from Example 8.1.9. If O× denotes the sheaf of nonvanishing holomorphic functions on C, then there is a morphism of × × 2πif sheaves ϕ : O → O given locally by ϕU : O(U) → O(U) , f 7→ e . In fact, one can see that ϕ is an isomorphism on stalks (using the complex logarithm). However, on U = C×, the map O(C×) → O(C×)× is not surjective, since f(z) = z is not in the image. Thus on the level of stalks we have exact sequences

ψx ϕx × Zx −→Ox −→Ox where Z denotes the constant sheaf, but the cokernel of the map ψ : Z → O is not O×, but rather (O/Z)sh.

150 8.2 The Category of Sheaves 8 Sheaf Theory

Definition. The sheaf cokernel of a morphism of sheaves ϕ : F → G is the sheafification of the presheaf cokernel:

U 7−→ (coker ψ)(U) := (G(U)/ϕ(F(U)))sh.

Theorem 8.2.4. For any space X, ShX is an abelian category. Let ϕ : F → G be a morphism of presheaves on X. Then for any x ∈ X, since the stalks Fx and Gx are defined via direct limits, there is an induced morphism ϕx : Fx → Gx. Lemma 8.2.5. Let F be a sheaf on X and U ⊆ X an open set with sections s, t ∈ F(U). Then s = t if and only if for all x ∈ U, s|x = t|x in Fx. In other words, the morphism Q F(U) → x∈U Fx is injective.

Proof. If s|x = t|x, then there is some neighborhood Ux ⊆ U of x such that s|U = t|U . Since S x x this holds for all x ∈ U and U = x∈U Ux, we have s|Ux = s|Ux for all Ux, but by the sheaf axioms this implies s = t.

Corollary 8.2.6. Let F be a presheaf, G a sheaf and ϕ, ψ : F → G two morphisms of presheaves. Then ϕ = ψ if and only if ϕx = ψx : Fx → Gx for all x ∈ X.

Proof. The ( =⇒ ) implication is clear, so suppose ϕx = ψx for all x. Let U ⊆ X be an open set and take s ∈ F(U). Then the elements ϕU (s), ψU (s) ∈ G(U) restrict on stalks to:

ϕU (s)|x = ϕx(s|x) since ϕ is a morphism

= ψx(s|x) by hypothesis

= ψU (s)|x since ψ is a morphism.

Therefore by Lemma 8.2.5, ϕU (s) = ψU (s). Since this holds for all U, we get ϕ = ψ. Definition. Let F be a sheaf of abelian groups on X and s ∈ Γ(X, F) a global section. Then the support of s is the set

supp(s) = {x ∈ X | s|x 6= 0 in Fx}.

Lemma 8.2.7. For any s ∈ Γ(X, F), supp(s) is a closed set in X.

Proposition 8.2.8. Let ϕ : F → G be a morphism of presheaves, with F a sheaf. Then

(1) The morphisms ϕx : Fx → Gx are injective for all x ∈ X if and only if the morphisms ϕU : F(U) → G(U) are injective for all open sets U ⊆ X.

(2) The morphisms ϕx : Fx → Gx are bijective for all x ∈ X if and only if the morphisms ϕU : F(U) → G(U) are bijective for all open sets U ⊆ X.

Proof. (1) Assume ϕx are injective for all x ∈ X and suppose s, t ∈ F(U) are sections such that ϕU (s) = ϕU (t) in G(U). Then for all x ∈ X,

ϕx(s|x) = ϕU (s)|x = ϕU (t)|x = ϕx(t|x)

151 8.2 The Category of Sheaves 8 Sheaf Theory

since ϕ is a morphism, so injectivity implies s|x = t|x. Finally, Lemma 8.2.5 shows that s = t in F(U). Conversely, assume that each ϕU is injective. For a fixed x ∈ X, take sx, tx ∈ Fx such that ϕx(sx) = ϕx(tx) in Gx. Then there is a neighborhood U of x such that sx = s|x and tx = t|x for some s, t ∈ F(U). Since Gx = lim G(U) over all neighborhoods U of x, the −→ condition that ϕx(sx) = ϕx(tx) in Gx means there is some smaller neighborhood V ⊆ U containing x on which ϕU (s)|V = ϕU (t)|V . Since ϕ is a morphism, this is equivalent to ϕV (s|V ) = ϕV (t|V ), but by injectivity of ϕV , we get s|V = t|V . Passing to the direct limit, we have sx = tx in Fx. Hence ϕx is injective. (2) The ( ⇒ = ) implication is trivial. For ( =⇒ ), suppose each ϕx is bijective. Fix U ⊆ X and t ∈ G(U); we must find s ∈ F(U) so that ϕU (s) = t. For any x ∈ U, consider the image tx of t in the stalk Gx. By hypothesis, there is some sx ∈ Fx such that

ϕx(sx) = tx. Then there is a neighborhood Ux ⊆ U of x on which sx = sUx |x for some

sUx ∈ F(Ux). By construction, ϕUx (sUx ) = t in Gx, so there is a neighborhood Vx ⊆ Ux of x with ϕV (sU |V ) = ϕU (sU )|V = tV . Set sV = sU |V . Letting x range over the points of x x x xS x x x x x x U, we get a cover U = x∈U Vx and moreover, if x, y ∈ U, then

ϕVx∩Vy (sVx |Vx∩Vy ) = ϕVx (sVx )|Vx∩Vy since ϕ is a morphism

= t|Vx∩Vy

= ϕVy (sVy )|Vx∩Vy

= ϕVx∩Vy (sVy |Vx∩Vy ) for the same reason.

We are assuming each ϕx is bijective, so in particular (1) implies each ϕVx∩Vy is injective, and

thus sVx |Vx∩Vy = sVy |Vx∩Vy . Now the second sheaf axiom guarantees that there is a section

s ∈ F(U) such that s|Vx = sVx for all x ∈ U. Finally, observe that

ϕU (s)|Vx = ϕVx (s|Vx ) = ϕVx (sVx ) = t|Vx

for all x ∈ U. Thus the first sheaf axiom implies ϕU (s) = t.

Remark. Notice that the proof of surjectivity in (2) crucially relies on the fact that the ϕx and ϕU are already injective. It is not true in general that the ϕx are all surjective if and only if the ϕU are all surjective.

Proposition 8.2.9. For any x ∈ X, the stalk functor PreshX → Ab, F 7→ Fx, is exact.

ϕ ψ Proof. Suppose F 0 −→F −→F 00 is an exact sequence of presheaves on X. We must show the 0 ϕx ψx 00 induced sequence of stalks Fx −→Fx −→Fx is exact. First, since ψ ◦ ϕ = 0 globally, on each stalk we get ψx ◦ ϕx = 0 so im ϕx ⊆ ker ψx. On the other hand if sx ∈ ker ψx, then for some neighborhood U of x, there is a section s ∈ F(U) such that s|x = sx and ψU (s)|x = ψx(sx) = 0. Thus on some smaller neighborhood V ⊆ U of x, ψV (s|V ) = ψU (s)|V = 0, so 0 0 0 s|V ∈ ker ψV . By exactness, this means s|V = ϕV (s ) for some s ∈ F (V ). Then we have 0 0 0 0 sx = ϕV (s )x = ϕx(sx) for sx = s |x in Fx. Hence ker ψx ⊆ im ϕx so the sequence is exact. This gives us a good definition for what it means for a sequence of sheaves to be exact.

152 8.2 The Category of Sheaves 8 Sheaf Theory

Definition. A sequence of sheaves F 0 → F → F 00 is exact if for all x ∈ X, the sequence of 0 00 stalks Fx → Fx → Fx is exact.

ϕ ψ Theorem 8.2.10. Let 0 → F 0 −→F −→F 00 → 0 be a short exact sequence of sheaves on X. Then for every open set U ⊆ X, the sequence

0 → F 0(U) → F(U) → F 00(U) is exact. That is, the forgetful functor ShX → PreshX is left exact.

0 ϕx ψx 00 Proof. By definition, 0 → Fx −→ Fx −→ Fx → 0 is exact for all x ∈ X, so Proposi- tion 8.2.8(1) implies 0 → F 0(U) → F(U) is exact for all open sets U ⊆ X. Set K(U) = ker(F(U) → F 00(U)). Then by Lemma 8.2.1, K : U 7→ K(U) is a sheaf and it is easy to 00 see that the stalks are Kx = ker(Fx → Fx ). Now we have (ψ ◦ ϕ)x = ψx ◦ ϕx = 0 for all x ∈ X by exactness of the sequence of stalks, so (ψ ◦ ϕ)U = ψU ◦ ϕU = 0 for all U, and thus im ϕU ⊆ K(U). Finally, ϕ can be written as a morphism F → K, but on the level of stalks ∼ this is a bijection ϕx : Fx −→ Kx = im ϕx, so by Proposition 8.2.8(2), ϕU : F(U) → K(U) is also bijective. Hence F ∼= im ϕ = K so the sequence 0 → F 0(U) → F(U) → F 00(U) is exact. In general, if F → F 00 is surjective, it need not be true that F(U) → F 00(U) is surjective. 00 00 By definition, surjectivity of sheaves means surjectivity of stalks, so an element sx ∈ Fx is the image of some sx ∈ Fx but this sx need only be defined on some neighborhood of x, not necessarily on arbitrary U. Example 8.2.3 gives a counterexample.

Definition. A sheaf F on X is called flasque (or flabby) if for every open inclusion V ⊆ U, the restriction map F(U) → F(V ) is surjective.

ϕ ψ Theorem 8.2.11. If 0 → F 0 −→F −→F 00 → 0 is a short exact sequence of sheaves with F 0 flasque, then the sequence

0 → F 0(U) → F(U) → F 00(U) → 0 is exact for any open U ⊆ X. If in addition F is flasque, then so is F 00.

