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Cohomology maps induced by cochain maps

PDF p. 4: Definition 2.2

Statement

A cochain map $f : C \to D$ induces a homomorphism $H^{i} (f) : H^{i} (C) \to H^{i} (D)$ for every integer $i$ . These homomorphisms preserve identities and composition. A cochain map is a family of homomorphisms $f^{i} : C^{i} \to D^{i}$ satisfying $d_{D}^{i} f^{i} = f^{i + 1} d_{C}^{i}$ .

Facts & Assumptions

[F1]

Cohomology is the group of cocycles modulo coboundaries, as in Cohomology of a cochain complex.

Proof

Given: Cochain complexes and cochain maps between them.

Goal: Construct the induced maps and establish their functoriality.

1.1F1construct

The defining commutation relation sends cocycles to cocycles and coboundaries to coboundaries. Thus $[x] \mapsto [f^{i} (x)]$ is a well-defined homomorphism on the quotient in [F1].

2.1step 1.1∎

Applying the identity or a composite to a cocycle gives respectively the same cocycle or the successive images. Hence the induced maps preserve identities and composition.

Depends on

Cohomology of a cochain complex

Used by · 2

Dependency tree · 3 levels

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