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Cohomology of a cochain complex

PDF p. 4: Definition 2.2

Statement

For a cochain complex $C$ of abelian groups and an integer $i$ , the quotient $H^{i} (C) = \ker (d^{i} : C^{i} \to C^{i + 1}) / im (d^{i - 1} : C^{i - 1} \to C^{i})$ is an abelian group.

A cochain complex consists of abelian groups $C^{i}$ and homomorphisms $d^{i}$ satisfying $d^{i + 1} d^{i} = 0$ . Elements of the kernel are cocycles and elements of the image are coboundaries.

Facts & Assumptions

[F1]

A cochain differential squares to zero, as in Derived category of an abelian category.

Proof

Given: A cochain complex $C$ and an integer $i$ .

Goal: Show that the displayed quotient is an abelian group.

1.1F1

By [F1], every coboundary in degree $i$ is a cocycle. The kernel and image are subgroups of the abelian group $C^{i}$ , so the image is a normal subgroup of the kernel.

2.1step 1.1∎

The quotient by that subgroup therefore inherits a well-defined abelian group operation. This is $H^{i} (C)$ .

Depends on

Derived category of an abelian category

Used by · 1

Cohomology maps induced by cochain maps

Dependency tree · 3 levels

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