00 Proof. By Theorem 8.2.10, it’s enough to show ψU : F(U) → F (U) is surjective. Further, after restriction of sheaves, it’s enough to consider the case U = X. Let t ∈ F 00(X). Then for any x ∈ X, there is a neighborhood Ux ⊆ X of x and a section sUx ∈ F(Ux) such that

ψUx (sUx ) = t|Ux , by surjectivity of stalks. Consider the set of pairs

C = {(U, s) | U ⊆ X, s ∈ F(U), ψU (s) = t|U }

of open sets which admit an extension of t. Then C is a partially ordered set via (U1, s1) 

(U2, s2) if and only if U1 ⊆ U2 and s2|U1 = s1. For a linearly ordered subset (Uα, sα) in C, the S ∗ ∗ S pair ( Uα, s ) is an upper bound, where s ∈ F ( Uα) is defined using the sheaf conditions on F. Thus by Zorn’s Lemma, C contains a maximal element, say (U, s). Assume x ∈ X rU.

153 8.2 The Category of Sheaves 8 Sheaf Theory

Then there is a neighborhood V of x and a section sV,0 ∈ F(U) with ψV (sV,0) = t|V by surjectivity on stalks. Note that

ψU∩V (s|U∩V − sV,0|U∩V ) = ψU∩V (s|U∩V ) − ψU∩V (sV,0|U∩V ) = t|U∩V − t|U∩V = 0

so s|U∩V −sV,0|U∩V ∈ ψU∩V = im ϕU∩V by exactness at F(U ∩V ). That is, s|U∩V −sV,0|U∩V = 0 0 0 0 0 0 0 0 ϕU∩V (s0) for some s0 ∈ F (U ∩ V ). Now since F is flasque, s0 = s |U∩V for some s ∈ F (V ). 0 Set sV = sV,0 + ϕV (s ). By construction, s|U∩V = sV |U∩V so by the sheaf axioms on F over U ∪ V , there is a section sU∪V ∈ F(U ∪ V ) with sU∪V |U = s and sU∪V |V = sV . This shows that (U, s) ≺ (U ∪ V, sU∪V ), contradicting the fact that (U, s) is a maximal element in C. Hence X r U = ∅, or U = X, and s ∈ F(X) satisfies ψX (s) = t by definition. For the last statement, consider the following diagram with exact rows: 0 F 0(U) F(U) F 00(U) 0

0 F 0(V ) F(V ) F 00(V ) 0

If F is flasque, then the middle column is surjective, so it follows that the right column is also surjective. Hence F 00 is also flasque. Proposition 8.2.12. Let ϕ : F → G be a morphism of presheaves on X. Then (1) ϕ is surjective over all open sets U ⊆ X if and only if ϕ is an epimorphism in the category PreshX . (2) If F and G are sheaves, then ϕ is surjective on stalks if and only if ϕ is an epimorphism in the category ShX . Proof. (1) Let coker ϕ be the presheaf cokernel of ϕ. Then ϕ is an epimorphism if and only if coker ϕ = 0. For any open U ⊆ X, (coker ϕ)(U) is the cokernel of ϕU : F (U) → G(U) in the category of abelian groups, so it is 0 if and only if ϕU is surjective. On the level of sheaves, coker ϕ = 0 if and only if (coker ϕ)(U) = 0 for all U, thus ϕ is surjective on open sets if and only if it’s an epimorphism. (2) First suppose ϕ is surjective on stalks. If

ϕ f F G H g

is a diagram of sheaves such that f ◦ ϕ = g ◦ ϕ, then for all x ∈ X, we have

fx ◦ ϕx = (f ◦ ϕ)x = (g ◦ ϕ)x = gx ◦ ϕx

but since ϕx is surjective, this implies fx = gx. Applying Corollary 8.2.6 gives us f = g, so ϕ is an epimorphism. Conversely, if ϕ is an epimorphism, let C = coker ϕ be the presheaf cokernel of ϕ. If p : G → G/ϕ(F ) ∼= C is the natural projection of presheaves, consider the diagram

154 8.2 The Category of Sheaves 8 Sheaf Theory

ϕ p F G C 0

Then p ◦ ϕ = 0 = 0 ◦ ϕ so by assumption, p = 0. That is, C = 0, which means Cx = 0 for all x ∈ X. But Cx = coker(ϕx : Fx → Gx), so Cx = 0 implies ϕx is surjective for all x ∈ X. op Definition. Let X be a topological space and ShX (C) the category of sheaves TopX → C with values in a category C. Then the section functor is defined by

Γ: ShX (C) −→ C F 7−→ Γ(X, F) = F(X). For any open set U ⊆ X, the sections over U are simply the restricted sections Γ(U, F) = F(U). Proposition 8.2.13. For any X, the section functor Γ is left exact. Proof. Follows from Theorem 8.2.10. Example 8.2.14. When C = Ab is the category of abelian groups, the section functor is representable: ∼ Γ(X, F) = HomShX (ZX , F) where ZX is the constant sheaf on X with stalks Z. Recall from Corollary 3.2.8 that Hom(ZX , −) is left exact; this gives an easier proof of Proposition 8.2.13 in this case.

Theorem 8.2.15. Suppose C has products and enough injectives. Then ShX (C) has enough injectives.

Proof. Given x ∈ X and an object A ∈ A, let x∗A denote the skyscraper sheaf at x with coefficients A: ( A, x ∈ U x∗A(U) = 0, x 6∈ U.

Then Hom(G, x∗A) consists of collections of morphisms G(U) → A in C for each U containing x which are compatible with the restrictions G(U) → G(V ) for V,→ U. By the universal property of stalks (they are limits), this is the same as a map Gx → A. Thus ∼ HomShX (C)(G, x∗A) = HomC(Gx,A).

If A is an injective of C, this shows that x∗A is an injective of ShX (C). Moreover, since Hom commutes with products (Theorem 3.2.10), we have ! Y ∼ Y ∼ Y HomShX (C) G, x∗Ax = HomShX (C)(G, x∗Ax) = HomC(Gx,Ax) x∈X x∈X x∈X

for any family of objects (Ax)x∈X in C. Since C has enough injectives, for each x ∈ X we may choose a monic ϕx : Gx ,→ Ax with Ax injective, so that the induced morphism Y G −→ x∗Ax x∈X Q is monic and x∈X x∗Ax is injective.

155 8.3 Sheaf Cohomology 8 Sheaf Theory

Thus for a category C with products and enough injectives, the right derived functors of the section functor Γ : ShX (C) → C are defined.

8.3 Sheaf Cohomology

Theorem 8.2.15 allows us to construct the right derived functors of the global section functor.

Definition. Let X be a topological space and C a category with products and enough in- jectives. Then the sheaf cohomology of X is the sequence of right derived functors of Γ: ShX (C) → C, denoted Hi(X, F) := RiΓ(X, F).

One of the main phenomena to exploit is that ShX has many acyclic objects which are not injective. This is important because of the following result.

Theorem 8.3.1. Let F be a sheaf on X and suppose L• is an acyclic resolution of F , i.e. a complex of sheaves for which each Ln is acyclic. Then for all i ≥ 0,

Hi(X,F ) ∼= Hi(Γ(X,L•)).

Proof. The statement is trivial for i = 0. Let

f f 0 → F → L0 −→0 L1 −→1 L2 → · · ·

0 i i be our acyclic resolution of F . Set K = F and for each i ≥ 1, K = im fi−1 ⊆ L . Then there are short exact sequences of sheaves

g 0 → Ki −→ei Li −→i Ki+1 → 0 where ei is the inclusion of a subsheaf and the gi satisfy ei+1 ◦ gi = fi. Since Γ is left exact (Theorem 8.2.10), each sequence

Γg 0 → Γ(X,Ki) −−→Γei Γ(X,Li) −−→i Γ(X,Ki+1)

i is exact. Note that Γei identifies Γ(X,K ) with ker Γgi, which is equal to ker Γfi since Γei is i ∼ injective. Likewise, im Γgi−1 = im Γfi−1. Therefore Γ(X,K )/ im Γgi−1 = ker Γfi/ im Γfi−1 = Hi(Γ(X,L•)). On the other hand, there is a long exact sequence

Γg 0 → Γ(X,Ki) → Γ(X,L•) −−→i Γ(X,Ki+1) → H1(X,Ki) → H1(X,Li) = 0 with a zero on the end because Li is acyclic. Hence

1 i ∼ i+1 i+1 • H (X,K ) = Γ(X,K )/ im Γgi = H (Γ(X,L )).

The same argument shows that there is an isomorphism Hj−1(X,Ki+1) ∼= Hj(X,Ki) for each j ≥ 2. In particular, H1(X,Ki) ∼= Hi+1(X,K0) = Hi+1(K,F ) so we obtain Hi+1(X,F ) ∼= Hi+1(Γ(X,L•)) for all i ≥ 0.

156 8.3 Sheaf Cohomology 8 Sheaf Theory

In fact, Theorem 8.3.1 holds in much greater generality, namely for the derived functors of a left exact functor out of an abelian category with enough injectives, but we will only need it for sheaf cohomology computations in these notes.

Example 8.3.2. Let X be a smooth, paracompact manifold and define the sheaf of rings

U 7−→ OX (U) = {f : U → R | f is smooth}.

The pair (X, OX ) is called a ringed space. A sheaf of abelian groups F on X is called an OX -module if for all U ⊆ X, F(U) is an OX (U)-module such that the module structure OX (U) × F(U) → F(U) is compatible with restrictions F(U) → F(V ). Examples of OX - k modules include the sheaves ΩX (differential k-forms), VF X (smooth vector fields), Γ(E) (sections of any vector bundle E), etc. Using this language, we have our first of several ‘vanishing theorems’:

Theorem 8.3.3. For a smooth, paracompact manifold X and any OX -module F,

Hi(X, F) = 0 for all i > 0.

