LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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The horseshoe construction stays short exact after applying a right exact functor

Statement

Assume the Axiom of Dependent Choice.

Let $F : A \to B$ be a right exact functor between abelian categories, and let $0 \to A^{'} \to A \to A^{''} \to 0$ be a short exact sequence in $A$ . If $0 \to P_{∙}^{'} \to H_{∙} \to P_{∙}^{''} \to 0$ is a horseshoe short exact sequence of projective resolutions of $A^{'}, A, A^{''}$ , then $0 \to F (P_{∙, d e l}^{'}) \to F (H_{∙, d e l}) \to F (P_{∙, d e l}^{''}) \to 0$ is a short exact sequence of complexes in $B$ .

Facts & Assumptions

Given: A horseshoe short exact sequence of projective resolutions over $0 \to A^{'} \to A \to A^{''} \to 0$ .

[L1]

A right exact functor is additive (A left or right exact functor between abelian categories is automatically additive).

[L2]

The horseshoe lemma produces a degreewise split short exact sequence of projective resolutions (The horseshoe lemma for projective resolutions).

[L3]

Additive functors apply degreewise to complexes and chain maps (An additive functor applies degreewise to complexes and chain maps).

[L4]

Exactness of a sequence of complexes is equivalent to exactness in each degree, and that is the definition of a short exact sequence of complexes (A sequence of chain maps is exact exactly when it is exact degreewise, Short exact sequence of complexes).

Proof

technique · direct

1.1L1L2givenalgebra

By [L2], each degree of the horseshoe row is a split short exact sequence $0 \to P_{n}^{'} \to H_{n} \to P_{n}^{''} \to 0.$ Because $F$ is additive by [L1], it preserves the biproduct decomposition carried by that split sequence, so each degree remains exact after applying $F$ .

2.1L3L4step 1.1construct∎

By [L3], the degreewise images from step 1.1 assemble into a sequence of chain maps $0 \to F (P_{∙, d e l}^{'}) \to F (H_{∙, d e l}) \to F (P_{∙, d e l}^{''}) \to 0.$ Since it is exact in every degree, [L4] identifies it as a short exact sequence of complexes.

Depends on

Used by

The connecting map for left derived functors Definition

Dependency tree · two levels

23 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, Chapter 2 `Derived Functors` (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)