Homological Algebra — Chapter 2 Abelian Categories

Chapter 2
Abelian Categories

2.1. Additive categories

Definition 2.1.1. §

An additive category $𝒞$ is a category with the following properties:

i.

for $A, B \in Obj (𝒞)$ , the set of morphisms ${Hom}_{𝒞} (A, B)$ in $𝒞$ has an abelian group law (addition) with the property that for any diagram

Additivity of composition. A full diagram description follows.

Diagram description: Additivity of composition

This diagram records the composable morphisms in the distributivity identity: h composed with (g subscript 1 plus g subscript 2) composed with f equals the sum of h composed with g subscript 1 composed with f and h composed with g subscript 2 composed with f.

Objects, listed by row and column:

Row 1, from left to right: column 1: A; column 2: B; column 3: C; column 4: D.

Arrows and lines:

An arrow from A to B, labelled f.
An arrow from B to C, labelled g subscript (1).
An arrow from B to C, labelled g subscript (2).
An arrow from C to D, labelled h.

in $𝒞$ , we have

h \circ (g_{1} + g_{2}) \circ f = h \circ g_{1} \circ f + h \circ g_{2} \circ f,

ii.

$𝒞$ has a zero object $0$ ,

iii.

$𝒞$ admits finite coproducts.

Note that between any two objects $A$ and $B$ in an additive category $𝒞$ , there always exists the zero morphism $0 : A \to B$ , the unique map factoring through $0$ . This is the identity element in the abelian group ${Hom}_{𝒞} (A, B)$ .

Examples 2.1.2. §

a.: The categories $𝐀𝐛$ and $R -mod$ are additive categories, with the usual addition of homomorphisms.
b.: The full subcategory $R -mod$ of finitely generated $R$ -modules is an additive category.
c.: The category of topological Hausdorff abelian groups (with continuous group homomorphisms) is an additive category.

In an additive category, we denote the coproduct of two objects $A_{1}$ and $A_{2}$ by $A_{1} \oplus A_{2}$ .

Lemma 2.1.3. §

Finite products exist in an additive category, and there are natural isomorphisms

A_{1} ∐ A_{2} ≅ A_{1} \times A_{2}

for $A_{1}, A_{2} \in Obj (𝒞)$ . The resulting inclusion morphisms $ι_{i} : A_{i} \to A_{1} \oplus A_{2}$ and projection morphisms and $p_{i} : A_{1} \oplus A_{2} \to A_{i}$ obtained by viewing $A_{1} ∐ A_{2}$ as a product and coproduct, respectively, satisfy $p_{i} \circ ι_{i} = {id}_{A_{i}}$ and $p_{i} \circ ι_{j} = 0$ for $i \neq j$ , while

ι_{1} \circ p_{1} + ι_{2} \circ p_{2} = {id}_{A_{1} ∐ A_{2}} .

Proof.

We have maps $ι_{i} : A_{i} \to A_{1} \oplus A_{2}$ by definition of the direct sum. We also have maps $p_{i} : A_{1} \oplus A_{2} \to A_{i}$ defined by

p_{i} \circ ι_{j} = {\begin{matrix} {id}_{A_{i}} & if i = j \\ 0 & if i \neq j \end{matrix}

and the universal property of the coproduct.

We then have

(ι_{1} \circ p_{1} + ι_{2} \circ p_{2}) \circ ι_{i} = ι_{i},

and hence

ι_{1} \circ p_{1} + ι_{2} \circ p_{2} = {id}_{A_{1} ∐ A_{2}} .

Given an object $B \in Obj (𝒞)$ and maps $g_{i} : B \to A_{i}$ , we then have a map

ψ = ι_{1} \circ g_{1} + ι_{2} \circ g_{2} : B \to A_{1} \oplus A_{2},

which is unique such that

p_{i} \circ ψ = g_{i} .

Hence $A_{1} \oplus A_{2}$ satisfies the universal property of the product. □

Definition 2.1.4. §

An object $A$ in an additive category $𝒞$ together with objects $A_{i}$ , inclusion morphisms $ι_{i} : A_{i} \to A$ , and projection morphisms $p_{i} : A \to A_{i}$ for $i \in {1, 2}$ for which $p_{i} \circ ι_{j}$ is zero if $i \neq j$ and ${id}_{A_{i}}$ if $i = j$ and for which $ι_{1} \circ p_{1} + ι_{2} \circ p_{2} = {id}_{A}$ is called a biproduct of the objects $A_{1}$ and $A_{2}$ , and we write it as $A_{1} \oplus A_{2}$ .

The reader will verify the following.

Lemma 2.1.5. §

In an additive category $𝒞$ , the biproduct $A_{1} \oplus A_{2}$ is a coproduct of the $A_{i}$ via the inclusion morphisms $ι_{i}$ and to the product of the $A_{i}$ via the morphisms $p_{i}$ .

The notion of a biproduct allows us to reinterpret addition in an additive category. First, note the following definitions.

Definition 2.1.6. §

Let $A$ be an object in an additive category $𝒞$ .

a.: The diagonal morphism $Δ_{A} : A \to A \oplus A$ in $𝒞$ is the unique morphism induced by two copies of ${id}_{A} : A \to A$ and the universal property of the product.
b.: The codiagonal morphism $\nabla_{A} : A \oplus A \to A$ in $𝒞$ is the unique morphisms induced by two copies of ${id}_{A} : A \to A$ and the universal property of the coproduct.

Definition 2.1.7. §

Let $𝒞$ be an additive category, and let $f_{1} : A_{1} \to B_{1}$ and $f_{2} : A_{2} \to B_{2}$ be morphisms in $𝒞$ . The biproduct, or direct sum, $f_{1} \oplus f_{2}$ of the maps $f_{1}$ and $f_{2}$ is the morphism $A_{1} \oplus A_{2} \to B_{1} \oplus B_{2}$ induced as the (morphism defined by the universal property of the) coproduct of the composite maps $A_{i} \to B_{i} \to B_{1} \oplus B_{2}$ , the latter morphisms being inclusions.

Remark 2.1.8. §

Equivalently, the direct sum of $f_{1}$ and $f_{2}$ as in Definition 2.1.7 is induced as the product of the composite maps $A_{1} \oplus A_{2} \to A_{i} \to B_{i}$ , the initial morphisms being projections.

Of course, we could make these definitions in an arbitrary category using products and coproducts.

Lemma 2.1.9. §

Let $f, g : A \to B$ be two morphisms in an additive category $𝒞$ . Then we have

f + g = \nabla_{B} \circ (f \oplus g) \circ Δ_{A} .

Proof.

Let $ι_{i}^{A}$ and $p_{i}^{A}$ respectively denote the inclusion maps and projection maps for the biproduct $A \oplus A$ , and similarly for $B$ . We have

\begin{matrix} \nabla_{B} \circ (f \oplus g) \circ Δ_{A} = \nabla \circ (f \oplus g) \circ (ι_{1}^{A} \circ p_{1}^{A} + ι_{2}^{A} \circ p_{2}^{A}) \circ Δ_{A} \\ = \nabla_{B} \circ (f \oplus g) \circ ι_{1}^{A} \circ p_{1}^{A} \circ Δ_{A} + \nabla_{B} \circ (f \oplus g) \circ ι_{2}^{A} \circ p_{2}^{A} \circ Δ_{A} . \end{matrix}

Taking the first term without loss of generality, we have

\nabla_{B} \circ ((f \oplus g) \circ ι_{1}^{A}) \circ (p_{1}^{A} \circ Δ_{A}) = \nabla_{B} \circ (ι_{1}^{B} \circ f) \circ {id}_{A} = {id}_{B} \circ f = f .

□

Definition 2.1.10. §

A functor $F : 𝒞 \to 𝒟$ between additive categories is called additive if for each $A, B \in Obj (𝒞)$ , the map

{Hom}_{𝒞} (A, B) \to {Hom}_{𝒟} (F (A), F (B))

is a group homomorphism.

Example 2.1.11. §

Let $𝒞$ be an additive category. Then for any $A \in Obj (𝒞)$ , the functors $h^{A}$ and $h_{A}$ may be considered as functors to $𝐀𝐛$ , rather than $𝐒𝐞𝐭$ . The resulting functors are additive.

Example 2.1.12. §

Let $R$ be a ring, and let $A$ be a right $R$ -module. Then the functor

t_{A} : R -mod \to 𝐀𝐛

given by $t_{A} (B) = A \otimes_{R} B$ and $t_{A} (f) = {id}_{A} \otimes f$ for $f : B \to C$ is additive.

Lemma 2.1.13. §

A functor $F : 𝒞 \to 𝒟$ of additive categories is additive if and only if $F$ preserves biproducts, which is to say that the natural morphisms $F (A_{1}) \oplus F (A_{2}) \to F (A_{1} \oplus A_{2})$ and $F (A_{1} \oplus A_{2}) \to F (A_{1}) \oplus F (A_{2})$ are inverse isomorphisms for all objects $A_{1}, A_{2}$ in $𝒞$ .

Proof.

Suppose first that $F$ is an additive functor. Note that $F (ι_{i} \circ p_{i}) = {id}_{F (A_{i})}$ and $F (ι_{i} \circ p_{j}) = F (0)$ for $i \neq j$ , but $F (0) = 0$ by additivity of $F$ . Again by additivity of $F$ , we have

F (ι_{1}) \circ F (p_{1}) + F (ι_{2}) \circ F (p_{2}) = F ({id}_{A_{1} \oplus A_{2}}) = {id}_{F (A_{1} \oplus A_{2})} .

It follows that $F (A_{1} \oplus A_{2})$ is a biproduct of $F (A_{1})$ and $F (A_{2})$ in $𝒟$ , so in particular it is a coproduct.

On the other hand, if $F$ preserves biproducts and $f, g : A \to B$ are morphisms in $𝒞$ , then it is easy to see that $F (f \oplus g) = F (f) \oplus F (g)$ , and Lemma 2.1.9 tells us that

F (f + g) = F (\nabla_{B} \circ (f \oplus g) \circ Δ_{A}) = \nabla_{F (B)} \circ (F (f) \oplus F (g)) \circ Δ_{F (A)} = F (f) + F (g) .

