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The connecting morphism vanishes for a chain-split short exact sequence

Statement

Let $0 \to A_{∙} \overset{i}{\to} B_{∙} \overset{p}{\to} C_{∙} \to 0$ be a short exact sequence of complexes. If it admits either a chain section $s : C_{∙} \to B_{∙}$ with $p s = 1_{C}$ or a chain retraction $r : B_{∙} \to A_{∙}$ with $r i = 1_{A}$ , then $\partial_{n} = 0 : H_{n} (C) \to H_{n - 1} (A)$ for every $n$ .

Facts & Assumptions

Given: A short exact sequence $0 \to A_{∙} \overset{i}{\to} B_{∙} \overset{p}{\to} C_{∙} \to 0$ of complexes.

[L1]

A chain map is a degreewise morphism commuting with the differentials (Chain map).

[L2]

The connecting morphism is induced from the preconnecting arrow on cycles (The connecting morphism in homology, The preconnecting arrow on cycles).

[L3]

The associated homology sequence is exact (The long exact sequence in homology).

Proof

technique · direct

1.1L1L3givenalgebra

Suppose first that there is a chain section $s$ with $p s = 1_{C}$ . Functoriality of homology gives $H_{n} (p) H_{n} (s) = 1_{H_{n} (C)}$ , so $H_{n} (p)$ is epic. Exactness in [L3] gives $\ker (\partial_{n}) = im (H_{n} (p)) = H_{n} (C)$ , hence $\partial_{n} = 0$ .

2.1L1L2L3givenalgebra∎

Suppose instead that there is a chain retraction $r$ with $r i = 1_{A}$ . Then $H_{n - 1} (r) H_{n - 1} (i) = 1_{H_{n - 1} (A)}$ , so $H_{n - 1} (i)$ is monic. Exactness in [L3] gives $im (\partial_{n}) = \ker (H_{n - 1} (i)) = 0$ , and a morphism with zero image in an abelian category is zero. Thus the connecting morphism vanishes in either chain-split situation.

Depends on

Used by

Homology of a chain-split direct-sum sequence Corollary

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)