Proof. We induct on i > 0. For an OX -module F, the proof of Theorem 8.2.15 shows that Q I = x∈X Fx is an injective sheaf and F ,→ I is injective. It’s easy to check that I is also an OX -module. Thus we obtain a short exact sequence of OX -modules

0 → F → I → I/F → 0.

By definition H1(X, F) = coker(Γ(X, I) → Γ(X, I/F)). Take a section s ∈ Γ(X, I/F). By S the sheaf axioms and paracompactness, we may choose a locally finite open cover Ui = X and sections ti ∈ I(Ui) such that ti 7→ s|Ui . Fix a partition of unity fi subordinate to the Ui, i.e. functions fi ∈ OX (X) such that for each Ui, X fi|XrUi = 0 and fj ≡ 1. j

Then fiti extends to a global section of I vanishing away from Ui, so by the sheaf axioms, P P the section t = j fjtj maps to s = j fjs. This shows that Γ(X, I) → Γ(X, I/F) is surjective, so H1(X, F) = 0. The inductive step follows from analyzing the same short exact sequence, which gives Hk(X, F) ∼= Hk−1(X, I/F). In fact, the proof of Theorem 8.3.3 shows:

Corollary 8.3.4. If F is a flasque sheaf on X, then Hi(X, F) = 0 for all i > 0.

This shows that flasque sheaves are acyclic, so we may compute Hi(X, G) using a flasque resolution of G. In general, cohomology is hard to compute from the derived functors definition. However, we may begin by recovering ordinary (singular) cohomology with coefficients:

157 8.3 Sheaf Cohomology 8 Sheaf Theory

Theorem 8.3.5. Let X be a topological space, A an abelian group and AX the constant sheaf on X with stalks A. Then i ∼ i H (X,AX ) = H (X; A) where Hi(X; A) denotes the ordinary cohomology of X with coefficients in A.

Proof. For each open set U ⊆ X, let Cn(U; A) denote the A-module spanned by singular simplices on U and let ∂ : Cn(U; A) → Cn−1(U; A) denote the simplicial boundary map. n Dualizing gives us the modules of singular cosimplices C (U; A) := HomA(Cn(U; A),A) with coboundary map given by the adjoint, d = ∂∗ : Cn(U; A) → Cn+1(U; A). The assignment U 7→ Cn(U; A) defines a presheaf Cn on X for each n ≥ 0. In fact by construction, for any inclusion V,→ U, the restriction map Cn(U; A) → Cn(V ; A) is surjective, so Cn is a flasque presheaf on X. Note that Cn satisfies the gluing axiom of a sheaf, but it is not separated. However, sheafifying a flasque presheaf which satisfies the gluing axiom yields a flasque sheaf in general, so setting Cn := (Cn)sh, we get a complex of flasque sheaves C•. The differential n n+1 n n+1 d : C → C is induced by the coboundary maps dU : C (U; A) → C (U; A). In fact, n n n one can show that C is isomorphic to the quotient sheaf C /C0 , where

n n C0 (U) = {ϕ ∈ C (U) | there exists an open cover U = {Ui} of U such that ϕ|Ui ≡ 0}.

To verify that C• is an exact complex, it suffices to compute exactness on stalks. On a neighborhood of each point, however, this sequence becomes C•(U) which is exact by construction in algebraic topology. Note that ker(d : C0 → C1) is precisely the constant • sheaf AX , so C is a flasque resolution of X. Since flasque sheaves are acyclic, Theorem 8.3.1 implies that i ∼ i • H (X,AX ) = H (Γ(X, C )) for all i ≥ 0. Finally, singular cohomology is computed by the complex of abelian groups C•(X) but we have a short exact sequence of complexes of presheaves

• • • 0 → C0 → C → C → 0

i i • ∼ i • i so to prove H (X; A) = H (Γ(X,C )) = H (Γ(X, C )) = H (X,AX ), it remains to show that • U C0 is acyclic. Fix an open cover U of X and let Cn (U; A) denote the submodule of Cn(U; A) n U spanned by simplices lying in an element of U. Set CU (U; A) = HomA(Cn (U; A),A) and let • CU be the resulting complex of presheaves on X. There is a natural morphism of presheaves n n n C → CU which is surjective; call its kernel CU,0. Then we can see that

n n C0 = lim CU,0 −→ where the limit is taken over all open covers U of X. This identification is compatible with • • differentials, so we get C0 = lim CU,0. By a result in algebraic topology, there exists a cover −→ • U for which CU,0 is acyclic, which finishes the proof. Now the strategy for computing general sheaf cohomology is clear: we may take an in- jective resolution of AX and compute the cohomology of the resulting complex. Of course,

158 8.3 Sheaf Cohomology 8 Sheaf Theory

different ways of resolving a constant sheaf can give different ways of computing cohomol- ogy, and these observations will produce beautiful comparisons between different flavors of cohomology theory. k Let ΩX be the sheaf of differential k-forms on a smooth manifold X. This defines a sheaf • L∞ k of graded rings (differential graded algebras) ΩX = k=0 ΩX . Explicitly, there is a complex

0 d 1 d 2 d OX = ΩX −→ ΩX −→ ΩX −→· · · where d is the exterior derivative.

• Definition. The sheaf cohomology of ΩX is called the de Rham cohomology of X, written

i i • HdR(X) := H (Γ(X, ΩX )). Theorem 8.3.6. For a smooth, paracompact manifold X, the complex of sheaves

i 0 d 1 d 2 d 0 → RX −→ ΩX −→ ΩX −→ ΩX −→· · · is acyclic, where RX denotes the constant sheaf on X with stalks R, and i is the inclusion of constant functions. Proof. It is enough to compute this on stalks, where we have for each x ∈ X a coordinate ∼ n i n chart U = R . By Poincar´e’s Lemma in algebraic topology, HdR(R ) = 0 for all i > 0, so i HdR(U) = 0 for all i > 0 as well. Taking direct limits commutes with cohomology, so

i • i • i H (X;ΩX,x) = lim H (U;ΩX (U)) = lim HdR(U) = 0. −→ −→

Corollary 8.3.7. For a smooth manifold X, there is an isomorphism

• ∼ • HdR(X) = H (X, RX )

where RX is the constant sheaf. 0 1 2 Proof. Theorem 8.3.6 shows that ΩX → ΩX → ΩX → · · · is a deleted acyclic resolution of RX , so this follows from the definition of sheaf cohomology as the right derived functors of Γ(X, RX ). Corollary 8.3.8 (De Rham’s Theorem). For a smooth manifold X, there is an isomorphism between de Rham cohomology and singular cohomology:

• ∼ • HdR(X) = H (X; R). Proof. Apply Corollary 8.3.7 and Theorem 8.3.5. Theorem 8.3.9 (Leray). If F is a sheaf on X which is acyclic for an open cover U, then the maps Hˇ p(U,F ) −→ Hp(X,F ) are isomorphisms for all p ≥ 0.

159 8.3 Sheaf Cohomology 8 Sheaf Theory

Proof. We prove this by induction on p. For p = 0, this is clear since both Hˇ 0(U,F ) and H0(X,F ) are Γ(X,F ). Now assume Hˇ j(U,G) → Hj(X,G) is an isomorphism for all j ≤ p and all acyclic sheaves G. For our given F , take a short exact sequence of sheaves

0 → F → E → Q → 0

i where E is injective. Set U = Ui0,...,ip . Then by the hypothesis on F , H (U, F |U ) = 0 for all i i > 0. Since E is injective, and thus flasque, we have H (U, E|U ) = 0 as well. Thus in the long exact sequence induced by the above sequence, we can see

i i i+1 0 = H (U, E|U ) → H (U, Q) → H (U, F ) = 0.

i This shows H (U, Q) = 0 for all i ≥ 1, so by taking the product over all such Ui0,...,ip , we conclude that Q is acyclic for U. Therefore, the sequence of complexes of sheaves

0 → C•(U,F ) → C•(U,E) → C•(U,Q) → 0

is exact and induces a long exact sequence

· · · → Hˇ p−1(U,E) → Hˇ p−1(U,Q) → Hˇ p(U,F ) → Hˇ p(U,E) → · · ·

Now since E is flasque, Hˇ p−1(U,E) = Hˇ p(U,E) = 0. Therefore we have a diagram with exact rows 0 Hˇ p−1(U,Q) Hˇ p(U,F ) 0

0 Hp−1(X,Q) Hp(X,F ) 0

By induction, the left column is an isomorphism, so it follows that Hˇ p(U,F ) → Hp(X,F ) is an isomorphism.

i Corollary 8.3.10. For a smooth, compact manifold X, HdR(X) is finite dimensional for all i ≥ 0.

Proof. Endow X with a Riemannian metric. Then every point in X admits a neighborhood which is geodesically convex, meaning any two points in the neighborhood lie on a geodesic which is contained in the neighborhood. Since X is compact, we may choose a finite cover U = {U1,...,Um} consisting of these geodesically convex open sets. Note that for any i1, . . . , ir ∈ {1, . . . , m}, the intersection Ui1,...,ir = Ui1 ∩ · · · ∩ Uir is also geodesically convex. A geodesically convex set has vanishing de Rham cohomology, so de Rham’s theorem says that Hi(U , | ) ∼= Hi (U ) = 0 for all i > 0. Therefore is acyclic for the i1,...,ir RX Ui1,...,ir dR i1,...,ir RX i ˇ i i cover U, so by Leray’s theorem, H (X, RX ) = H (U, RX ). Now observe that each C (U, RX ) in the Cechˇ complex is a finite product of one-dimensional vector spaces, hence itself finite ˇ i dimensional, so H (U, RX ) is finite dimensional.