□

Corollary 2.1.14. §

Let $F : 𝒞 \to 𝒟$ be a fully faithful functor of additive categories. Then $F$ is an additive functor.

For additive functors, we may consider a finer notion of representability.

Remark 2.1.15. §

If $F : 𝒞 \to 𝐀𝐛$ is an additive contravariant (resp., covariant) functor of additive categories, then we may consider it to be representable if there exists an object $X \in Obj (𝒞)$ and a natural isomorphism $η : h^{X} ⇝ F$ (resp., $η : h_{X} ⇝ F$ ). In this case, the morphisms $η_{A}$ for $A \in Obj (𝒞)$ will be isomorphisms of groups.

2.2. Kernels and cokernels

Remark 2.2.1. §

In an additive category, any morphism to $0$ is an epimorphism, and any morphism from $0$ is a monomorphism.

Definition 2.2.2. §

Let $𝒞$ be an additive category, and let $f : A \to B$ be a morphism in $𝒞$ .

a.: The kernel $\ker f$ of $f$ is the equalizer $eq (f, 0)$ , when it exists.
b.: The cokernel $coker f$ is the coequalizer $coeq (f, 0)$ , when it exists.

Example 2.2.3. §

In the category $R -mod$ , these definitions agree with the classical ones.

Lemma 2.2.4. §

A morphism in an additive category that admits kernels is a monomorphism if and only if it has zero kernel. A morphism in an additive category that admits cokernels is an epimorphism if and only if it has zero cokernel.

Proof.

Let $f : A \to B$ be a monomorphism, and let $h : \ker f \to A$ be the induced morphism. Since $f \circ h = 0$ by definition of the kernel, we have $h = 0$ , as $f$ is a monomorphism. This forces $\ker f$ to be $0$ , since $h$ factors through $0$ . (Or, one could just apply Lemma 1.7.3.) On the other hand, suppose that $f$ has trivial kernel, and let $g, h : C \to A$ be maps with $f \circ g = f \circ h$ . Then $f \circ (g - h) = 0$ , and by universal property of the kernel, $g - h$ factors through $0$ , i.e., is $0$ .

The proof for cokernels is similar, or is the result on kernels in the opposite (additive) category. □

Remark 2.2.5. §

Let $𝒞$ be an additive category that admits kernels (resp., cokernels). Then there is a functor

\ker : Mor (𝒞) \to 𝒞 (resp., coker : Mor (𝒞) \to 𝒞)

which takes an object in $Mor (𝒞)$ to its kernel (resp., cokernel) and a morphism in $Mor (𝒞)$ to the natural morphism between kernels (resp., cokernels).

Proposition 2.2.6. §

Let $𝒞$ be an additive category that admits kernels and cokernels. Let $f : A \to B$ be a morphism in $𝒞$ . Then

im f ≅ \ker (B \to coker f)

and

coim 𝑓≅ coker (\ker f \to A) .

Proof.

We prove the first isomorphism. Let $g : B \to coker f$ . By Yoneda’s lemma, it suffices to show that $h^{im f}$ and $h^{\ker g}$ are naturally isomorphic. For $C \in Obj (𝒞)$ , we have a map

{Hom}_{𝒞} (C, im f) \to_{}^{\sim} {α : C \to B ∣ ι_{1} \circ α = ι_{2} \circ α}

that takes a morphism $C \to im f$ and composes it with the morphism $im f \to B$ given by definition of the equalizer of the maps $ι_{i} : B \to B ∐_{A} B$ . It is a bijection by the universal property of the equalizer.

For any $D \in Obj (𝒞)$ and morphisms $ϕ_{1}, ϕ_{2} : B \to D$ such that $ϕ_{1} \circ f = ϕ_{2} \circ f$ , note that there exists a unique morphism $k : B ∐_{A} B \to D$ with $ϕ_{i} = k \circ ι_{i}$ . Any $α : C \to B$ such that $ι_{1} \circ α = ι_{2} \circ α$ then satisfies $ϕ_{1} \circ α = ϕ_{2} \circ α$ for any such $ϕ_{i} : B \to D$ and any $D$ . On the other hand, note that the $ι_{i}$ themselves satisfy the property that $ι_{1} \circ f = ι_{2} \circ f$ and are morphisms $ι_{i} : B \to D$ with $D = B ∐_{A} B$ . In other words, we have

\begin{matrix} {α : C \to B ∣ ι_{1} \circ α = ι_{2} \circ α} \\ = {α : C \to B ∣ ϕ_{1} \circ α = ϕ_{2} \circ α if ϕ_{1} \circ f = ϕ_{2} \circ f for some ϕ_{1}, ϕ_{2} : B \to D (for some D \in Obj (𝒞))} . \end{matrix}

Now, we are in an additive category, so this equals

{α : C \to B ∣ ϕ \circ α = 0 if ϕ \circ f = 0 for some ϕ : B \to D} .

(2.2.1)

By the universal property of $coker f$ , for any $ϕ : B \to D$ with $ϕ \circ f = 0$ , there is a morphism $j : coker f \to D$ with $j \circ g = ϕ$ . If $g \circ α = 0$ , then $ϕ \circ α = j \circ g \circ α = 0$ , and this works for any $ϕ : B \to D$ with $ϕ \circ f = 0$ . On the other hand, $g$ itself satisfies $g \circ f = 0$ so is such a $ϕ$ . It follows that the set in (2.2.1) equals

{α : C \to B ∣ g \circ α = 0} .

By the universal property of $\ker g$ , this is in bijection with ${Hom}_{𝒞} (C, \ker g)$ , taking an $α$ in the set to the unique morphism to $\ker g$ through which it factors. Clearly, the composition of these bijections is natural in $C$ , so we have the desired natural isomorphism. □

2.3. Abelian categories

Definition 2.3.1. §

An abelian category is an additive category $𝒞$ in which

i.: every morphism in $𝒞$ admits a kernel and a cokernel and
ii.: every morphism in $𝒞$ is strict.

Examples 2.3.2. §

a.: The category $R -mod$ is abelian.
b.: The full subcategory $𝒞$ of $R -mod$ of finitely generated $R$ -submodules is not abelian in all cases. E.g., when $R$ is commutative and non-Noetherian, we can take $I$ to be an ideal of $R$ that is not finitely generated, and so the kernel of $R \to R ∕ I$ is not in $𝒞$ .
c.: The category of topological Hausdorff abelian groups is not abelian, but it is additive and admits kernels and cokernels. For instance, consider the inclusion map $ι : ℚ \to ℝ$ with $ℝ$ having its usual topology and $ℚ$ having the subspace topology. Then $\ker ι = 0$ and $coker ι = 0$ (since $ℚ$ is dense in $ℝ$ , and thus every continuous map from $ℝ$ is determined by its values on $ℚ$ ). By Proposition 2.2.6, we have $im ι ≅ ℝ$ but $coim ι ≅ ℚ$ .

Remark 2.3.3. §

Note that if $𝒞$ is an abelian category, then so is $𝒞^{op}$ . The roles of mono- and epimorphisms, kernels and cokernels, and images and coimages switch in $𝒞$ and $𝒞^{op}$ .

Proposition 2.3.4. §

The functor category $𝐅𝐮𝐧𝐜 (𝒞, 𝒟)$ from a small category $𝒞$ to an abelian category $𝒟$ is abelian.

Proof.

We sketch the proof. First, note that it is additive: we have the zero functor which sends all objects to the zero object and all morphisms to the zero (identity) morphism of the zero object, and if $F, G : 𝒞 \to 𝒟$ are functors, then $F \oplus G$ is given by $(F \oplus G) (C) = F (C) \oplus G (C)$ and $(F \oplus G) (f) = F (f) \oplus G (f)$ for $f : A \to B$ in $C$ . This can be used to define the addition on morphisms (i.e., natural transformations) as before.

Next, the kernel of a natural transformation $η : F ⇝ G$ is defined by $(\ker η) (C) = \ker η_{C}$ and $(\ker η) (f)$ for $f : A \to B$ is the kernel of the induced morphism $\ker η_{A} \to \ker η_{B}$ . The cokernel is defined similarly. Note that

{(coim η)}_{A} ≅ coker {(\ker η ⇝ η)}_{A} ≅ coker (\ker η_{A} ⇝ η_{A}) ≅ coim η_{A},

and similarly for images. Finally, since $𝒟$ is abelian, the natural map $coim η ⇝ im η$ is an isomorphism $coim η_{A} \to im η_{A}$ on objects $A$ in $𝒞$ , hence has a natural inverse determined by the inverses of these morphisms. □

We discuss a related class of examples.

Definition 2.3.5. §

Let $X$ be a topological space and $𝒞$ an abelian category. Consider the category $𝒰_{X}$ with objects the open sets in $X$ and morphisms the inclusion maps between subspaces to the category $𝒞$ .

a.

A presheaf $F$ on $X$ with values in $𝒞$ is a contrvariant functor $F : 𝒰_{X} \to 𝒞$ . The category of presheaves is the functor category $𝐅𝐮𝐧𝐜 (𝒰, 𝒞)$ .

b.

Suppose that $𝒞$ admits arbitrary products. A sheaf $F$ on $X$ with values in $𝒞$ is a presheaf $F : 𝒰_{X} \to 𝒞$ such that each $F (U)$ for $U \in 𝒰_{X}$ is the equalizer of the diagram

The sheaf equalizer condition. A full diagram description follows.

Diagram description: The sheaf equalizer condition

The two arrows are induced by restriction to intersections of open sets. The sheaf condition says that F(U) is the equalizer of these two arrows for the given open covering. The parallel arrows themselves are not required to be equal.

Objects, listed by row and column:

Row 1, from left to right: column 1: product subscript (V in script V) F(V); column 2: product subscript (V subscript (1), V subscript (2) in script V) F(V subscript (1) intersection V subscript (2)).

