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

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The horseshoe lemma for injective resolutions

Statement

Assume the Axiom of Dependent Choice.

Let $0 \to A^{'} \to A \to A^{''} \to 0$ be a short exact sequence, and let injective resolutions of $A^{'}$ and $A^{''}$ be given. Then there exists an injective resolution of $A$ whose degree- $n$ term is a finite product, equivalently biproduct, $I^{' n} \oplus I^{'' n}$ .

Facts & Assumptions

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

[L1]

The projective horseshoe lemma holds (The horseshoe lemma for projective resolutions).

[L2]

Injective resolutions are the cochain objects to be dualized (Injective resolutions in an abelian category).

[L3]

The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).

Proof

technique · direct

1.1L1L2L3construct

By [L3], pass to the opposite abelian category. The given injective resolutions from [L2] become projective resolutions there, so [L1] supplies the dual horseshoe resolution in the opposite category.

2.1step 1.1∎

Translating back to the original category reverses arrows again and turns the opposite-category coproducts into finite products, which in an abelian category are the same biproducts. Hence one obtains the asserted injective horseshoe resolution.

Depends on

Used by

Classical derived functors are the cohomology objects of the total derived functor Proposition
The two Ext long exact sequences agree under balance Proposition
Right derived functors form a cohomological delta functor Theorem

Dependency tree · two levels

13 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)