160 8.4 Cechˇ Cohomology 8 Sheaf Theory

8.4 Cechˇ Cohomology

For a general space X, the sheaf cohomology groups may be quite difficult to compute – how does one produce a flasque or even injective resolution resolution in general? Fortunately, there is another construction of sheaf cohomology which, though cumbersome to define, is much more amenable to computation. We motivate the definition of Cechˇ cohomology with a classical problem originally studied by Mittag-Leffler. Let X be a Riemann surface, that is, a 2-manifold admitting a complex structure with holomorphic transition functions. We do not assume X to be closed. Suppose E is a closed, discrete subset of X and for each point a ∈ E, we are given a function za : Ua → C on some neighborhood Ua ⊆ X of a such that za(a) = 0. Consider a function

−m −m+1 −1 pa(za) = α−mza + α−m+1za + ... + α−1za , αj ∈ C.

−1 That is, pa is a polynomial in za , or a Laurent polynomial on Ua centered at a. The Mittag- Leffler problem is to find a meromorphic function f : X → C such that f is holomorphic on X r E and for all a ∈ E, the function f − pa has a removable discontinuity at a. Then pa will be the principal part of f on Ua. In short, the Mittag-Leffler problem asks us to extend meromorphic functions defined on open sets in X to a meromorphic function on the whole Riemann surface. The following restatement will be useful for later generalization. Let U = {Ui}i∈I be an open cover of X and suppose {fi : Ui → C} is a collection of meromorphic functions such that each fi is either holomorphic on Ui or has a single pole ai ∈ Ui, with ai 6∈ Uj for any j 6= i. The Mittag-Leffler problem is then to find a meromorphic function f : X → C such that for each i ∈ I, f|Ui − fi is holomorphic. First notice that if the fi agree on all overlaps Ui ∩ Uj, then the sheaf condition on M (the sheaf of meromorphic functions on X) guarantees that there is a global meromorphic function f ∈ M(X) so that f|Ui − fi = 0 for all i, a much stronger conclusion than the Mittag-Leffler problem asks for. Thus in some sense, the problem is to study how far one can lift local meromorphic functions that do not glue together on overlaps. In any case, one way to find such a function f : X → C in the previous paragraph is to find a family

{hi : Ui → C} of holomorphic functions on each Ui such that (fi + hi)|Ui∩Uj = (fj + hj)|Ui∩Uj for all i, j. This can be rewritten as

fi|Ui∩Uj − fj|Ui∩Uj = hj|Ui∩Uj − hi|Ui∩Uj .

Set tij = fi|Ui∩Uj − fj|Ui∩Uj . Then if the above equation is satisfied, we have tij ∈ O(Ui ∩ Uj), where O is the sheaf of holomorphic functions on X and

tjk − tik + tij = 0.

Thus we want to find holomorphic functions hi ∈ O(Ui) satisfying:

(1) tij = hj − hi on Ui ∩ Uj (the boundary condition); and

(2) tjk − tik + tij = 0 on Ui ∩ Uj ∩ Uk (the cycle condition).

This generalizes to any space X with open cover U = {Ui}i∈I .

161 8.4 Cechˇ Cohomology 8 Sheaf Theory

Q Definition. Let F be a presheaf on X. A family of sections (tij) ∈ i,j F (Ui ∩ Uj) is called ˇ a Cechˇ 1-cocycle if for all i, j, k, tjk − tik + tij = 0 on Ui ∩ Uj ∩ Uk. The group of Cech 1-cocycles is written Zˇ1(U,F ). Q Definition. A family (tij) ∈ F (Ui ∩ Uj) is called a Cechˇ 1-coboundary if there is a Q i,j family (hi) ∈ i F (Ui) such that tij = hj − hi on each Ui ∩ Uj. This forms a subgroup of Zˇ1(U,F ) which is denoted Bˇ1(U,F ).

Definition. The first Cechˇ cohomology group of the cover U with coefficients in F is the quotient group Hˇ 1(U,F ) := Zˇ1(U,F )/Bˇ1(U,F ).

Now to solve the Mittag-Leffler problem, it’s enough to show that for a Riemann surface X with cover U, Hˇ 1(U, O) = 0 where O is the sheaf of holomorphic functions on X. Note that these Cechˇ cohomology groups depend on the cover U. To package together all of the information about holomorphic functions on covers of X, we introduce the notion of a refinement of covers.

Definition. Let U = {Ui}i∈I and V = {Vj}j∈J be open covers of X. We say V is a refine- ment of U if there exists a function τ : J → I such that for all j ∈ J, Vj ⊆ Uτ(j). Notice that such a refinement τ induces a map

τ 1 : Zˇ1(U,F ) −→ Zˇ1(V,F )
(tik)i,k∈I tτ(j)τ(`)|Vj ∩V` j,`∈J .

Lemma 8.4.1. If τ is a refinement from U to V, then τ 1(Bˇ1(U,F )) ⊆ Bˇ1(V,F ).

Thus there is an induced map on Cechˇ cohomology groups, Hˇ 1(U,F ) → Hˇ 1(V,F ) and one can check that this does not depend on the choice of map τ : J → I.

Definition. The first Cechˇ cohomology of X with coefficients in F is the direct limit

Hˇ 1(X,F ) = lim Hˇ 1(U,F ) −→ over all open covers U of X, ordered by refinement.

Lemma 8.4.2. For any sheaf F on a Riemann surface X, the maps

Hˇ 1(U,F ) −→ Hˇ 1(X,F )

are injective for every open cover U of X.

ˇ 1 Proof. Let U = {Ui}i∈I be an open cover of X. If g = (gik)i,k∈I ∈ H (U,F ) becomes 0 in ˇ 1 H (X,F ), this means there is some open cover V = {Vj}j∈J which is a refinement of U, 0 0
ˇ1 0 say by τ : J → I, such that g = (gj`)j,` = gτ(j)τ(`)|Vj ∩V` j,` lies in B (V,F ). Since g is a

162 8.4 Cechˇ Cohomology 8 Sheaf Theory

Q 0 1-coboundary, there is a family (hj) ∈ j F (Vj) such that gj` = hj − h` on Vj ∩ V`. For a fixed index i ∈ I, observe that

0 gj` = gτ(j)τ(`)

= gτ(j)i + giτ(`) by the cocycle condition

= giτ(`) − giτ(j) by the cocycle condition again

= hj − h` on Vj ∩ V`. This can be rewritten as

giτ(j) + hj = giτ(`) + h` on Ui ∩ Vj ∩ V`.

Consider the cover {Ui ∩ Vj}j∈J of Ui. By the sheaf axioms for F , there exists a section ti ∈ F (Ui) such that for each Vj, ti|Ui∩Vj = (giτ(j) + hj)|Ui∩Vj . Now let i, k ∈ I be arbitrary and fix some j ∈ J. Then

gik = giτ(j) + gτ(j)k = giτ(j) − gkτ(j)by the cocycle condition

= (ti − hj) − (tk − hj) by the above

= ti − tk on Ui ∩ Uk ∩ Vj.

Since j was arbitrary and {Ui ∩Uk ∩Vj}j∈J covers Ui ∩Uk, the sheaf axioms imply gik −ti −tk ˇ1 ˇ 1 on Ui ∩ Uk. Hence gik ∈ B (U,F ) so g = 0 in H (U,F ). Theorem 8.4.3 (Mittag-Leffler). For a non-compact Riemann surface X with sheaf of holo- morphic functions O, Hˇ 1(U, O) = 0 for every open cover U of X. Proof. We will show that Hˇ 1(X, O) = 0 in this situation, so Lemma 8.4.2 implies that Hˇ 1(U, O) = 0.

Definition. Let U = {Ui}i∈I be an open cover of X and let F be a sheaf of abelian groups on X. The Cechˇ complex of F with respect to U is the cochain complex C•(U,F ) defined by p Y C (U,F ) = F (Ui0 ∩ · · · ∩ Uip )

i0,...,ip∈I with differential

d : Cp(U,F ) −→ Cp+1(U,F ) p+1 ! X α 7−→ (−1)kα | i0,...,ik−1,ik+1,...,ip+1 Ui0,...,ip+1 k=0

where Ui0,...,ip+1 = Ui0 ∩ · · · ∩ Uip+1 . Lemma 8.4.4. d2 = 0; that is, C•(U,F ) is a cochain complex. Definition. The pth Cechˇ cohomology with respect to an open cover U of a space X with coefficients in a sheaf F is the pth cohomology of the Cechˇ complex: ˇ p p • HU (X, F) := H (C (U, F)).

163 8.4 Cechˇ Cohomology 8 Sheaf Theory

ˇ 0 0 Lemma 8.4.5. For any sheaf F and cover U of X, HU (X, F) = H (X, F) = Γ(X, F). ˇ 0 0 1 0 Proof. By definition, HU (X, F) = ker(d : C (U, F) → C (U, F)). For α = (αi) ∈ C (U, F), we have dα = (αi − αj)i,j which is zero if and only if αi = αj on Ui ∩ Uj for all i, j. Thus ker d = Γ(X, F).

0 0 0 Suppose U is a refinement of U, that is, U = {Uj}j∈J is a cover of X and there is 0 a function λ : J → I such that for all j ∈ J, Uj ⊆ Uλ(j). Then there is a chain map C•(U 0, F) → C•(U, F) given by

Cp(U 0, F) −→ Cp(U, F), (α ) 7−→ (α | ). j0,...,jp λ(j0),...,λ(jp) Uj0 ∩···∩Ujp

This in turn induces maps on Cechˇ cohomology:

ˇ p ˇ p HU 0 (X, F) −→ HU (X, F) for all p.