Arrows and lines:

An arrow from product subscript (V in script V) F(V) to product subscript (V subscript (1), V subscript (2) in script V) F(V subscript (1) intersection V subscript (2)), without a label.
An arrow from product subscript (V in script V) F(V) to product subscript (V subscript (1), V subscript (2) in script V) F(V subscript (1) intersection V subscript (2)), without a label.

where $𝒱$ is an open covering of $U$ by open subsets via the two maps on products induced by the application of $F$ to inclusion maps $V_{1} \cap V_{2} \to V_{i}$ for each pair $(V_{1}, V_{2})$ of open sets in $𝒱$ . The category of sheaves is the full subcategory of the category of presheaves with objects the sheaves.

Example 2.3.6. §

Let $X$ be a topological space and $𝒞$ an abelian category. For any $A \in Obj (𝒞)$ , we can view $A$ as a discrete space and consider the constant sheaf

F (U) = {f : U \to A ∣ f continuous}

for all $U \in 𝒰_{X}$ . If $U$ is connected, then $F (U) = A$ .

Example 2.3.7. §

Let $X$ be a topological space and for $U \in 𝒰_{X}$ . Then $𝒪_{X}$ is the sheaf with values in $𝐀𝐛$ of continuous functions to $ℂ$ that for $U \in 𝒰_{X}$ satisfies

𝒪_{X} (U) = {continuous maps U \to ℂ} .

Example 2.3.8. §

Let $X = ℝ$ , and let $F : 𝒰_{ℝ} \to 𝐀𝐛$ be the presheaf such that $F (U)$ is the abelian group of bounded continuous functions on $U$ . Then $F$ is not a sheaf, as we may consider the open covering of $ℝ$ by intervals $I_{n} = (- n, n)$ for $n \in ℤ$ , and then the function $f (x) = x$ is contained in $F (I_{n})$ for each $n$ , so in the equalizer, but it is not in $F (ℝ)$ .

Terminology 2.3.9. §

In an abelian category $𝒞$ , we will typically refer to a coproduct (when it exists) as a direct sum, and we write $\oplus_{i \in I} A_{i}$ in place of $∐_{i \in I} A_{i}$ .

2.4. Exact sequences

Suppose that $f : A \to B$ and $g : B \to C$ are morphisms in an abelian category $𝒞$ with $g \circ f = 0$ . Note that $f : κ \circ λ$ , where $λ : A \to im f$ is an epimorphism and $κ : im f \to B$ is a monomorphism. We have $g \circ κ \circ λ = 0$ , so the fact that $λ$ is an epimorphism tells us that $g \circ κ = 0$ . The universal property of the kernel then provides a morphism $α : im f \to \ker g$ such that when composed with the canonical monomorphism $β : \ker g \to B$ is $β \circ α = κ$ . Since $β$ and $κ$ are both monomorphisms, which is to say have trivial kernel, we have that $α$ is a monomorphism as well.

Definition 2.4.1. §

We say that a diagram

A \to_{}^{f} B \to_{}^{g} C

in an abelian category $𝒞$ is exact if $g \circ f = 0$ and the induced monomorphism $im f \to \ker g$ is an isomorphism. We call such a diagram a three term exact sequence.

We generalize this notion. First, we define a sequence.

Definition 2.4.2. §

A diagram of the form

\dots \to A_{n + 1} \to_{}^{d_{n + 1}^{A}} A_{n} \to_{}^{d_{n}^{A}} A_{n - 1} \to \dots

in a category $𝒞$ is a sequence, with the understanding that the diagram may terminate (i.e., be of finite length) on either, both, or neither side. We denote the data of a sequence by $A_{\cdot} = (A_{i}, d_{i}^{A})$ , where $i$ runs for $A_{i}$ over the interval of integers on which $A_{i}$ is defined and for $d_{i}^{A}$ over every such $i$ but the left endpoint, if it exists.

Definition 2.4.3. §

Let $𝒞$ be an abelian category.

a.: A chain complex in $𝒞$ is a sequence $A_{\cdot} = {(A_{i}, d_{i}^{A})}_{i \in ℤ}$ in $𝒞$ such that $d_{i}^{A} \circ d_{i + 1}^{A} = 0$ for all $i \in ℤ$ .
b.: For a chain complex $A_{\cdot}$ and $i \in ℤ$ , the morphism $d_{i}^{A} : A_{i} \to A_{i - 1}$ is called the $i$ th differential in the complex $A_{\cdot}$ .

Notation 2.4.4. §

Unless otherwise specified, the $i$ th object in a chain complex $A_{\cdot}$ will be denoted $A_{i}$ and the $i$ th differential by $d_{i} : A_{i} \to A_{i - 1}$ . If we have multiple complexes, we will use $d_{i}^{A}$ to specify the differential on $A_{\cdot}$ .

Notation 2.4.5. §

In an abelian category, if $A$ is a subobject of an object $B$ , we will often write $A \subseteq B$ to denote this and $B ∕ A$ for the cokernel of the inclusion morphism $A \to B$ . For $f : B \to C$ , we will let $f (A)$ denote the image of the composite of the inclusion with $f$ .

Definition 2.4.6. §

A sequence

\dots \to A_{n + 1} \to_{}^{d_{n + 1}} A_{n} \to_{}^{d_{n - 1}} A_{n - 1} \to \dots

in an abelian category $𝒞$ is exact if the subdiagram $A_{m + 1} \to A_{m} \to A_{m - 1}$ is exact for each $m$ such that there are terms $A_{m + 1}$ and $A_{m - 1}$ in the diagram.

Definition 2.4.7. §

A long exact sequence in an abelian category $𝒞$ is an exact sequence $A_{\cdot} = (A_{i}, d_{i}^{A})$ , where $i$ runs over all integers.

Definition 2.4.8. §

A complex is often said to be acyclic if it is an exact sequence.

Definition 2.4.9. §

Let $𝒞$ be an abelian category.

a.: A short exact sequence in $𝒞$ is an exact sequence in $𝒞$ of the form
$0 \to A \to B \to C \to 0 .$
b.: A left short exact sequence in $𝒞$ is an exact sequence in $𝒞$ of the form
$0 \to A \to B \to C .$
c.: A right short exact sequence in $𝒞$ is an exact sequence in $𝒞$ of the form
$A \to B \to C \to 0 .$

Definition 2.4.10. §

Let $𝒞$ be an abelian category.

a.

We say that an epimorphism $g : B \to C$ is split if there exists a morphism $t : C \to B$ in $𝒞$ with $g \circ t = {id}_{C}$ . In this case, we say that $t$ is a splitting of $g$ .

b.

We say that a monomorphism $f : A \to B$ is split if there exists a morphism $s : B \to A$ in $𝒞$ with $s \circ f = {id}_{A}$ . In this case, we say that $s$ is a splitting of $f$ .

c.

We say that a short exact sequence

0 \to A \to_{}^{f} B \to_{}^{g} C \to 0

(2.4.1)

in an abelian category $𝒞$ splits if there exists an isomorphism $A \oplus C \to_{}^{w} B$ in $𝒞$ with $w \circ ι_{A} = f$ and $w \circ ι_{C} \circ g = {id}_{B}$ , where $ι_{A} : A \to A \oplus C$ and $ι_{C} : C \to A \oplus C$ are the inclusion morphisms.

Example 2.4.11. §

Any exact sequence

0 \to ℤ ∕ 3 ℤ \to ℤ ∕ 6 ℤ \to ℤ ∕ 2 ℤ \to 0

is split in Ab, but

0 \to ℤ ∕ 2 ℤ \to ℤ ∕ 4 ℤ \to ℤ ∕ 2 ℤ \to 0

is not.

Lemma 2.4.12. §

The following conditions on a short exact sequence (2.4.1) are equivalent:

The sequence (2.4.1) splits.
The monomorphism $f : A \to B$ splits.
The epimorphism $g : B \to C$ splits.

Proof.

Suppose we have a splitting map $t : C \to B$ . Then define $s : B \to A$ by $s (b) = a$ where $f (a) = b - t (g (b))$ . This is well-defined as $f$ is injective, and such an $a$ exists since
$g (b - t (g (b))) = g (b) - g (t (g (b))) = g (b) - g (b) = 0 .$
It splits $f$ as
$s (f (a)) = s (b) - s (t (g (b))) = s (b),$
the latter step using the fact that $s \circ t = 0$ , which follows in turn from
$f (s (t (c))) = t (c) - t (g (t (c))) = t (c) - t (c) = 0 .$
Suppose that we have a splitting map $s : B \to A$ . Then define $t : C \to B$ by $t (c) = b - f (s (b))$ where $g (b) = c$ . To see this is well defined, note that $f (a) - f (s (f (a))) = 0$ for any $a \in A$ .
Define $w (a, c) = f (a) + t (c)$ . Its inverse is $w^{'} (b) = (s (b), g (b))$ . We check
$w^{'} \circ w (a, c) = (s (f (a) + t (c)), g (f (a) + t (c))) = (s (f (a)), g (t (c))) = (a, c)$
and
$w \circ w^{'} (b) = w (s (b), g (b)) = f (s (b)) + t (g (b)) .$
It remains to see that $f \circ s + t \circ g = {id}_{B}$ . It is easy to see this on elements of the form $f (a)$ or $t (c)$ and that an arbitrary element of $b$ is a sum of these.
Set $t (b) = w (0, b)$ .

□

2.5. Exact functors

Definition 2.5.1. §

Let $F : 𝒞 \to 𝒟$ be an additive functor between abelian categories.

a.

We say that $F$ is left exact if it preserves exact sequences of the form

0 \to A \to B \to C,

which is to say that the sequence

0 \to F (A) \to F (B) \to F (C)

is exact.

b.

We say that $F$ is right exact if it preserves exactness of exact sequences of the form

A \to B \to C \to 0 .

c.

We say that $F$ is exact if it is both left and right exact.

The reader will verify the following.

Lemma 2.5.2. §

An additive functor $F : 𝒞 \to 𝒟$ between abelian categories is exact (resp., left exact, resp., right exact) if and only if it takes short exact sequences in $𝒞$ to short exact (resp., left short exact, resp. right short exact) sequences in $𝒟$ . Moreover, $F$ is exact if and only if it preserves three term exact sequences.

