Homological algebra MOC

Chain map

A chain map¹ between chain complexes in is a sequence of homomorphisms such that the following diagram commutes in for all :² homology

It follows that each maps -cycles to -cycles and -boundaries to -boundaries, and hence there is an induced homomorphism between chain homologies, defined by .

Proof

Let so . Then and thus . Now let so for some . Then and thus .

Chain maps form morphisms in Category of chain complexes.

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Footnotes

1. German Kettenabbildung ↩

2. 2010, Algebraische Topologie, ¶3.1.4, p. 128 ↩