Definition. The pth Cechˇ cohomology of a space X with coefficients in a sheaf F is the direct limit Hˇ p(X,F ) := lim Hˇ p(U,F ) −→ taken over all covers U ordered with respect to refinement.

Theorem 8.4.6. If X is paracompact, then there is a natural isomorphism

Hˇ p(X, F) ∼= Hp(X, F)

for all p and F.

Fix a space X, an open cover U = {Ui}i∈I and a sheaf F on X. We construct a complex of sheaves of abelian groups Cp(U,F ) on X as follows. For an open set V ⊆ X, define

p p Y C (U,F ): V 7−→ Γ(V, C (U,F )) := F (V ∩ Ui0,...,ip )

i0,...,ip∈I

where as usual Ui0,...,ip = Ui0 ∩ · · · ∩ Uip . Proposition 8.4.7. If F is a sheaf on X, then Cp(U,F ) is a sheaf for all p ≥ 0.

We give C•(U,F ) the structure of a complex of sheaves by letting

p p+1 dV : Γ(V, C (U,F )) −→ Γ(V, C (U,F ))

be the map induced from the Cechˇ differential d : Cp(U,F ) → Cp+1(U,F ) restricted to each

V ∩ Ui0,...,ip . Proposition 8.4.8. For an open cover U and a sheaf F , there is an exact sequence of sheaves

0 → F → C0(U,F ) → C1(U,F ) → · · ·

164 8.4 Cechˇ Cohomology 8 Sheaf Theory

Proof. The sheaf axioms on F imply F → C0(U,F ) is injective. To prove exactness at Cp(U,F ), we need to check that the sequence of stalks

p−1 α p β p+1 C (U,F )x −→C (U,F )x −→C (U,F )x

p is exact for all x ∈ X. Since U covers X, we know x ∈ Uj for some Uj. Take fx ∈ C (U,F )x p and let V be a neighborhood of x and f ∈ Γ(V, C (U,F )) with f|x = fx. We may assume V ⊆ Uj. Then for each p ≥ 1,

p p p−1 θ : C (U,F )x −→ C (U,F )x

fx 7−→ fj,i0,...,ip−1 |x

is well-defined and independent of the choice of V and Uj. Now with the same fx and f as above, we have

p+1 θ (df)x = (df)j,i0,...,ip |x p ! X = f − (−1)kf i0,...,ip j,i0,...,ik−1,ik+1,...,ip k=0 x p = fx − d(θ fx).

p+1 p p Hence θ d + dθ is the identity on C (U,F )x, which shows the identity map on this stalk is chain homotopic to the zero map and thus ker β = im α, proving exactness.

Definition. For a sheaf F on X, the exact sequence

0 → F → C0(U,F ) → C1(U,F ) → · · ·

is called the Cechˇ resolution of F with respect to the cover U.

Theorem 8.4.9. For any sheaf F on X and any open cover U of X, there is a natural map

Hˇ p(U,F ) −→ Hp(X,F )

for all p ≥ 0 which is compatible with refinement. In particular, there is a natural map

Hˇ p(X,F ) −→ Hp(X,F )

for all p ≥ 0.

Proof. Applying the global sections functor to the Cechˇ resolution of F yields

0 → F (X) → Cˇ0(U,F ) → Cˇ1(U,F ) → · · ·

Clearly the cohomology groups of this sequence are precisely the Cechˇ cohomology groups Hˇ p(U,F ). Therefore there is a chain map

165 8.4 Cechˇ Cohomology 8 Sheaf Theory

0 F C0(U,F ) C1(U,F ) ···

id

0 F E0 E1 ···

for any injective resolution E• of F . This induces the maps on cohomology, Hˇ p(U,F ) → Hp(X,F ), and one can check that they commute with the maps coming from refinements of open covers. The second statement follows.

The sheafification functor PreshX → ShX (Theorem 8.1.15) can alternatively be con- structed using the Cechˇ resolution. In fact, this construction works in a much more general context known as Grothendieck topology. Let F be a presheaf on a space X, U = {Ui} an open cover of X and n o ˇ 0 Y H (U,F ) = f = (fi) ∈ F (Ui): fi|Ui∩Uj = fj|Ui∩Uj for all i, j .

For an open set U ⊆ X, let

ˇ 0 ˇ 0 0 H (U, F |U ) = lim H (U ,F |U ) −→

0 # ˇ 0 where the limit is over all open covers U of U. Define the set F (U) = H (U, F |U ). For ˇ 0 an inclusion of open sets V,→ U in X, there is a natural restriction map H (U, F |U ) → ˇ 0 # H (V,F |V ) coming from restriction of an open cover. This makes U 7→ F (U) into a presheaf on X. Moreover, Proposition 8.4.8 shows that F → F # is injective.

Definition. We call a presheaf F on X a separated presheaf if it satisfies the first sheaf S axiom; that is, if U = Ui is an open cover and there are sections s, t ∈ F (U) such that s|Ui = t|Ui for all Ui, then s = t. Theorem 8.4.10. Let F be a presheaf on X. Then

(1) F # is a separated presheaf on X.

(2) If F is a separated presheaf, then F # is a sheaf. In particular, F ## is always a sheaf on X.

(3) F ## is a sheafification of F .

# Proof. (1) Suppose s, t ∈ F (U) and there is an open cover {Vj} of U such that s|Vj = t|Vj # for all Vj. We must show s = t in F (U). Since they are elements of a direct limit, s and t have representatives ˇ 0 sU = (si), tU = (ti) ∈ H (U,F |U ) on some open cover U of U – a priori there may be two different covers supporting s and t, but after a common refinement of the covers we may assume this takes place on just a single cover. By hypothesis, there is an open cover Wj = {Wjk}k of each Vj that is a refinement of

{Vj ∩ Ui}i such that si|Wjk = ti|Wjk for all k. Then W = {Wjk}j,k is an open cover of U on

166 8.4 Cechˇ Cohomology 8 Sheaf Theory

ˇ 0 which si|W = ti|W on every element. Therefore the images of sU and tU in lim H (U,F |U ) jk jk −→ are equal, so s = t as required. (2) Now assume F itself is separated; we must show that F # satisfies the second sheaf axiom. As an intermediate step, we claim that for any open cover U = {Ui}i∈I of U, the map ˇ 0 ˇ 0 H (U,F |U ) −→ H (U, F |U ) ˇ 0 is injective (this is conditioned on F being separated). Notice that if s, t ∈ H (U,F |U ) ˇ 0 become equal in the direct limit H (U, F |U ), then there exists a cover V = {Vj}j∈J of U such that s|Vj = t|Vj for all j ∈ J. We may assume V is a refinement of U, say by a map ˇ τ : J → I. Then sτ(j)|Vj = tτ(j)|Vj for all j ∈ J. Since s and t are Cech 0-cocycles, we have

si|Ui∩Uτ(j) = sτ(j)|Ui∩Uτ(j) and ti|Ui∩Uτ(j) = tτ(j)|Ui∩Uτ(j) for all i, j. Since Vj ⊆ Uτ(j) for all j, this means

si|Ui∩Vj = ti|Ui∩Vj

for all i, j. Fixing i ∈ I, the collection {Ui ∩Vj}j is an open cover of Ui and si|Ui∩Vj = ti|Ui∩Vj , ˇ 0 so by the separated condition, si = ti on Ui. This implies s = t in H (U,F |U ), proving the claim. # Now take si ∈ F (Ui) for some cover U = {Ui}i∈I of U such that si|Ui∩Uj = sj|Ui∩Uj for # all i, j ∈ I. We must construct a section s ∈ F (U) restricting to si on each Ui. Each si is ˇ 0 represented by some sVi = (sik) ∈ H (Vi,F |Ui ) for an open cover Vi = {Vik}k of Ui. Then V = {Vik}i,k is an open cover of U. Fixing i, j ∈ I, consider the cover W = {Vik ∩ Vj`}k,` of Ui ∩ Uj. The sections

ˇ 0 s˜i = (sik|Vik∩Vj` ), s˜j = (sj`|Vik∩Vj` ) ∈ H (W,F |Ui∩Uj )

ˇ 0 become equal in H (Ui ∩Uj,F |Ui∩Uj ) by assumption, so because F is separated, the previous ˇ 0 paragraph showss ˜i =s ˜j in H (W,F |Ui∩Uj ). Therefore thes ˜i give a well-defined element ˇ 0 # s ∈ H (U, F |U ) restricting to si on each Ui. Hence F is a sheaf. (3) Suppose F → G is a morphism of presheaves, where G is a sheaf. This determines a chain map C•(U,F ) −→ C•(U,G) for any open cover U of X. Hence there is a map on Cechˇ cohomology

Hˇ p(U,F ) −→ Hˇ p(U,G) for all p ≥ 0, which is compatible with refinement. In particular, there is a map

Hˇ 0(X,F ) −→ Hˇ 0(X,G).