Remark 2.5.3. §

We may extend the definition of left and right exact functors to contravariant functors. The requirement that a contravariant functor $F : 𝒞 \to 𝒟$ be left exact is that it preserves exactness of sequences of the form $0 \to A \to B \to C$ , which is to say that $F (C) \to F (B) \to F (A) \to 0$ is exact.

Examples 2.5.4. §

a.

If $𝒞$ is abelian, then the functor $F : 𝒞 \to 𝒞$ by $F (A) = A \oplus A$ with $F (f) = f \oplus f$ is exact.

b.

In $R -mod$ for a commutative ring $R$ , the functor $t_{A}$ is right exact for any $R$ -module $A$ .

c.

Let $G$ be a group. For a $ℤ [G]$ -module $A$ , consider the abelian group

A^{G} = {a \in A ∣ 𝑔𝑎 = a for all g \in G}

of $G$ -invariants in $A$ , with $ℤ [G]$ -module homomorphisms $A \to B$ restricting to homomorphisms $A^{G} \to B^{G}$ . The resulting functor $ℤ [G] -mod \to 𝐀𝐛$ is left exact but not in general right exact.

Recall that for an additive category $𝒞$ , the functors $h^{X}$ (and $h_{X}$ ) may be viewed as taking values in $𝐀𝐛$ , and clearly such functors are additive. In fact, they are also left exact.

Lemma 2.5.5. §

Let $𝒞$ be an abelian category, and let $X$ be an object of $𝒞$ .

a.: The functor $h_{X} : 𝒞 \to 𝐀𝐛$ is left exact.
b.: The functor $h^{X} : 𝒞^{op} \to 𝐀𝐛$ is left exact.

Proof.

Let

0 \to A \to_{}^{f} B \to_{}^{g} C \to 0

be an exact sequence in $𝒞$ . Applying $h_{X}$ , we obtain homomorphisms

0 \to {Hom}_{𝒞} (X, A) \to_{}^{h_{X} (f)} {Hom}_{𝒞} (X, B) \to_{}^{h_{X} (g)} {Hom}_{𝒞} (X, C)

of abelian groups, and we claim this sequence is exact. If $h_{X} (f) (α) = 0$ , then $f \circ α = 0$ , but $f$ is a monomorphism, so $α = 0$ . Since $h_{X}$ is a functor, we have $h_{X} (g) \circ h_{X} (f) = 0$ , and if $β \in \ker h_{X} (g)$ , then $g \circ β = 0$ . Naturality of the kernel implies that $β$ factors through a morphism $X \to \ker g$ . But we have canonical isomorphisms

A \to_{}^{\sim} coim f \to_{}^{\sim} im f \to_{}^{\sim} \ker g,

the first as $f$ is a monomorphism, and the composite of the composite of these with the canonical morphism $\ker g \to B$ is $g$ . Therefore, we obtain a morphism $α : X \to A$ satisfying $f \circ α = g$ . This proves part a, and part b is just part a with $𝒞$ replaced by $𝒞^{op}$ . □

Lemma 2.5.6. §

Let $𝒞$ be an abelian category. A sequence

A \to_{}^{f} B \to_{}^{g} C

is exact if every sequence

{Hom}_{𝒞} (X, A) \to_{}^{h_{X} (f)} {Hom}_{𝒞} (X, B) \to_{}^{h_{X} (g)} {Hom}_{𝒞} (X, C)

is exact.

Proof.

For $X = A$ , we get

g \circ f = h_{X} (g) \circ h_{X} (f) ({id}_{A}) = 0,

so we have a monomorphism $s : im f \to \ker g$ . For $X = \ker g$ and $β : \ker g \to B$ the natural monomorphism defined by the kernel, we have $h^{X} (g) (β) = g \circ ι = 0$ , so there exists $α : \ker g \to A$ with $f \circ α = β$ . We then have that $β$ factors a morphism $t : \ker g \to im f$ inverse to $s$ . □

Proposition 2.5.7. §

Suppose that $G : 𝒞 \to 𝒟$ is an additive functor between abelian categories that admits a left (resp., right) adjoint. Then $F$ is left (resp., right) exact.

Proof.

We treat the case of left exactness, the other case simply being the corresponding statement in opposite categories. Suppose that

0 \to A \to_{}^{f} B \to_{}^{g} C

is a left exact sequence in $𝒞$ . Let $F$ be a left adjoint to $G$ . Then for any $D \in Obj (𝒟)$ , the sequence

0 \to h_{F (D)} (A) \to_{}^{h_{F (D)} (f)} h_{F (D)} (B) \to_{}^{h_{F (D)} (g)} h_{F (D)} (C)

is left exact. Since $F$ is left adjoint to $G$ , this sequence is isomorphic to

0 \to h_{D} (G (A)) \to_{}^{h_{D} (G (f))} h_{D} (G (B)) \to_{}^{h_{D} (G (g))} h_{D} (G (C))

as a sequence of abelian groups. Since this holds for all $D$ , the sequence

0 \to G (A) \to_{}^{G (f)} G (B) \to_{}^{G (g)} G (C)

is exact. □

Example 2.5.8. §

The inclusion functor from the category of sheaves on a space $X$ to that of presheaves on $X$ has a left adjoint called sheafification. The inclusion functor is therefore left exact, and the sheafification functor is in fact exact.

The following embedding theorem allows us to do most of the homological algebra that can be done in the category of $R$ -modules for any $R$ in an arbitrary abelian category.

Theorem 2.5.9 (Freyd-Mitchell). §

If $𝒞$ is a small abelian category, then there exists a ring $R$ and an exact, fully faithful functor $𝒞 \to R$ -mod.

In other words, $𝒞$ is equivalent to a full, abelian subcategory of $R -mod$ for some ring $R$ . We can use this as follows: suppose there is a result we can prove about exact diagrams in $R$ -modules for all $R$ , like the snake lemma. We then have the result in all abelian categories, since we can take a small full, abelian subcategory containing the objects in which we are interested and embed it into some category of left $R$ -modules. If the result holds in that category, then by exactness of the embedding, the result will hold in the original category.

While we do not prove the Freyd-Mitchell embedding theorem, we will give a few remarks on its details (which hopefully are not so far off from the truth).

Remark 2.5.10. §

To prove Theorem 2.5.9, the first step is to embed $𝒞$ in the (opposite category of the) full subcategory of $𝐅𝐮𝐧𝐜 (𝒞, 𝐀𝐛)$ consisting of left exact, additive functors, which is an abelian category using $h_{𝒞}$ . While $h_{𝒞}$ is not exact as an embedding into the category of additive functors in $𝐅𝐮𝐧𝐜 (𝒞, 𝐀𝐛)$ , it is exact as an embedding into the latter category. (Another method is to embed $𝒞$ in the category of $𝐏𝐫𝐨 (𝒞)$ of cofiltered limits of the objects of $𝒞$ , viewed as a subcategory of $𝐅𝐮𝐧𝐜 {(𝒞, 𝐀𝐛)}^{op}$ .) In either case, the category $E$ in question has what is called a projective generator $G$ which we can choose so that every object in $𝒞$ is a quotient object of $G$ , and the point is that $R_{G} = {Hom}_{E} (G, G)$ is a ring under its usual addition and composition. Then $h_{G}$ induces a functor $𝒞 \to R_{G} -mod$ , which is the exact, fully faithful functor in question.

2.6. Standard lemmas

We begin with the following extremely useful result.

Theorem 2.6.1 (Snake lemma). §

Suppose that we have a commutative diagram

The snake lemma's input diagram. A full diagram description follows.

Diagram description: The snake lemma's input diagram

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: D; column 2: A; column 3: B; column 4: C; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: A prime; column 3: B prime; column 4: C prime; column 5: D prime.

Arrows and lines:

An arrow from D to A, labelled e.
An arrow from A to B, labelled f.
An arrow from A to A prime, labelled alpha.
An arrow from B to C, labelled g.
An arrow from B to B prime, labelled beta.
An arrow from C to 0 (row 1, column 5), without a label.
An arrow from C to C prime, labelled gamma.
An arrow from 0 (row 2, column 1) to A prime, without a label.
An arrow from A prime to B prime, labelled f prime.
An arrow from B prime to C prime, labelled g prime.
An arrow from C prime to D prime, labelled h prime.

(2.6.1)

with exact rows. Then there is a homomorphism $ψ : \ker γ \to coker α$ fitting into a larger commutative diagram

The snake lemma's connecting path. A full diagram description follows.

Diagram description: The snake lemma's connecting path

The middle two rows are exact, and the structural squares commute. The dashed path from D through kernel alpha, kernel beta, kernel gamma, cokernel alpha, cokernel beta, cokernel gamma, and D prime is an exact eight-term sequence. The long curved arrow is the connecting map psi from kernel gamma to cokernel alpha.

Objects, listed by row and column:

Row 1, from left to right: column 3: kernel alpha; column 4: kernel beta; column 5: kernel gamma.
Row 3, from left to right: column 2: D; column 3: A; column 4: B; column 5: C; column 6: 0.
Row 5, from left to right: column 2: 0; column 3: A prime; column 4: B prime; column 5: C prime; column 6: D prime.
Row 7, from left to right: column 3: cokernel alpha; column 4: cokernel beta; column 5: cokernel gamma.

Arrows and lines:

A dashed arrow from kernel alpha to kernel beta, without a label.
An arrow from kernel alpha to A, without a label.
A dashed arrow from kernel beta to kernel gamma, without a label.
An arrow from kernel beta to B, without a label.
An arrow from kernel gamma to C, without a label.
A dashed curved arrow from kernel gamma to cokernel alpha, without a label.
A dashed curved arrow from D to kernel alpha, without a label.
An arrow from D to A, without a label.
An arrow from A to B, without a label.
An arrow from A to A prime, without a label.
An arrow from B to C, without a label.
An arrow from B to B prime, without a label.
An arrow from C to 0 (row 3, column 6), without a label.
An arrow from C to C prime, without a label.
An arrow from 0 (row 5, column 2) to A prime, without a label.
An arrow from A prime to B prime, without a label.
An arrow from A prime to cokernel alpha, without a label.
An arrow from B prime to C prime, without a label.
An arrow from B prime to cokernel beta, without a label.
An arrow from C prime to D prime, without a label.
An arrow from C prime to cokernel gamma, without a label.
A dashed arrow from cokernel alpha to cokernel beta, without a label.
A dashed arrow from cokernel beta to cokernel gamma, without a label.
A dashed curved arrow from cokernel gamma to D prime, without a label.

and the resulting eight-term sequence

D \to \ker α \to \ker β \to \ker γ \to coker α \to coker β \to coker γ \to D^{'}

is exact.

