Arun Ram: Complexes and Homology

Homology and derived functors

Arun Ram
Department of Mathematics
University of Wisconsin
Madison, WI 53706 USA
ram@math.wisc.edu

Complexes

Let $A$ be a ring and let $𝒜$ be the category of $A$ -modules. More generally, $𝒜$ could be any abelian category.

A complex of $A$ -modules is a $ℤ$ -graded $A$ -module $C$ , with a morphism $d : C ⟶ C$ such that

$\dots ⟶ C_{n + 1} \overset{d_{n + 1}}{⟶} C_{n} \overset{d_{n}}{⟶} C_{n - 1} ⟶ \dots, such that \deg d = - 1 and d \cdot d = 0 .$

A morphism is a graded $A$ -module homomorphism $u : C ⟶ C^{'}$ such that $\deg u = 0$ and $d^{'} \cdot u = u \cdot d$ .

The homology of a complex $C$ is

$H (C) = \frac{Z (C)}{B (C)}, where Z (C) = \ker d and B (C) = im d .$

A quasiisomorphism is a morphism $f : C \to C^{'}$ such that $H (f) : H (C) \to H (C^{'})$ is an isomorphism.

A complex is exact if $H (C) = 0$ . The Grothendieck group of $𝒜$ is the group generated by

$[M], for M \in 𝒜, with relations [M_{1}] = [M_{2}], if M_{1} ≅ M_{2}$

and

$[M] = [M_{1}] + [M_{2}], if there exists an exact sequence 0 ⟶ M_{1} ⟶ M ⟶ M_{2} ⟶ 0$ .

The Euler characteristic of a graded $A$ -module $M$ is

$χ (M) = \sum_{n \in ℤ} (– 1)^{n} [M_{n}] and P_{m} (t) = \sum_{n \in ℤ} t^{n} [M_{n}]$

is the Poincare polynomial of $M$ .

Let

$0 ⟶ C^{'} \overset{u}{⟶} C \overset{v}{⟶} C^{''} ⟶ 0$

be an exact sequence of complexes. The long exact sequence in homology is the exact triangle

$\begin{matrix} H (C) \\ H (u) ↗ ↘ H (v) \\ H (C) \overset{\partial (u, v)}{⟵} H (C^{''}) \\ [t^{'}] ⟵ α \end{matrix}$

$\cdot\cdot\cdot ⟶ H_{n + 1} (C^{''}) \overset{\partial_{n + 1} (u, v)}{⟶} H_{n} (C^{'}) \overset{H_{n} (u)}{⟶} H_{n} (C) \overset{H_{n} (v)}{⟶} H_{n} (C^{''}) \overset{\partial_{n} (u, v)}{⟶} H_{n - 1} (C^{'}) \overset{H_{n - 1} (u)}{⟶} H_{n - 1} (C) \overset{H_{n - 1} (v)}{⟶} H_{n - 1} (C^{''}) ⟶ \cdot\cdot\cdot$

where: if $z^{''} \in Z (C^{''})$ such that $[z^{''}] = α$ , and; $x \in C_{n}$ such that $v (x) = z^{''}$ , then; $t^{'} \in C_{n - 1}^{'}$ such that $u (t^{'}) = d x$ .

Derived functors

A complex is exact if $H (C) = 0$ . An exact functor is a functor $F : 𝒞 ⟶ 𝒞^{'}$ such that if

if $0 ⟶ X \overset{f}{⟶} Y \overset{g}{⟶} Z ⟶ 0 is exact then 0 ⟶ F (X) \overset{F (f)}{⟶} F (Y) \overset{F (g)}{⟶} F (Z) ⟶ 0 is exact.$

A left exact functor is a functor $F : 𝒞 ⟶ 𝒞^{'}$ such that if

if $0 ⟶ X \overset{f}{⟶} Y \overset{g}{⟶} Z is exact then 0 ⟶ F (X) \overset{F (f)}{⟶} F (Y) \overset{F (g)}{⟶} F (Z) is exact.$

A projective object is an object $Y \in 𝒞$ such that $Hom (Y, \cdot)$ is an exact functor.
An injective object is an object $Y \in 𝒞$ such that $Hom (\cdot, Y)$ is an exact functor.
A flat $A$ -module is an $A$ -module $Y$ such that $\cdot \otimes_{A} Y$ is an exact functor.
A free $A$ -module is an ???.
A torsion free $A$ -module is an ???

A presentation of an $A$ -module $M$ is an exact sequence

$R ⟶ X ⟶ M ⟶ 0,$ where $R$ and $X$ are free modules.