ˇ 0 ˇ 0 Let U ⊆ X be an open set. Then there is a map H (U, F |U ) → H (U, G|U ) which fits into a diagram

167 8.4 Cechˇ Cohomology 8 Sheaf Theory

F (U) G(U)

∼=

ˇ 0 ˇ 0 H (U, F |U ) H (U, G|U )

The right column is an isomorphism since G is a sheaf. Thus we may defined the dashed ˇ 0 # arrow H (U, F |U ) → G(U) which induces a morphism of presheaves F → G through which F → G factors. Repeating the argument shows that F → G further factors through a morphism F ## → G. Uniqueness follows from a similar argument as the one in the proof of Theorem 8.1.15. Changing topics, we next describe some useful applications of sheaf cohomology (using Cechˇ complexes for computation). Theorem 8.4.11. For any paracompact space X and sheaf F, there exists an injective sheaf I on X such that Hn(X, F) ∼= Hn−1(X, I/F) for all n. Let us illustrate this for n = 1. Note that H1(X, F) = Ext1 ( , F) where is a ShX (Ab) ZX ZX 1 constant sheaf. Explicitly, elements of Ext (ZX , F) are isomorphism classes of short exact sequences of sheaves 0 → F → G → ZX → 0. ∼ Assume X is connected, so that Γ(X, ZX ) = Z. For any short exact sequence 0 → F → G → ZX → 0, we have a long exact sequence

δ 1 0 → Γ(X, F) → Γ(X, G) → Z −→ H (X, F) → · · ·

Set ε = δ(1). Then ε = 0 if and only if there is a splitting Z → Γ(X, G) of δ. There are open sets {Ui} in X such that Γ(Ui, G) → Γ(Ui, ZX ) is surjective for all i (although this need not be true for global sections). So on Ui, the short exact sequence

0 → F|Ui → G|Ui → ZUi → 0

splits, i.e. there is some σi ∈ G(Ui) mapping to 1 ∈ Z. For each i, j, set τij = σi − σj ∈ G(Ui ∩ Uj). Then τij maps to 1 − 1 = 0 in Γ(Ui ∩ Uj, ZX ), so by exactness τij ∈ F(Ui ∩ Uj). 0 Also note that each σi is unique up to adding any bi ∈ F(Ui), but if σi = σi + bi, we get 0 0 ˇ τij = τij + bi − bj. Viewing (τij) and (τij) as Cech 1-cocycles, the previous statement shows 0 ˇ 1 that [(τij)] = [(τij)] in H (X, F). Q Conversely, given a 1-cocycle (τij) in F(Ui ∩ Uj), we show how to reconstruct the extension 0 → F → G → ZX → 0. We define G by its ´espace´etale:

´et G = {(x, f, n, Ui) | x ∈ Ui, f ∈ Fx, n ∈ Z}/ ∼ ´et ∼ ´et where (x, f, n, Ui) ∼ (x, f +nτij, n, Uj) for every x ∈ Ui ∩Uj. Thus locally, G |Ui = F |Ui ×Z. Over Ui, the section σi : x 7→ (x, 0, 1,Ui) gives a splitting of δ : Γ(X, G) → Z, since

168 8.4 Cechˇ Cohomology 8 Sheaf Theory

(x, 0, 1,Ui) ∼ (x, τij, 1,Uj) whenever x ∈ Ui ∩ Uj. This defines G as a sheaf of sets on X, but with a corresponding action of F given locally by F(U) × (F(U) × Z) → F(U) × Z. In other words, G is a torsor for F on X. In general, torsors are in correspondence with cohomology classes in H1(X, F), as we have seen in this case.

Example 8.4.12. Suppose X is a manifold (possibly with extra structure), G is a topological group and P → X is a principal G-bundle – this means that over certain open sets U ⊆ X, there is a G-equivariant isomorphism

∼= P |U U × G

U

making the diagram commute. The data of a principal bundle is equivalent to specifying an open cover {Ui} of X and “transition functions” τij : Ui ∩ Uj → G (which preserve the extra structure on the manifold). Explicitly, if x ∈ Ui ∩ Uj then (x, g) 7→ (x, gτij(x)) is an isomorphism on (Ui ∩Uj)×G as a trivial bundle over Ui ∩Uj. It follows from this description −1 ˇ that τji = τij and τijτjk = τik for all i, j, k. So (τij) is a Cech cocycle for some sheaf, but which one? Let ΓG be the sheaf on X given locally by

ΓG(U) = {σ : U → U × G | σpσ = idU }.

Then the above description shows that principal G-bundles up to isomorphism are in one- 1 to-one correspondence with elements of H (X, ΓG).

Example 8.4.13. Let F be a field and G = GLn(F ). Then principal G-bundles are in one-to-one correspondence with rank n vector bundles over X, so the isomorphism classes 1 of rank n vector bundles are determined by classes in H (X, ΓGLn(F )). If F = R and n = 1, × so that GL1(R) = R , we have a short exact sequence of abelian groups

exp × sgn 0 → R −−→ R −−→ Z/2Z → 0.

1 2 By de Rham’s theorem (8.3.8), H (X, ΓR) = H (X, ΓR) = 0, so the long exact sequence in 1 ∼ 1 sheaf cohomology implies H (X, ΓR× ) = H (X; Z/2Z). That is, isomorphism classes of line bundles (rank 1 vector bundles) on X are in one-to-one correspondence with cohomology classes H1(X; Z/2Z). The class ω(L) ∈ H1(X; Z/2Z) determined by a line bundle L → X is called the Stiefel-Whitney class of L. A similar calculation for F = C and the short exact sequence

exp × 0 → Z −→ C −−→ C → 0

1 ∼ shows that complex line bundles up to isomorphism are determined by classes in H (X, ΓC× ) = H2(X; Z), called Chern classes, denoted c(L).

169 8.5 Direct and Inverse Image 8 Sheaf Theory

8.5 Direct and Inverse Image

Let ShX denote the category of sheaves (of abelian groups) on a topological space X. For a continuous map f : X → Y , there are various functors between ShX and ShY which may illuminate the structure of these categories. The easiest to define is the direct image.

Definition. For a morphism f : X → Y and a presheaf F on TopX , define the direct op image (or pushforward) of F along f to be the presheaf f∗F : TopY → Ab sending V 7→ −1 (f∗F )(V ) := F (f (V )) for all open V ⊆ Y .

Proposition 8.5.1. For any continuous map f : X → Y , f∗ is a functor ShX → ShY . Proof. Suppose ϕ : F → G is a morphism of presheaves on X. Then for all open V ⊆ Y , −1 −1 define (f∗ϕ)V :(f∗F )(V ) → (f∗G)(V ) to be the map ϕf −1(V ) : F (f (V )) → G(f (V )). S −1 This shows that f∗ is a functor. To finish, take a covering V = Vi in Y , set U = f (V ) and S −1 consider the covering U = Ui where Ui = f (Vi). Suppose s, t ∈ (f∗F )(V ) = F (U) such that s|Vi = t|Vi for all Vi. This means s|Vi = t|Vi in (f∗F )(Vi) = F (Ui), so by the first sheaf axiom for F , s = t in F (U) = (f∗F )(V ). On the other hand, suppose si ∈ (f∗F )(Vi) = F (Ui)

for all i such that si|Vi∩Vj = sj|Vi∩Vj for all Vi,Vj. This means si and sj are equal in each −1 (f∗F )(Vi ∩ Vj) = F (f (Vi ∩ Vj)) = F (Ui ∩ Uj) so by the second sheaf axiom on F , there is

a section s ∈ F (U) restricting to si on each Ui. That is, s ∈ (f∗F )(V ) such that s|Vi = si in each (f∗F )(Vi). Hence f∗F is a sheaf. Example 8.5.2. If i : {x} → X is a point of X, then any sheaf on {x} is an abelian group, say A, and i∗A = Ax is the skyscraper sheaf on X supported at x with stalk A. This explains the usual notation x∗A for skyscraper sheaves.

Example 8.5.3. Consider the inclusion j : D2 ,→ R2 and let F be the sheaf of locally 2 2 constant functions on D with values in S. Then for all open sets U ⊆ R ,(j∗F )(U) = 2 2 ×n F (U ∩ D ). If U ∩ D has n connected components, then (j∗F )(U) = S . Example 8.5.4. Let p : X → Y be a finite-sheeted covering space of degree n > 1 and let F = FS be the sheaf of locally constant functions on Y with values in S, |S| > 1. Then for −1 `n V ⊆ Y sufficiently small, f (V ) = i=1 Ui for disjoint neighborhoods U1,...,Un which are ×n each homeomorphic to V . Thus (p∗F )(V ) = S . This shows that for each y ∈ Y , the stalk at y of F is ×n (p∗F )y = S . ×n Another sheaf on Y with stalk S is the locally constant sheaf G = FS×n . However, p∗F is not isomorphic to G, as we have

×n (p∗F )(Y ) = S 6= S = G(Y ).

Proposition 8.5.5. For any map f : X → Y , the functor f∗ : ShX → ShY is left exact.

Proof. Suppose 0 → F → G → H → 0 is a short exact sequence in ShX . By Theorem 8.2.10, the sequence 0 → F (U) → G(U) → H(U)

170 8.5 Direct and Inverse Image 8 Sheaf Theory

is exact for all open sets U ⊆ X. In particular, if V ⊆ Y is open, then the above applied to U = f −1(V ) shows that

0 → (f∗F )(V ) → (f∗G)(V ) → (f∗H)(V ) is exact. Passing to stalks proves the statement.

This allows us to define the right derived functors of f∗. • Definition. Let E be an injective resolution of a sheaf F ∈ ShX and suppose f : X → Y is a continuous map. The higher direct images of f are the right derived functors of the direct image functor: n n • R f∗F := H (f∗E ) where for a complex of sheaves G• on Y and n ≥ 0, Hn(G•) denotes the quotient sheaf ker(Gn → Gn+1)/ im(Gn−1 → Gn).

f g Proposition 8.5.6. Suppose X −→ Y −→ Z are continuous maps. Then (g ◦ f)∗ = g∗ ◦ f∗.

Lemma 8.5.7. Let f : X → Y be a continuous map, V = {Vj} an open cover of Y and −1 U = {f (Vj)} the induced open cover of X. Then for all sheaves F on X and all p ≥ 0, ˇ p ˇ p H (U,F ) = H (V, f∗F ).