Proof.

We define $ψ$ as follows. For $c \in \ker γ$ , find $b \in B$ with $g (b) = c$ . Then $g^{'} \circ β (b) = γ (c) = 0$ , so $β (b) = f^{'} (a^{'})$ for some $a^{'} \in A^{'}$ . Let $ψ (c)$ denote the image ${\bar{a}}^{'}$ of $a^{'}$ in $coker α$ . To see that this is well-defined, note that if $b_{2} \in B$ also satisfies $g (b_{2}) = c$ , then $g (b - b_{2}) = 0$ , so $b_{2} - b = f (a)$ for some $a \in A$ . We then have

β (b_{2}) = β (b) + β \circ f (a) = β (b) + f^{'} \circ α (a),

so $b_{2} = f^{'} (a^{'} + α (a))$ . But $a^{'} + α (a)$ has image ${\bar{a}}^{'}$ in $coker α$ , so $ψ$ is well-defined.

We now check that the other maps are well-defined. Since

f^{'} \circ α \circ e = β \circ f \circ e = 0

and $f^{'}$ is injective, we have $α \circ e = 0$ , so $im e \subseteq \ker α$ . Since

β \circ f (\ker α) = f^{'} \circ α (\ker α) = 0,

we have $f (\ker α) \subseteq \ker β$ . Similarly, $g (\ker β) \subset \ker γ$ . Also, if ${\bar{a}}^{'} \in coker α$ , then we may lift it to $a^{'} \in A^{'}$ , map to $b^{'} \in B^{'}$ , and then project to ${\bar{b}}^{'} \in coker β$ . This is well-defined as any other choice of $a^{'}$ differs by some $a \in A$ , which causes ${\bar{b}}^{'}$ to change by the image of $β (f (a))$ , which is zero. Thus $f^{'}$ induces a well-defined map

{\bar{f}}^{'} : coker α \to coker β .

Similarly, we have a well-defined

{\bar{g}}^{'} : coker β \to coker γ .

Finally, $im g = C$ , so $im γ \subseteq im g^{'}$ , and therefore $h^{'} (im γ) = 0$ . Thus $h^{'}$ induces a map

{\bar{h}}^{'} : coker γ \to D^{'} .

It is clear from the above definitions that the $8$ -term sequence is a complex at all the terms but $\ker γ$ and $coker α$ . Let $b \in \ker β$ . Then $ψ (g (b))$ is given by considering $β (b) = 0$ , lifting it to some $a^{'} \in A^{'}$ , which we may take to be $0$ , and projecting to $coker α$ . Hence $ψ (g (\ker β)) = 0$ . On the other hand, if $c \in \ker γ$ , then ${\bar{f}}^{'} (ψ (c))$ is given by definition by projecting $β (b)$ to $coker β$ , where $g (b) = c$ , hence is zero.

We now check exactness at each term. We have

\ker α \cap \ker f = \ker α \cap im e = im e,

so the sequence is exact at $\ker α$ . Next, if $b \in \ker β \cap \ker g$ , then there exists $a \in A$ with $f (a) = b$ .

Since $f^{'} \circ α (a) = β \circ f (a) = 0$ and $f^{'}$ is injective, we have $α (a) = 0$ , or $a \in \ker α$ . Hence

f (\ker α) = \ker (\ker β \to \ker γ),

and we have exactness at $\ker β$ .

If $c \in \ker ψ$ , then whenever $g (b) = c$ and $f^{'} (a^{'}) = β (b)$ , we have ${\bar{a}}^{'} = 0$ , letting ${\bar{a}}^{'}$ denote the image of $a^{'} \in coker α$ . We then have $a^{'} = f (a)$ for some $a \in A$ , so $b_{2} = b - f (a)$ still satisfies $f (b_{2}) = c$ , but $β (b_{2}) = 0$ . So $b_{2} \in \ker β$ , and we have exactness at $\ker γ$ .

If ${\bar{a}}^{'} \in coker α$ is the image of $a^{'} \in A^{'}$ and ${\bar{f}}^{'} ({\bar{a}}^{'}) = 0$ , then there exists $b \in B$ with $β (b) = f^{'} (a^{'})$ . Now

γ (g (b)) = g^{'} (β (b)) = g^{'} (f^{'} (a^{'})) = 0,

so $g (b) \in \ker γ$ , and $ψ (g (b)) = {\bar{a}}^{'}$ . Hence, we have exactness at $coker α$ .

If ${\bar{b}}^{'} \in coker β$ is the image of $b^{'} \in B^{'}$ and ${\bar{g}}^{'} ({\bar{b}}^{'}) = 0$ , then there exists $c \in C$ with $g (b^{'}) = γ (c)$ . Now $c = g (b)$ for some $b \in B$ . And $b_{2}^{'} = b^{'} - b$ has image ${\bar{b}}^{'}$ in $coker β$ . On the other hand, $f^{'} (b_{2}^{'}) = 0$ , so $b_{2}^{'} = f^{'} (a^{'})$ for some $a^{'} \in A^{'}$ . If ${\bar{a}}^{'} \in coker α$ is the image of $a^{'}$ , then ${\bar{f}}^{'} ({\bar{a}}^{'}) = {\bar{b}}^{'}$ as the image of $b_{2}^{'}$ in $coker β$ . Thus, we have exactness at $coker β$ .

Finally, let ${\bar{c}}^{'} \in coker γ$ be the image of $c^{'} \in C^{'}$ . Note that ${\bar{h}}^{'} ({\bar{c}}^{'}) = h (c^{'})$ . If this is zero, then $c^{'} = g^{'} (b^{'})$ for some $b^{'} \in B^{'}$ , and ${\bar{c}}^{'}$ is the image of the projection ${\bar{b}}^{'}$ of $b^{'}$ to $coker β$ under ${\bar{g}}^{'}$ . □

Lemma 2.6.2. §

Suppose we have a commutative diagram as in (2.6.1) and that both $g$ and $f^{'}$ are split. Then the snake map $ψ : \ker γ \to coker α$ is zero.

Proof.

Let $c \in \ker γ$ . Let $t$ split $g$ and $s^{'}$ split $f^{'}$ . By Lemma ??, we also have a splitting map

t^{'} : \ker h^{'} \to A^{'} .

It is easy to check that $t^{'} \circ γ = β \circ t$ . It follows that $t (\ker γ) \subseteq \ker β$ . Thus $ψ (c)$ , which is the image of $s^{'} (β (t (c)))$ in $coker α$ , is zero. □

2.7. Complexes

Let $𝒞$ be an abelian category. We have already defined the notion of a chain complex in $𝒞$ above. In addition to chain complexes, we will also deal with cochain complexes, which are likewise defined, but with increasing superscripts replacing decreasing subscripts. This is primarily a notational convenience, but it is a very useful one. When the choice of chain or cochain complexes matters little, we will typically choose to work with cochain complexes.

Definition 2.7.1. §

A cochain complex $A^{\cdot} = (A^{i}, d_{A}^{i})$ is a diagram

\dots \to A^{i - 1} \to_{}^{d_{A}^{i - 1}} A^{i} \to_{}^{d_{A}^{i}} A^{i + 1} \to \dots,

such that $d_{A}^{i} \circ d_{A}^{i - 1} = 0$ for all $i \in ℤ$ . Again, the $d_{A}^{i}$ are referred to as differentials.

Remark 2.7.2. §

We refer to both chain complexes and cochain complexes as complexes, though we will usually mean the latter if we do not specify. We will often state facts simply for cochain complexes that have a direct translation to the setting of chain complexes.

Definition 2.7.3. §

Let $A^{\cdot} = (A^{i}, d_{A}^{i})$ and $B_{\cdot} = (B^{i}, d_{B}^{i})$ be complexes. A morphism of complexes $f^{\cdot} : A^{\cdot} \to B^{\cdot}$ is a sequence of morphisms $f^{i} : A^{i} \to B^{i}$ commuting with the differentials of the complexes in the sense that

d_{B}^{i} \circ f^{i} = f^{i - 1} \circ d_{A}^{i}

for all $i \in ℤ$ .

Remark 2.7.4. §

We may consider the category $𝐂𝐡 (𝒞)$ of cochain complexes in $𝒞$ , where morphisms of chain complexes $A^{\cdot} = (A^{i}, d^{i})$ and $B^{\cdot} = (B^{i}, {d^{'}}^{i})$ are morphisms $A^{i} \to B^{i}$ for each $i$ commuting with the differentials.

The reader can easily check the following.

Proposition 2.7.5. §

Let $𝒞$ be an abelian category. Then the category $𝐂𝐡 (𝒞)$ is an abelian category as well.

Definition 2.7.6. §

Let $A_{\cdot}$ (resp., $A^{\cdot}$ ) be a chain complex (resp., cochain complex).

a.: We say that $A_{\cdot}$ (resp., $A^{\cdot}$ ) is bounded below if $A_{i} = 0$ (resp., $A^{i} = 0$ ) for all $i < N$ for some $N \in ℤ$ .
b.: We say that $A_{\cdot}$ (resp., $A^{\cdot}$ ) is bounded above if $A_{i} = 0$ (resp., $A^{i} = 0$ ) for all $i > N$ for some $N \in ℤ$ .
c.: We say that $A_{\cdot}$ (resp., $A^{\cdot}$ ) is bounded if it is both bounded below and bounded above.

In most examples that we shall explore, our chain complexes (resp., cochain complexes) will be bounded below (resp., bounded above), usually with $N = 0$ .

