Cohomology

Cohomology measures the failure of exactness in a complex. The $n$ -th cohomology $H^n(A^\bullet)$ is the quotient of cocycles (elements in the kernel of $d^n$ ) by coboundaries (elements in the image of $d^{n-1}$ ). It is a functor from complexes to the underlying abelian category, and its behavior under short exact sequences gives rise to long exact sequences.

Definition

Definition4.4Cohomology

For a complex $A^\bullet$ in an abelian category $\mathcal{A}$ , the $n$ -th cohomology is

$H^n(A^\bullet) = \ker d^n / \mathrm{im}\, d^{n-1} = \ker(A^n \to A^{n+1}) / \mathrm{im}(A^{n-1} \to A^n)$

Elements of $\ker d^n$ are called $n$ -cocycles ( $Z^n$ ), and elements of $\mathrm{im}\, d^{n-1}$ are called $n$ -coboundaries ( $B^n$ ).

Definition4.5Acyclic complex

A complex $A^\bullet$ is acyclic (or exact) if $H^n(A^\bullet) = 0$ for all $n$ , i.e., the complex is an exact sequence.

Examples

ExampleCohomology of a short complex

For $0 \to A \xrightarrow{f} B \to 0$ (concentrated in degrees 0 and 1): $H^0 = \ker f$ and $H^1 = \mathrm{coker}\, f = B / \mathrm{im}\, f$ .

ExampleDe Rham cohomology

$H^n_{\mathrm{dR}}(M) = \ker(d : \Omega^n \to \Omega^{n+1}) / \mathrm{im}(d : \Omega^{n-1} \to \Omega^n)$ is the $n$ -th de Rham cohomology. For $M = \mathbb{R}^n$ , all cohomology vanishes except $H^0 \cong \mathbb{R}$ (Poincare lemma).

ExampleSingular cohomology

Applying $\mathrm{Hom}(-, G)$ to the singular chain complex gives the singular cochain complex. Its cohomology is the singular cohomology $H^n(X; G)$ .

ExampleGroup cohomology

For a group $G$ and $G$ -module $M$ , $H^n(G, M)$ is the cohomology of the complex $\mathrm{Hom}_G(P_\bullet, M)$ where $P_\bullet \to \mathbb{Z}$ is a projective resolution. $H^0(G, M) = M^G$ (fixed points), $H^1(G, M)$ classifies extensions, $H^2(G, M)$ classifies central extensions.

ExampleSheaf cohomology

$H^n(X, \mathcal{F}) = R^n\Gamma(X, \mathcal{F})$ is the $n$ -th right derived functor of global sections. It can be computed via injective resolutions or Cech cohomology.

ExampleCohomology of the Koszul complex

For a regular sequence $x_1, \ldots, x_n$ in a Noetherian ring $R$ , the Koszul complex $K(x_1, \ldots, x_n)$ is acyclic except in degree $0$ , where $H^0 = R/(x_1, \ldots, x_n)$ .

ExampleExt as cohomology

$\mathrm{Ext}^n_R(M, N) = H^n(\mathrm{Hom}_R(P_\bullet, N))$ where $P_\bullet \to M$ is a projective resolution. This is the cohomology of the Hom complex.

ExampleTor as homology

$\mathrm{Tor}_n^R(M, N) = H_n(P_\bullet \otimes_R N)$ where $P_\bullet \to M$ is a projective resolution. This is the homology of the tensor complex.

ExampleCohomology as a functor

$H^n : \mathbf{Ch}(\mathcal{A}) \to \mathcal{A}$ is an additive functor. A chain map $f : A^\bullet \to B^\bullet$ induces $H^n(f) : H^n(A^\bullet) \to H^n(B^\bullet)$ . Chain homotopic maps induce the same map on cohomology.

ExampleEuler characteristic

For a bounded complex with finite-length cohomology, the Euler characteristic is $\chi(A^\bullet) = \sum_n (-1)^n \ell(H^n(A^\bullet))$ . It is additive on short exact sequences of complexes.

ExampleCohomology of the mapping cone

For $f : A^\bullet \to B^\bullet$ , the long exact sequence of the mapping cone gives $\cdots \to H^n(A) \xrightarrow{H^n(f)} H^n(B) \to H^n(\mathrm{Cone}(f)) \to H^{n+1}(A) \to \cdots$ . Thus $f$ is a quasi-isomorphism iff $\mathrm{Cone}(f)$ is acyclic.

ExampleHypercohomology

For a complex of sheaves $\mathcal{F}^\bullet$ on $X$ , the hypercohomology $\mathbb{H}^n(X, \mathcal{F}^\bullet)$ is computed by taking an injective resolution of each sheaf and forming the total complex. It generalizes both sheaf cohomology and de Rham cohomology.

Theorem4.2Long exact sequence in cohomology

A short exact sequence of complexes $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ induces a long exact sequence:

$\cdots \to H^n(A) \to H^n(B) \to H^n(C) \xrightarrow{\delta^n} H^{n+1}(A) \to H^{n+1}(B) \to \cdots$

The connecting homomorphism $\delta^n$ is constructed via the Snake Lemma. See the detailed proof.

RemarkCohomology and derived categories

In the derived category $D(\mathcal{A})$ , two complexes are isomorphic if and only if they have the same cohomology in a strong sense (they are connected by a zigzag of quasi-isomorphisms). The cohomology functor $H^n$ factors through $D(\mathcal{A})$ .