−1 Proof. Set Vj0,...,jp = Vj0 ∩ · · · ∩ Vjp and Uj0,...,jp = f (Vj0,...,jp ). Then (f∗F )(Vj0,...,jp ) = p p F (Uj0,...,jp ) so it follows that C (U,F ) = C (V, f∗F ) for all p ≥ 0. Hence the cohomologies of these complexes are the same. Example 8.5.8. Let i : X,→ Y be a closed embedding and let F be a sheaf on X. Then for any y ∈ Y , the stalk of i∗F at y is ( Fx, y = i(x) ∈ i(X) (i∗F )y = 0, y 6∈ i(X). In particular, this proves:

Lemma 8.5.9. If i : X,→ Y is a closed embedding, then i∗ : ShX → ShY is an exact functor. Therefore, if E• is an injective resolution of a sheaf F on X, then for a closed embedding • i : X,→ Y the cohomology of i∗F may be computed via the resolution i∗E :

p p p • H (X,F ) = H (Y, i∗F ) = H (Γ(Y, i∗E )). Let f : X → Y be any continuous map. There is also an inverse image functor f ∗ : ShY → ShX which is a little harder to construct, since f(U) need not be open in Y for all open U ⊆ X. Definition. Let f : X → Y be a continuous map, F a sheaf on Y and define a presheaf f −1F by (f −1F )(U) = lim F (V ) −→ where the direct limit is taken over all open sets V ⊆ Y containing f(U). The inverse image (or pullback) of F along f, denoted f ∗F , is the sheafification of f −1F .

171 8.5 Direct and Inverse Image 8 Sheaf Theory

Remark. The inverse image sheaf may also be defined on the level of ´etalespaces as follows. ∗ Let EF → Y be the ´etalespace of F . Then f F is the sheaf of sections of the map E = Ef ∗F → X whose total space is the pullback in the diagram

E X

f

EF Y

Example 8.5.10. If f : X → Y is an open map, then for all U ⊆ X,(f −1F )(U) = F (f(U)) and this already defines a sheaf, so f −1F = f ∗F .

Example 8.5.11. Let i : {x} ,→ X be the inclusion of a point in X and let F be a sheaf on ∗ X. Then i F = Fx, the (constant sheaf on {x} given by the) stalk of F at x. More generally: Proposition 8.5.12. For any continuous map f : X → Y , sheaf F on Y and point x ∈ X, ∗ ∼ there is a natural isomorphism (f F )x = Ff(x). Proof. Let f −1F be the presheaf U 7→ lim F (V ) as in the definition of inverse image, set −→ y = f(x) and suppose U ⊆ X is an open neighborhood of x and V ⊆ Y is an open set in Y containing f(U). Then y lies in every such V so we get morphisms F (V ) → Fy which assemble into a morphism out of the direct limit,

−1 λU :(f F )(U) −→ Fy.

−1 Taking the limit over all such neighborhoods U of x gives a morphism λ :(f F )x → Fy. We claim this is an isomorphism. For sy ∈ Fy, choose a representative sV ∈ F (V ) on some open neighborhood V of y. Then U = f −1(V ) is an open neighborhood of x in X such that f(U) ⊆ V . Hence there are maps

−1 λU F (V ) → (f F )(U) −→ Fy

−1 whose composition is the stalk map F (V ) → Fy. Thus λU (f sV ) maps to sy in the stalk at y, so λ is surjective. −1 On the other hand, if t1, t2 ∈ (f F )x with λ(t1) = λ(t2), then there is some neighborhood −1 U of x such that t1 = tU,1|x and t2 = tU,2|x for sections tU,1, tU,2 ∈ (f F )(U). (A priori, these happen on different open neighborhoods of x but we may pass to the intersection.) In turn these tU,1, tU,2 are represented by sections sV,1, sV,2 ∈ F (V ) for some open set V containing f(U) and the fact that λ(t1) = λ(t2) ensures that sV,1|y = sV,2|y in Fy. Thus by 0 shrinking V , we have sV,1|V 0 = sV,2|V 0 for some open neighborhood V ⊆ V containing y. Let 0 −1 0 0 −1 U = f (V ) ⊆ U. Then x ∈ U and tU,1|U 0 = tU,2|U 0 so t1 = t2 in (f F )x. Thus λ is an isomorphism. −1 ∗ −1 ∗ Finally, the sheafification map f F → f F induces a map on stalks (f F )x → (f F )x −1 ∗ ∼ such that the composition (f F )x → (f F )x = Fy is precisely the isomorphism λ. This ∗ ∼ shows (f F )x = Fy as desired.

172 8.5 Direct and Inverse Image 8 Sheaf Theory

∗ Proposition 8.5.13. For any continuous map f : X → Y , f is a functor ShY → ShX . Proof. By definition, f ∗F is a sheaf on X for all sheaves F on Y , so we need only prove naturality. Let ϕ : F → G be a morphism of sheaves on Y and fix an open set U ⊆ X. Then the maps ϕV : F (V ) → G(V ) over all open V ⊆ Y containing U assemble into a map

(f −1F )(U) = lim F (V ) −→ lim G(V ) = (f −1G)(U). −→ −→ Passing to the sheafification, we get a map f ∗ϕ : f ∗F → f ∗G.

∗ Corollary 8.5.14. For any f : X → Y , the pullback functor f : ShY → ShX is exact. Proof. Let 0 → F → G → H → 0 be an exact sequence of sheaves on Y . Then for all y ∈ Y , the sequence on stalks 0 → Fy → Gy → Hy → 0 is exact. Thus Proposition 8.5.12 says that ∗ ∗ ∗ 0 → (f F )x → (f G)x → (f H)x → 0 is exact for all x ∈ X, with f(x) = y. By definition this means 0 → f ∗F → f ∗G → f ∗H → 0 is exact. Proposition 8.5.15. For any f : X → Y and any sheaf F on Y , f ∗F is the sheaf of sections of the map EF ×Y X → X, where EF → Y is the ´etalespace of F .

∗ −1 sh Proof. Set E = EF ×Y X and let Ef ∗F be the ´etalespace of f F = (f F ) . Then −1 Ef ∗F = Ef −1F so it’s enough to show that Ef −1F = E. The maps λ :(f F )x → Ff(x) for all x ∈ X in the proof of Proposition 8.5.12 induce a map of sets

a −1 a Φ: Ef −1F = (f F )x −→ Fy = EF . x∈X y∈Y

It’s easy to check that Φ is continuous with respect to the topology on the ´etale spaces. Moreover, the diagram

Ef −1F EF

f X Y

commutes by construction, so by the universal property of pullbacks, we get a map Ψ : Ef −1F → E = EF ×Y X. Notice that the fibres of Ef −1F and E are the same and Ψ induces bijections on these fibres. Hence Ψ is a continuous bijection. Finally, the fact that Ef −1F → X and E → X are both local homeomorphisms ensures that Ψ is a homeomorphism.

Example 8.5.16. Let f : X → Y be continuous and F = FS the sheaf of locally constant functions on Y with values in S. Then EF = S × Y → Y where S has the discrete topology. By Proposition 8.5.12, Ef ∗F = EF ×Y X = (S ×Y )×Y X = S ×X, which by Theorem 8.1.12 has sheaf of sections FS → X, the sheaf of locally constant functions on X. So pullback of a locally constant sheaf is again locally constant. Example 8.5.17. Let i : X,→ Y be the inclusion of a subspace. Then i∗ is just the restriction functor F 7→ F |X .

173 8.5 Direct and Inverse Image 8 Sheaf Theory

∗ Theorem 8.5.18. For a continuous map f : X → Y , (f , f∗) is an adjoint pair of functors. Proof. Let F be a sheaf on X and G be a sheaf on Y . Then since f ∗G = (f −1G)sh, Theorem 8.1.15 gives a natural isomorphism ∗ ∼ −1 HomShX (f G, F ) = HomPreshX (f G, F ) so it suffices to construct a natural isomorphism −1 ∼ HomPreshX (f G, F ) = HomShY (G, f∗F ). Notice that for any open V ⊆ Y ,

−1 −1 −1 0 (f∗f G)(V ) = (f G)(f (V )) = lim G(V ) −→ over all open sets V 0 ⊆ Y containing f(f −1(V )). But since V itself is an open set containing f(f −1(V )), we get a map G(V ) → lim G(V 0) which is compatible with restriction. This −→ defines a natural transformation

−1 η : idShY −→ f∗f which will be the unit of our adjunction. Next, we construct a counit of the adjunction. For any open U ⊆ X, we have

−1 −1 (f f∗F )(U) = lim(f∗F )(V ) = lim F (f (V )) −→ −→ where the limit is over all open V ⊆ Y containing f(U). Well for every such V , f −1(V ) ⊇ U so the restriction maps F (f −1(V )) → F (U) assemble to give a map lim F (f −1(V )) → F (U). −→ This defines the counit −1 ε : f f∗ −→ idPreshX . Finally, by Lemma 3.2.6 it remains to show that η and ε satisfy the triangle identities. For an open set V ⊆ Y , we have

−1 f∗εF ◦ ηf∗F (V ): f∗F (V ) → f∗f f∗F (V ) → f∗F (V ) which is F (f −1(V )) → lim F (f −1(V 0)) → F (f −1(V )) −→ where the limit in the middle is over all open V 0 ⊆ Y containing f(f −1(V )). Since V ⊇ −1 −1 f(f (V )), this composition is clearly the identity on F (f (V )). Therefore f∗εF ◦ ηf∗F is the identity on f∗F . Likewise, for U ⊆ X open,

−1 −1 −1 −1 −1 εf −1G ◦ f ηG(U): f G(U) → f f∗f G(U) → f G(U) which is lim G(V ) → lim G(V 0) → lim G(V ) −→ −→ −→ where the outer limits are over all open V ⊆ Y containing f(U) and the middle limit is over all open V 0 ⊆ Y containing f(f −1(V )) for any V containing f(U). Any such V in the first limit also appears in the second limit, so the composition is given by the trivial restrictions G(V ) → G(V ) for all V . That is, the composition is the identity on lim G(V ) = −→ f −1G(U).