Definition 2.7.7. §

a.

We define $i$ th homology (object) of a chain complex $A_{\cdot}$ by

H_{i} (A_{\cdot}) = \ker d_{i}^{A} ∕ im d_{i + 1}^{A}

for each $i \in ℤ$ .

b.

We define the $i$ th cohomology (object) of a cochain complex $A^{\cdot}$ by

H^{i} (A^{\cdot}) = \ker d_{A}^{i} ∕ im d_{A}^{i - 1}

for $i \in ℤ$ .

Remark 2.7.8. §

For a complex of abelian groups, we speak of homology and cohomology groups, as opposed to objects.

Example 2.7.9. §

Consider the chain complex $A^{\cdot} = (ℤ \oplus ℤ, d_{A}^{i})$ with $d_{A}^{i} = 3 {id}_{ℤ} \oplus 0$ and $d_{A}^{i + 1} = 0 \oplus 2 {id}_{ℤ}$ for $i$ even. Then $H_{i} (A_{\cdot}) = ℤ ∕ 2 ℤ$ and $H_{i + 1} (A_{\cdot}) = ℤ ∕ 3 ℤ$ for $i$ even.

Lemma 2.7.10. §

Let $f^{\cdot} : A^{\cdot} \to B^{\cdot}$ be a morphism of complexes. Then we have natural homomorphisms

f_{*}^{i} : H^{i} (A^{\cdot}) \to H^{i} (B^{\cdot})

for each $i \in ℤ$ .

Proof.

Let $A^{\cdot} = (A^{i}, d_{A}^{i})$ and $B^{\cdot} = (B^{i}, d_{B}^{i})$ . We have $f^{i} \circ d_{A}^{i + 1} = d_{B}^{i + 1} \circ f^{i + 1}$ for all $i$ , so

f^{i} (im d_{A}^{i + 1}) \subseteq im d_{B}^{i + 1},

and

d_{B}^{i} (f^{i} (\ker d_{A}^{i})) = f^{i - 1} (d_{A}^{i} (\ker d_{A}^{i})) = 0,

f^{i} (\ker d_{A}^{i}) \subseteq \ker d_{B}^{i} .

Hence, we have an induced morphism

f_{*}^{i} : \frac{\ker d_{A}^{i}}{im d_{A}^{i + 1}} \to \frac{\ker d_{B}^{i}}{im d_{B}^{i + 1}},

as desired. □

Remark 2.7.11. §

Of course, the analogous result holds in cohomology as a consequence.

Remark 2.7.12. §

We note that a sequence

0 \to A^{\cdot} \to_{}^{f^{\cdot}} B^{\cdot} \to_{}^{g^{\cdot}} C^{\cdot} \to 0

in $𝐂𝐡 (𝒞)$ is exact if and only if each sequence

0 \to A^{i} \to_{}^{f^{i}} B^{i} \to_{}^{g^{i}} C^{i} \to 0

is exact in $𝒞$ .

Theorem 2.7.13. §

Let

0 \to A^{\cdot} \to_{}^{f^{\cdot}} B^{\cdot} \to_{}^{g^{\cdot}} C^{\cdot} \to 0

be an exact sequence of cochain complexes of $R$ -modules. Then we have a long exact sequence in cohomology

\dots \to H^{i} (A^{\cdot}) \to_{}^{f_{*}^{i}} H^{i} (B^{\cdot}) \to_{}^{g_{*}^{i}} H^{i} (C^{\cdot}) \to_{}^{\partial^{i}} H^{i + 1} (A^{\cdot}) \to \dots,

where $\partial_{i}$ is the map given by the snake lemma applied to the diagram (2.7.1) below.

Proof.

Consider the following diagram

Kernels and cokernels for the cohomology sequence. A full diagram description follows.

Diagram description: Kernels and cokernels for the cohomology sequence

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 2: cokernel d superscript (i minus 1) subscript (A); column 3: cokernel d superscript (i minus 1) subscript (B); column 4: cokernel d superscript (i minus 1) subscript (C); column 5: 0.
Row 2, from left to right: column 1: 0; column 2: kernel d superscript (i plus 1) subscript (A); column 3: kernel d superscript (i plus 1) subscript (B); column 4: kernel d superscript (i plus 1) subscript (C).

Arrows and lines:

An arrow from cokernel d superscript (i minus 1) subscript (A) to cokernel d superscript (i minus 1) subscript (B), without a label.
An arrow from cokernel d superscript (i minus 1) subscript (A) to kernel d superscript (i plus 1) subscript (A), labelled d superscript (i) subscript (A).
An arrow from cokernel d superscript (i minus 1) subscript (B) to cokernel d superscript (i minus 1) subscript (C), without a label.
An arrow from cokernel d superscript (i minus 1) subscript (B) to kernel d superscript (i plus 1) subscript (B), labelled d superscript (i) subscript (B).
An arrow from cokernel d superscript (i minus 1) subscript (C) to 0 (row 1, column 5), without a label.
An arrow from cokernel d superscript (i minus 1) subscript (C) to kernel d superscript (i plus 1) subscript (C), labelled d superscript (i) subscript (C).
An arrow from 0 (row 2, column 1) to kernel d superscript (i plus 1) subscript (A), without a label.
An arrow from kernel d superscript (i plus 1) subscript (A) to kernel d superscript (i plus 1) subscript (B), without a label.
An arrow from kernel d superscript (i plus 1) subscript (B) to kernel d superscript (i plus 1) subscript (C), without a label.

(2.7.1)

By the snake lemma, the two rows are exact, and the diagram commutes since we have $d_{B}^{i} \circ f^{i} = f^{i + 1} \circ d_{A}^{i}$ and $d_{C}^{i} \circ g^{i} = g^{i + 1} \circ d_{B}^{i}$ as morphisms $A^{i} \to B^{i + 1}$ and $B^{i} \to C^{i + 1}$ , respectively, by naturality of the cokernel and kernel. Since we have natural isomorphisms

\begin{array}{rcl} H^{i} (A^{\cdot}) ≅ \ker (A^{i} ∕ im d_{A}^{i - 1} \to A^{i + 1}) & and & H^{i + 1} (A^{\cdot}) ≅ coker (A^{i} \to \ker d_{A}^{i + 1}), \end{array}

we apply the snake lemma to obtain an exact sequence

H^{i} (A^{\cdot}) \to_{}^{f_{*}^{i}} H^{i} (B^{\cdot}) \to_{}^{g_{*}^{i}} H^{i} (C^{\cdot}) \to_{}^{\partial^{i}} H^{i + 1} (A^{\cdot}) \to_{}^{f_{*}^{i + 1}} H^{i + 1} (B^{\cdot}) \to_{}^{g_{*}^{i + 1}} H^{i + 1} (C^{\cdot}),

and the result follows by splicing together these sequences. □

Definition 2.7.14. §

Let $A^{\cdot} = (A^{i}, d_{A}^{i})$ and $B^{\cdot} = (B^{i}, d_{B}^{i})$ be chain complexes. Let $f^{\cdot}, g^{\cdot} : A^{\cdot} \to B^{\cdot}$ be morphisms of chain complexes.

a.

A chain homotopy from $f^{\cdot}$ to $g^{\cdot}$ is a sequence $s^{\cdot} = {(s^{i})}_{i \in ℤ}$ of morphisms $s^{i} : A^{i} \to B^{i - 1}$ satisfying

f^{i} - g^{i} = d_{B}^{i - 1} \circ s^{i} + s^{i + 1} \circ d_{A}^{i}

for all $i \in ℤ$ .

b.

We say that $f^{\cdot}$ and $g^{\cdot}$ are chain homotopic, and write $f^{\cdot} \sim g^{\cdot}$ , if there exists a homotopy from $f^{\cdot}$ to $g^{\cdot}$ .

c.

If $f^{\cdot}$ is (chain) homotopic to $0$ , then $f^{\cdot}$ is said to be null-homotopic.

The maps $s_{\cdot}$ defining a null-homotopy fit into a (not usually ) diagram

A null-homotopy. A full diagram description follows.

Diagram description: A null-homotopy

The diagonal maps are the components of a null-homotopy. This diagram is not generally commutative; the homotopy identity in the surrounding text is a sum of the two routes through a diagonal map.

Objects, listed by row and column:

Row 1, from left to right: column 1: ellipsis; column 2: A superscript (i minus 1); column 3: A superscript (i); column 4: A superscript (i plus 1); column 5: ellipsis.
Row 2, from left to right: column 1: ellipsis; column 2: B superscript (i minus 1); column 3: B superscript (i); column 4: B superscript (i plus 1); column 5: ellipsis.

Arrows and lines:

An arrow from ellipsis (row 1, column 1) to A superscript (i minus 1), without a label.
An arrow from A superscript (i minus 1) to A superscript (i), labelled d superscript (i minus 1) subscript (A).
An arrow from A superscript (i minus 1) to B superscript (i minus 1), labelled f superscript (i minus 1).
An arrow from A superscript (i) to A superscript (i plus 1), labelled d superscript (i) subscript (A).
An arrow from A superscript (i) to B superscript (i), labelled f superscript (i).
An arrow from A superscript (i) to B superscript (i minus 1), labelled s superscript (i).
An arrow from A superscript (i plus 1) to ellipsis (row 1, column 5), without a label.
An arrow from A superscript (i plus 1) to B superscript (i plus 1), labelled f superscript (i plus 1).
An arrow from A superscript (i plus 1) to B superscript (i), labelled s superscript (i plus 1).
An arrow from ellipsis (row 2, column 1) to B superscript (i minus 1), without a label.
An arrow from B superscript (i minus 1) to B superscript (i), labelled d superscript (i minus 1) subscript (B).
An arrow from B superscript (i) to B superscript (i plus 1), labelled d superscript (i) subscript (B).
An arrow from B superscript (i plus 1) to ellipsis (row 2, column 5), without a label.

Proposition 2.7.15. §

Assume that $f^{\cdot}$ and $g^{\cdot}$ are homotopic as maps $A^{\cdot} \to B^{\cdot}$ . Then the maps $f_{*}^{i}$ and $g_{*}^{i}$ on homology are equal for all $i \in ℤ$ .

