CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-01

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A split short exact sequence admits the direct-sum resolution

Statement

Assume the Axiom of Dependent Choice.

Let $0 \to A^{'} \to A \to A^{''} \to 0$ be a split short exact sequence, and let projective resolutions of $A^{'}$ and $A^{''}$ be given. Then the sequence admits a projective resolution of $A$ whose degree- $n$ term is the direct sum $P_{n}^{'} \oplus P_{n}^{''}$ of the chosen side terms.

Facts & Assumptions

Given: A split short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ and chosen projective resolutions of $A^{'}$ and $A^{''}$ .

[L1]

The horseshoe lemma produces a middle projective resolution (The horseshoe lemma for projective resolutions).

[L2]

A split short exact sequence identifies the middle object with the direct sum of the two ends (Split short exact sequence in an abelian category).

Proof

technique · direct

1.1L1L2construct

By [L2], identify $A$ with $A^{'} \oplus A^{''}$ . Applying [L1] to the chosen projective resolutions and the chosen splitting maps gives a middle resolution whose degree- $n$ term is $P_{n}^{'} \oplus P_{n}^{''}$ and whose augmentation is the direct-sum augmentation.

2.1step 1.1∎

Under that identification, the differentials are exactly the direct-sum differentials of the two side resolutions. Hence the split short exact sequence admits the direct-sum resolution.

Depends on

Used by

The horseshoe resolution of an extension of cyclic groups Example

Dependency tree · two levels

8 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, An Introduction to Homological Algebra (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)