174 8.5 Direct and Inverse Image 8 Sheaf Theory

f g Corollary 8.5.19. For any maps X −→ Y −→ Z, (g ◦ f)∗ = f ∗ ◦ g∗.

∗ ∗ ∗ Proof. Each of (g ◦ f) and f ◦ g is a left adjoint of (g ◦ f)∗ = g∗ ◦ f∗. Corollary 8.5.20. Let f : X → Y be a continuous map and E an injective sheaf on X. Then f∗E is an injective sheaf on Y .

∗ ∗ Proof. Suppose 0 → F → G is an exact sequence in ShY . Then since f is exact, 0 → f F → ∗ ∗ f G is an exact sequence in ShX and by adjointness of (f , f∗), we have a commutative diagram

Hom(G, f∗E) Hom(F, f∗E) 0

∼= ∼=

Hom(f ∗G, E) Hom(f ∗F,E) 0

with isomorphisms on the columns. Since E is injective, the bottom row is exact. Therefore the top row is also exact, which proves f∗E is injective. Remark. More generally, any functor with an exact left adjoint preserves injectives.

Corollary 8.5.21. For any continuous map f : X → Y and sheaf F on X, there is a homomorphism p p H (X,F ) −→ H (Y, f∗F ) for all p ≥ 0.

• • Proof. Take an injective resolution E of F in ShX . Then by Corollary 8.5.20, f∗E is an injective resolution of f∗F in ShY , so we can compute cohomology using this resolution:

p p • H (Y, f∗F ) = H (Γ(Y, f∗E )).

n n −1 Since each f∗E is defined locally by a direct limit lim E (f (V )), there is an induced map −→ n n of global sections Γ(X,E ) → Γ(Y, f∗E ) for each n ≥ 0. This induces the desired maps on p p cohomology: H (X,F ) −→ H (Y, f∗F ). Remark. To compute Hp(X, f ∗G) for a sheaf G on Y is harder in general, since if E• ∗ • is an injective resolution of G in ShY , the inverse images f E need not form an injective resolution of f ∗G on X. However, by the theory of derived categories, there exists an injective complex of sheaves J • on X and an embedding f ∗E• → J • which is a quasi-isomorphism, i.e. it induces isomorphisms on all cohomology groups. Therefore the cohomology of f ∗G can be computed with this resolution:

Hp(X, f ∗G) = Hp(Γ(X,J •)).

Further, there are chain maps

Γ(Y,E•) → Γ(X, f ∗E•) → Γ(X,J •)

175 8.5 Direct and Inverse Image 8 Sheaf Theory

which induce a homomorphism on cohomology

Hp(Y,G) −→ Hp(X, f ∗G)

for all p ≥ 0. In principle these can be defined directly from the definitions of sheaf coho- mology and f ∗, but the derived categorical perspective makes this straightforward.

One thing to note about the direct image functor is that it does not preserve stalks. Indeed, if f : X → Y is an arbitrary continuous map and y ∈ Y , then for any sheaf F on X,(f∗F )y = Fx if y = f(x) ∈ f(X), but if y 6∈ f(X), then (f∗F )y need not be 0 (the value of F over the empty set). One way to remedy this is to define a new type of pushforward functor that respects stalks. We start by giving the definition for open embeddings.

Definition. Let j : U,→ X be the inclusion of an open set. For a sheaf F on U, define the extension by zero of F along j to be the sheafification j!F of the presheaf P on X defined by ( F (V ),V ⊆ U P (V ) = 0,V 6⊆ U.

Lemma 8.5.22. For any open embedding j : U,→ X, the assignment F 7→ j!F is a functor ShU → ShX .

∗ Proposition 8.5.23. For any open embedding j : U,→ X, (j!, j ) is an adjoint pair of functors.

Proof. Let F be a sheaf on U and G a sheaf on X. First recall that the inverse image functor ∗ ∗ j is simply restriction: j G = G|U . Then for the presheaf P defined above, we have ∼ HomShX (j!F,G) = HomPreshX (P,G) by the universal property of sheafification ∼ = HomShU (F,G|U ) by definition of P ∗ = HomShU (F, j G).

∗ These isomorphisms are all natural in F and G, so (j!, j ) is an adjoint pair.

∗ Corollary 8.5.24. For any open embedding j : U,→ X, j! is exact and j preserves injec- tives.

Proof. Exactness of j! can be checked on stalks, and the second statement is a consequence of the remark following Corollary 8.5.20.

∗ If j : U,→ X is an open embedding, the above gives us a triple of functors (j!, j , j∗) in which each consecutive pair is an adjunction and j∗ is exact. In a similar fashion, we can ∗ extend the adjunction (i , i∗) for a closed embedding i : Z,→ X. Definition. Let i : Z,→ X be the inclusion of a closed subset. For a sheaf F on X, define the inverse image of F supported on Z to be the sheaf i!F whose value on an open set V ⊆ Z is ! (i F )(V ) = ΓZ (V,F ) := {s ∈ Γ(V,F ) | supp(s) ⊆ V }.

176 8.5 Direct and Inverse Image 8 Sheaf Theory

! Lemma 8.5.25. For any closed embedding i : Z,→ X, F 7→ i F is a functor ShX → ShZ . ! Proposition 8.5.26. For any closed embedding i : Z,→ X, (i∗, i ) is an adjoint pair of functors. ! Corollary 8.5.27. For any closed embedding i : Z,→ X, i∗ is exact and i preserves injectives. ∗ ! Therefore we have a triple of functors (i , i∗, i ) in which each consecutive pair is an ! adjunction and i∗ is exact. We remark that the functor f does not exist in general – although a derived analogue does exist and agrees with the derived functor of i! for a closed embedding. Let i : Z,→ X be a closed embedding, set U = X r Z and let j : U,→ X be the corresponding open embedding. Define a category C whose objects are triples (G, H, ϕ) ∗ where G is a sheaf on Z, H is a sheaf on U and ϕ : G → i j∗H is a morphism of sheaves, and whose morphisms are pairs of morphisms (ξ : G → G0, χ : H → H0) making the diagram ϕ ∗ G i j∗H

∗ ξ i j∗(χ)

0 ∗ 0 G i j∗H ϕ0

∗ ∗ commute. Let η : F → j∗j F be the unit of the adjunction (j , j∗). Then this induces a ∗ ∗ ∗ ∗ morphism of sheaves i (η): i F → i j∗j F . Theorem 8.5.28 (Recollment of Sheaves). For i : Z,→ X and j : U = X r Z,→ X as above,

ShX −→ C F 7−→ (i∗F, j∗F, i∗(η)) is an equivalence of categories. When i : Z,→ X is a closed embedding and G is a sheaf on Z, note that the stalks of the direct image sheaf F = i∗G are precisely ( Gx, x ∈ Z Fx = 0, x 6∈ Z.

∗ ∗ ∗ Since i preserves stalks, this shows that the counit ε : i i∗G → G of the adjunction (i , i∗) is in fact an isomorphism. Hence i∗G = F = i!G. On the other hand, for the open embedding j : U = X r Z,→ X, we have ( Gx, x ∈ U (j∗G)x = 0, x 6∈ U.

Therefore j∗G 6= j!G in general. This presents an obstacle to constructing the adjunctions ∗ ! (f!, f , f∗, f ) for more general maps f.

177 8.5 Direct and Inverse Image 8 Sheaf Theory

Definition. A subset W ⊆ X is locally closed if

(1) W is open in W .

(2) W = U ∩ C where U ⊆ X is open and C ⊆ X is closed.

(3) For all x ∈ W , there is a neighborhood V ⊆ X of x such that V ∩ W is closed in V .

∗ ! Proposition 8.5.29. The adjunctions (i!, i , i∗, i ) exist for any locally closed embedding i : W,→ X.

Proof. We have an open embedding W,→ W and a closed embedding W,→ X, so apply Theorem 8.5.28. Conversely:

Proposition 8.5.30. Let i : W,→ X be the inclusion of an arbitrary subset. If every sheaf on X is of the form i!G for some G ∈ ShW , then W is locally closed in X.

Theorem 8.5.31. Let i : Z,→ X be a closed embedding, set U = X r Z and let j : U,→ X be the corresponding open embedding. Then for any sheaf F on X, the sequence

∗ ∗ 0 → j!j F → F → i∗i F → 0

is exact.

Proof. By definition, it suffices to check this on stalks: for x ∈ X, the sequence is

( id 0 → Fx −→ Fx −→ 0 → 0, x ∈ U id 0 → 0 −→ Fx −→ Fx → 0, x ∈ Z.

Remark. Note that when j : Z,→ X is a closed embedding, the morphism of sheaves j!G → j∗G is injective for all G ∈ ShZ , so for every open set V ⊆ X, we may view Γ(V, j!G) as a subset of Γ(V, j∗G) = Γ(V ∩ U, G), where U = X r Z. Explicitly,

Γ(V, j!G) = {s ∈ Γ(V ∩ U, G) | supp(s) is closed in U}.

This explains another common name, ‘sections with compact support’, for j!. For a general map f : X → Y , this suggests defining the pushforward with compact support f! : ShZ → ShX by −1 (f!F )(V ) = {s ∈ Γ(f (V ),F ) | f|supp(s) : supp(s) → V is proper}. (Recall that a map is proper if the preimage of every compact set is compact.)

178