Proof.

It suffices to assume that $g = 0$ , since the $i$ th homology functor from $𝐂𝐡 (𝒞)$ to $𝒞$ is additive. So, we must show that $f_{*}^{i} = 0$ for all $i$ , which is to say that $f^{i} (\ker d_{A}^{i}) \subseteq im d_{A}^{i - 1}$ . Since $f^{i} = d_{B}^{i - 1} \circ s^{i} + s^{i + 1} \circ d_{A}^{i}$ , we have

f^{i} (\ker d_{A}^{i}) = d_{B}^{i - 1} (s^{i} (\ker d_{A}^{i})) \subseteq im d_{B}^{i - 1},

so $f_{*}^{i} = 0$ . □

Definition 2.7.16. §

A morphism of complexes $f^{\cdot} : A^{\cdot} \to B^{\cdot}$ is a homotopy equivalence if there exists a morphism $g^{\cdot} : B^{\cdot} \to A^{\cdot}$ such that $g^{\cdot} \circ f^{\cdot} \sim {id}_{A^{\cdot}}$ and $f^{\cdot} \circ g^{\cdot} \sim {id}_{B^{\cdot}}$ .

Definition 2.7.17. §

Let $A^{\cdot}$ be a complex, and let $j \in ℤ$ . The shift by $j$ of the complex $A$ is the complex $A {[j]}^{\cdot}$ with $A {[j]}^{i} = A^{i + j}$ and $d_{A [j]}^{i} = {(- 1)}^{j} d_{A}^{i}$ for all $i \in ℤ$ .

Notation 2.7.18. §

Let $n \geq 1$ , let $A_{i}, B_{i} \in 𝒞$ be objects in an abelian category $𝒞$ for $1 \leq i \leq n$ , and let $f_{i, j} : A_{i} \to B_{j}$ be morphisms in $𝒞$ for $1 \leq i, j \leq n$ . Let $A = \oplus_{i = 1}^{n} A_{i}$ and $B = \oplus_{i = 1}^{n} B_{i}$ , and let $ι_{i} : B_{i} \to B$ and $π_{i} : A \to A_{i}$ denote the canonical inclusions and projections. Then we use the matrix $(f_{i, j})$ to represent the morphism $f : A \to B$ in $𝒞$ that is the sum

f = \sum_{i = 1}^{n} \sum_{j = 1}^{n} ι_{j} \circ f_{i, j} \circ π_{j} .

Definition 2.7.19. §

Let $f^{\cdot} : A^{\cdot} \to B^{\cdot}$ be a morphism of complexes. The cone of $f^{\cdot}$ is the complex $Cone {(f)}^{\cdot}$ with

Cone {(f)}^{i} = A^{i + 1} \oplus B^{i}

and

d_{Cone (f)}^{i} = (\begin{matrix} - d_{A}^{i + 1} & 0 \\ f^{i + 1} & d_{B}^{i} \end{matrix}) : A^{i + 1} \oplus B^{i} \to A^{i + 2} \oplus B^{i + 1} .

Proposition 2.7.20. §

The cone of $f^{\cdot} : A^{\cdot} \to B^{\cdot}$ fits in a short exact sequence

0 \to B^{\cdot} \to Cone {(f)}^{\cdot} \to A {[1]}^{\cdot} \to 0,

and the resulting long exact sequence has $i$ th connecting homomorphism

H^{i} (A^{\cdot} [1]) = H^{i + 1} (A^{\cdot}) \to H^{i + 1} (B^{\cdot})

equal to $f_{*}^{i + 1}$ .

Proof.

The differential on $Cone {(f)}^{\cdot}$ preserves $B^{\cdot}$ and agrees with the differential on $B^{\cdot}$ , so $B^{\cdot}$ is a subcomplex. The quotient complex has $i$ th term $A^{i + 1}$ and differential induced by the negative of the differential on $A$ , so is canonically isomorphic to $A {[1]}^{\cdot}$ . Since $Cone {(f)}^{i} = A^{i + 1} \oplus B^{i}$ , we have a natural splitting of the surjection $Cone {(f)}^{i} \to A^{i + 1}$ given by the inclusion. The connecting homomorphism is then induced by the composition

\ker d_{A}^{i + 1} \to A^{i + 1} \to Cone {(f)}^{i} \to_{}^{d_{Cone (f)}^{i}} Cone {(f)}^{i + 1},

which is given by the sum of $- d_{A}^{i + 1}$ and $f^{i}$ as maps to $A^{i + 2} \oplus B^{i + 1}$ , but then has image in $B^{i + 1}$ (since $d_{A}^{i + 1}$ is zero on $- d_{A}^{i + 1}$ ) and therefore agrees with $f^{i + 1}$ as a morphism to $B^{i + 1}$ . In other words, the connecting homomorphism is canonically identified with $f_{*}^{i + 1}$ . □

2.8. Total complexes

Definition 2.8.1. §

A double (cochain) complex in an abelian category $𝒞$ is a complex in $𝐂𝐡 (𝒞)$ .

The data of a double complex $A^{⋅⋅}$ consists of complexes $(A^{i, \cdot}, d^{i, \cdot})$ of objects of $Ch (𝒞)$ , which is to say that each $A^{i, \cdot}$ is a complex in $𝒞$ with differentials of its own, and the $d^{i, \cdot}$ are morphisms between $A^{i, \cdot} \to A^{i + 1, \cdot}$ , so commute with the differentials on these complexes. We rephrase this as follows.

Remark 2.8.2. §

A double complex $(A^{⋅⋅}, d^{⋅⋅})$ in $𝒞$ is a diagram of the form

A double complex. A full diagram description follows.

Diagram description: A double complex

Each horizontal row and each vertical column is a complex, so consecutive horizontal differentials compose to zero and consecutive vertical differentials compose to zero. The squares commute: a horizontal step followed by a vertical step equals a vertical step followed by a horizontal step. The ellipses indicate continuation in both directions.

Objects, listed by row and column:

Row 1, from left to right: column 2: vertical ellipsis; column 3: vertical ellipsis; column 4: vertical ellipsis.
Row 2, from left to right: column 1: ellipsis; column 2: A superscript (i minus 1,j plus 1); column 3: A superscript (i,j plus 1); column 4: A superscript (i plus 1,j plus 1); column 5: ellipsis.
Row 3, from left to right: column 1: ellipsis; column 2: A superscript (i minus 1,j); column 3: A superscript (i,j); column 4: A superscript (i plus 1,j); column 5: ellipsis.
Row 4, from left to right: column 1: ellipsis; column 2: A superscript (i minus 1,j minus 1); column 3: A superscript (i,j minus 1); column 4: A superscript (i plus 1,j minus 1); column 5: ellipsis.
Row 5, from left to right: column 2: vertical ellipsis; column 3: vertical ellipsis; column 4: vertical ellipsis.

Arrows and lines:

An arrow from ellipsis (row 2, column 1) to A superscript (i minus 1,j plus 1), without a label.
An arrow from A superscript (i minus 1,j plus 1) to A superscript (i,j plus 1), labelled d subscript (h) superscript (i minus 1,j plus 1).
An arrow from A superscript (i minus 1,j plus 1) to vertical ellipsis (row 1, column 2), without a label.
An arrow from A superscript (i,j plus 1) to A superscript (i plus 1,j plus 1), labelled d subscript (h) superscript (i,j plus 1).
An arrow from A superscript (i,j plus 1) to vertical ellipsis (row 1, column 3), without a label.
An arrow from A superscript (i plus 1,j plus 1) to ellipsis (row 2, column 5), without a label.
An arrow from A superscript (i plus 1,j plus 1) to vertical ellipsis (row 1, column 4), without a label.
An arrow from ellipsis (row 3, column 1) to A superscript (i minus 1,j), without a label.
An arrow from A superscript (i minus 1,j) to A superscript (i,j), labelled d subscript (h) superscript (i minus 1,j).
An arrow from A superscript (i minus 1,j) to A superscript (i minus 1,j plus 1), labelled d subscript (v) superscript (i minus 1,j).
An arrow from A superscript (i,j) to A superscript (i plus 1,j), labelled d subscript (h) superscript (i,j).
An arrow from A superscript (i,j) to A superscript (i,j plus 1), labelled d subscript (v) superscript (i,j).
An arrow from A superscript (i plus 1,j) to ellipsis (row 3, column 5), without a label.
An arrow from A superscript (i plus 1,j) to A superscript (i plus 1,j plus 1), labelled d subscript (v) superscript (i plus 1,j).
An arrow from ellipsis (row 4, column 1) to A superscript (i minus 1,j minus 1), without a label.
An arrow from A superscript (i minus 1,j minus 1) to A superscript (i,j minus 1), labelled d subscript (h) superscript (i minus 1,j minus 1).
An arrow from A superscript (i minus 1,j minus 1) to A superscript (i minus 1,j), labelled d subscript (v) superscript (i minus 1,j minus 1).
An arrow from A superscript (i,j minus 1) to A superscript (i plus 1,j minus 1), labelled d subscript (h) superscript (i,j minus 1).
An arrow from A superscript (i,j minus 1) to A superscript (i,j), labelled d subscript (v) superscript (i,j minus 1).
An arrow from A superscript (i plus 1,j minus 1) to ellipsis (row 4, column 5), without a label.
An arrow from A superscript (i plus 1,j minus 1) to A superscript (i plus 1,j), labelled d subscript (v) superscript (i plus 1,j minus 1).
An arrow from vertical ellipsis (row 5, column 2) to A superscript (i minus 1,j minus 1), without a label.
An arrow from vertical ellipsis (row 5, column 3) to A superscript (i,j minus 1), without a label.
An arrow from vertical ellipsis (row 5, column 4) to A superscript (i plus 1,j minus 1), without a label.

such that (for all $i, j \in ℤ$ )

i.: each row is a complex: i.e., $d_{h}^{i + 1, j} \circ d_{h}^{i, j} = 0$ ,
ii.: each column is a complex: i.e., $d_{v}^{i, j + 1} \circ d_{v}^{i, j} = 0$ , and
iii.: the squares commute: i.e., $d_{h}^{i, j + 1} \circ d_{v}^{i, j} = d_{v}^{i + 1, j} \circ d_{h}^{i, j}$ .