An injective resolution of $M$ is an exact sequence

$0 ⟶ M ⟶ I^{0} ⟶ I^{1} ⟶ I^{2} ⟶ \dots,$ with all $I^{k}$ injective.

A projective resolution of $M$ is an exact sequence

$\dots ⟶ P^{- 2} ⟶ P^{- 1} ⟶ P^{0} ⟶ M ⟶ 0,$ with all $P^{- k}$ projective.

Let $F : 𝒞 ⟶ 𝒞^{'}$ be a left exact functor. The right derived functors of $F$ are

$R^{i} F : 𝒞 ⟶ 𝒞^{'} given by R^{i} F (M) = H^{i} (F (I)),$

where $I$ is an injective resolution of $M$ .

Let $F : 𝒞 ⟶ 𝒞^{'}$ be a right exact functor. The left derived functors of $F$ are

$L_{i} F : 𝒞 ⟶ 𝒞^{'} given by L_{i} F (M) = H^{- i} (F (P)),$

where $P$ is a projective resolution of $M$ .

Let

$0 ⟶ C^{'} ⟶ C ⟶ C^{''} ⟶ 0$

be an exact sequence. The long exact sequence is

$0 ⟶ F (C^{'}) ⟶ F (C) ⟶ F (C^{''}) \overset{δ^{0}}{⟶} R^{1} F (C^{'}) ⟶ R^{1} F (C) ⟶ R^{1} F (C^{''}) \overset{δ^{1}}{⟶} R^{2} F (C^{'}) ⟶ R^{2} F (C) ⟶ R^{2} F (C^{''}) ⟶ \dots$

A quasiisomorphism is a morphism of complexes $f : C \to C^{'}$ such that $H (f) : H (C) \to H (C^{'})$ is an isomorphism. The derived category of $𝒜$ is the category $D (𝒜)$ with a functor $Q : Kom (𝒜) \to D (𝒜)$ such that

if $f$ is a quasiisomorphism then $Q (f)$ is an isomorphism, and
if $F : Kom (𝒜) \to 𝒞$ is a functor that takes quasiisomorphisms to isomorphisms then there exists a unique functor $\tilde{F} : D (𝒜) \to 𝒞$ such that

$\begin{matrix} Kom (𝒜) & \overset{Q}{⟶} & D (𝒜) \\ F ↘ & ↓ \tilde{F} \\ 𝒞 \end{matrix}$

Let $D^{+} (𝒞)$ denote the left bounded derived category of $𝒞$ . Let $F : 𝒞 ⟶ 𝒞^{'}$ be a left exact functor. The derived functor of $F$ is the functor

$RF : D^{+} (𝒞) ⟶ D^{+} (C^{'})$ determined by $RF (K)^{i} = F (K^{i}),$ for $K \in Kom (𝒞)$ .

Then

$R^{i} F = H^{i} (RF)$ .

Examples

Ext: ${Ext}_{𝒞}^{i} (M, N) = {Hom}_{D (𝒞)} (M [0], N [i])$ , ${Ext}^{i} (M, \cdot) = R^{i} Hom (M, \cdot)$ , ${Ext}^{i} (\cdot, N) = R^{i} Hom (\cdot, N)$ .

Tor: Let $\otimes$ be the left adjoint functor to $Hom$ so that

$Hom (M \otimes V, N) ≃ Hom (M, Hom (N, V)), for all N .$

Then ${Tor}_{i} (\cdot, N) = R^{i} (\cdot \otimes_{A} N)$ , ${Tor}_{i} (M, \cdot) = R^{i} (M \otimes_{A} \cdot)$ .

Sheaf cohomology: Let $X$ be a topological space. The sheaf cohomology of $X$ is

$H^{i} (X; \cdot) = R^{i} Γ (X; \cdot),$ where $\begin{matrix} Γ : & {sheaves on X} & ⟶ & {abelian groups} \\ ℱ & \mapsto & Γ (X, ℱ) \end{matrix}$

is the global sections functor.

Group cohomology: Let $G$ be a group and let $M$ be a $G$ -module. The cohomology of $G$ is

$H^{i} (G; \cdot) = R^{i} (\cdot^{G}),$ where $M^{G} = {m \in M ∣ g m = m for all g \in G}$ .

is the invariants of $M$ .

Lie algebra cohomology: Let $𝔤$ be a Lie algebra and let $M$ be a $𝔤$ -module. The cohomology of $𝔤$ is

$H^{i} (𝔤; \cdot) = R^{i} (\cdot^{𝔤}),$ where $M^{𝔤} = {m \in M ∣ x m = m for all x \in 𝔤}$

is the invariants of $M$ .