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 #m/def/homology

https://q.uiver.app/#q=WzAsMTAsWzQsMCwiQV9rIl0sWzYsMCwiQV97aysxfSJdLFsyLDAsIkFfe2stMX0iXSxbMiwyLCJCX3trLTF9Il0sWzQsMiwiQl9rIl0sWzYsMiwiQl97aysxfSJdLFswLDAsIlxcY2RvdHMiXSxbMCwyLCJcXGNkb3RzIl0sWzgsMiwiXFxjZG90cyJdLFs4LDAsIlxcY2RvdHMiXSxbNSw0LCJkX3trKzF9Il0sWzQsMywiZF9rIl0sWzEsMCwiXFxwYXJ0aWFsX3trKzF9IiwyXSxbMCwyLCJcXHBhcnRpYWxfayIsMl0sWzksMSwiXFxwYXJ0aWFsX3trKzJ9IiwyXSxbOCw1LCJkX3trKzJ9Il0sWzIsNiwiXFxwYXJ0aWFsX3trLTF9IiwyXSxbMyw3LCJkX3trLTF9Il0sWzIsMywiZl97ay0xfSIsMl0sWzAsNCwiZl9rIiwyXSxbMSw1LCJmX3trKzF9IiwyXV0=

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 .

#state/tidy | #lang/en | #SemBr

Footnotes

  1. German Kettenabbildung ↩

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