Definition 2.8.3. §

The degree of the term $A^{i, j}$ of a double complex $A^{⋅⋅}$ is $i + j$ .

Definition 2.8.4. §

Let $A^{⋅⋅}$ be a double complex in an abelian category $𝒞$ .

a.

Suppose that $𝒞$ admits coproducts. The total sum complex ${Tot}^{\oplus} {(A)}^{\cdot}$ of $A^{⋅⋅}$ is the complex

{Tot}^{\oplus} {(A)}^{k} = ⨁_{i \in ℤ} A^{i, k - i}

with the differential

d^{k} = \sum_{i \in ℤ} (d_{h}^{i, k - i} + {(- 1)}^{i} d_{v}^{i, k - i}) : ⨁_{i \in ℤ} A^{i, k - i} \to ⨁_{i \in ℤ} A^{i, k + i - 1}

where the differentials $d_{h}^{i, k - i}$ and $d_{v}^{i, k - i}$ are taken to be zero on the $A^{j, k - j}$ with $j \neq i$ .

b.

Suppose that $𝒞$ admits products. The total product complex ${Tot}^{Π} {(A)}^{\cdot}$ of $A^{⋅⋅}$ is the complex

{Tot}^{Π} {(A)}^{k} = \prod_{i \in ℤ} A^{i, k - i}

with the differential

d^{k} = {(d_{h}^{i, k - i} + {(- 1)}^{i} d_{v}^{i, k - i})}_{i \in ℤ} : \prod_{i \in ℤ} A^{i, k - i} \to \prod_{i \in ℤ} A^{i, k + 1 - i},

where the differentials $d_{h}^{i, k - i}$ and $d_{v}^{i, k - i}$ are taken to be zero on the $A^{j, k - j}$ with $j \neq i$ .

Remark 2.8.5. §

We remark that ${Tot}^{\oplus} {(A)}^{\cdot}$ is in fact a complex (when it exists). That is,

\begin{array}{l} d^{k + 1} \circ d^{k} & = \sum_{i \in ℤ} \sum_{j \in ℤ} (d_{h}^{i, k + 1 - i} + {(- 1)}^{i} d_{v}^{i, k + 1 - i}) \circ (d_{h}^{j, k - j} + {(- 1)}^{j} d_{v}^{j, k - j}) \\ = \sum_{i \in ℤ} ({(- 1)}^{i} d_{h}^{i, k + 1 - i} \circ d_{v}^{i, k - i} + {(- 1)}^{i} d_{v}^{i, k + 1 - i} \circ d_{h}^{i - 1, k + 1 - i}) \\ = \sum_{i \in ℤ} ({(- 1)}^{i} d_{h}^{i, k + 1 - i} \circ d_{v}^{i, k - i} + {(- 1)}^{i + 1} d_{v}^{i + 1, k - i} \circ d_{h}^{i, k - i}) = 0 . \end{array}

Similarly, ${Tor}^{Π} {(A)}^{\cdot}$ is a complex as well.

Definition 2.8.6. §

An $n$ th quadrant double complex, for $1 \leq n \leq 4$ , is a double complex $A^{⋅⋅}$ such that $A^{i, j} = 0$ for $(i, j)$ not in the closed $n$ th quadrant of $ℝ^{2}$ . An upper, lower, left, or right half-plane double complex is one that is zero outside of said closed half-plane.

Remark 2.8.7. §

Recall that the closed 1st, 2nd, 3rd, and 4th quadrants of $ℝ^{2}$ are $ℝ_{\geq 0} \times ℝ_{\geq 0}$ , $ℝ_{\leq 0} \times ℝ_{\geq 0}$ , $ℝ_{\leq 0} \times ℝ_{\leq 0}$ , and $ℝ_{\geq 0} \times ℝ_{\leq 0}$ , respectively. The closed upper, lower, left, and right half-planes are $ℝ \times ℝ_{\geq 0}$ , $ℝ \times ℝ_{\leq 0}$ , $ℝ_{\leq 0} \times ℝ$ , and $ℝ_{\geq 0} \times ℝ$ , respectively.

Remark 2.8.8. §

If $A^{⋅⋅}$ is either a first or a fourth quadrant double complex, then

{Tot}^{\oplus} {(A)}^{\cdot} = {Tot}^{Π} {(A)}^{\cdot},

and it exists for any abelian category $𝒞$ , since there are only finitely many nonzero terms in the complex of a given degree. In this case, we simply write $Tot {(A)}^{\cdot}$ for either total complex.

Definition 2.8.9. §

Let $A^{⋅⋅}$ be a double complex. We define the truncated quotient complexes ${(^{I} τ^{\geq n} A)}^{⋅⋅}$ and ${(^{𝐼𝐼} τ^{\geq n} A)}^{⋅⋅}$ of $A^{⋅⋅}$ by

\begin{array}{rcl} {(^{I} τ^{\geq n} A)}^{i, j} = {\begin{matrix} A^{i, j} & if i > n \\ coker d_{h}^{n - 1, j} & if i = n \\ 0 & if i < n \end{matrix} & and & {(^{𝐼𝐼} τ^{\geq n} A)}^{i, j} = {\begin{matrix} A^{i, j} & if j > n \\ coker d_{v}^{j, n - 1} & if j = n \\ 0 & if j < n, \end{matrix} \end{array}

as well as subcomplexes ${(^{I} τ^{\leq n} A)}^{⋅⋅}$ and ${(^{𝐼𝐼} τ^{\leq n} A)}^{⋅⋅}$ by

\begin{array}{rcl} {(^{I} τ^{\leq n} A)}^{i, j} = {\begin{matrix} A^{i, j} & if i < n \\ \ker d_{h}^{n, j} & if i = n \\ 0 & if i > n \end{matrix} & and & {(^{𝐼𝐼} τ^{\leq n} A)}^{i, j} = {\begin{matrix} A^{i, j} & if j < n \\ \ker d_{v}^{j, n} & if j = n \\ 0 & if j > n . \end{matrix} \end{array}

Proposition 2.8.10. §

Suppose that $𝒞$ is an abelian category that admits direct products (resp., direct sums). Let $A^{⋅⋅}$ be an upper (or left) double half-plane complex in $𝒞$ with exact columns (resp., exact rows) or a right (or lower) double half-plane complex with exact rows (resp., exact columns). Then ${Tot}^{\oplus} {(A)}^{\cdot}$ (resp., ${Tot}^{Π} {(A)}^{\cdot}$ ) is exact.

Proof.

By interchanging rows and columns, we may suppose that $A^{⋅⋅}$ is an upper or lower half-plane complex with either exact rows or exact columns. By working in the opposite category (and changing the signs of the degrees), we may focus on the case of a lower half-plane complex. Suppose first that $A^{⋅⋅}$ has exact rows. For $n \in ℤ$ , consider the truncated complex ${(^{I} τ^{\leq n} A)}^{⋅⋅}$ , which then has exact rows as well. Then a shift by $n$ in the horizontal direction is a third quadrant double complex $B_{n}^{⋅⋅}$ . If we can show $Tot {(B_{n})}^{\cdot}$ to be exact (for all $n$ ), then ${Tot}^{\oplus} {(A)}^{\cdot}$ will be exact as well, as the reader may check that

{Tot}^{\oplus} {(A)}^{\cdot} ≅ \underset{n}{lim_{\to}} Tot {(B_{n})}^{\cdot},

the limit taken with respect to the morphisms induced by the inclusion morphisms $B_{n}^{\cdot} \to B_{n + 1}^{\cdot}$ . Thus, it suffices to consider the case of an exact left half-plane complex with exact rows, which is equivalent to the case of an exact lower half-plane complex $A^{⋅⋅}$ with exact columns, for which we must show that ${Tot}^{Π} {(A)}^{\cdot}$ is exact. Since we can always shift $A^{⋅⋅}$ horizontally, it suffices to check that $H^{0} ({Tot}^{Π} {(A)}^{\cdot}) = 0$ .

We now work in the category of $R$ -modules for a ring $R$ without loss of generality. Let

a = (a_{i}) \in {Tot}^{Π} {(A)}^{0} = \prod_{i = 0}^{\infty} A^{i, - i} .

with $d^{0} (a) = 0$ . Choose $b_{0} \in A^{0, - 1}$ with $d_{v}^{0, - 1} (b_{0}) = a_{0}$ by exactness of the $0$ th column, and set $b_{0}^{'} = 0 \in A^{- 1, 0}$ . Suppose we have defined $b_{j} \in A^{i, - i - 1}$ for $0 \leq j \leq i$ with

{(- 1)}^{i} d_{v}^{i, - i - 1} (b_{i}) + d_{h}^{i - 1, - i} (b_{i - 1}) = a_{i} .

Then

\begin{matrix} {(- 1)}^{i + 1} d_{v}^{i + 1, - i - 1} (a_{i + 1} - d_{h}^{i, - i - 1} (b_{i})) = {(- 1)}^{i + 1} d_{v}^{i + 1, - i - 1} (a_{i + 1}) + {(- 1)}^{i} d_{h}^{i, - i} \circ d_{v}^{i, - i - 1} (b_{i}) \\ = {(- 1)}^{i + 1} d_{v}^{i + 1, - i - 1} (a_{i + 1}) + d_{h}^{i, - i} (a_{i}), \end{matrix}

which is the $(i + 1, - i)$ -coordinate of $d^{i + 1} (a)$ , hence $0$ . By exactness of the $(i + 1)$ th column, we may then choose $b_{i + 1} \in A^{i + 1, - i}$ with

{(- 1)}^{i + 1} d_{v}^{i + 1, - i} (b_{i + 1}) = a_{i + 1} - d_{h}^{i, - i - 1} (b_{i}),

completing the induction. We then set $b = (b_{i}) \in {Tot}^{Π} {(A)}^{- 1}$ and note that $d^{- 1} (b) = a$ , finishing the proof. □