LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Hom from a projective makes injective-resolution columns exact

Statement

Let $P_{∙} \to M$ be a projective resolution and $0 \to N \to I^{0} \to I^{1} \to \dots$ an injective resolution. For every $p \geq 0$ , the augmented column $0 ⟶ Hom (P_{p}, N) ⟶ Hom (P_{p}, I^{0}) ⟶ Hom (P_{p}, I^{1}) ⟶ \dots$ is exact. These augmentations commute with precomposition by $d_{P}$ , so their edge complex is $Hom (P_{∙}, N)$ and maps naturally to the total Hom complex.

Facts & Assumptions

Given: The two resolutions in the statement and the Hom double complex $K^{p, q} = Hom (P_{p}, I^{q})$ .

Proof

technique · direct

1.1givenalgebra

Each $P_{p}$ is projective, so $Hom (P_{p}, -)$ is exact. Applying it to the augmented injective resolution gives the displayed exact column. Naturality of Hom shows that these augmentations commute with precomposition by $d_{P}$ .

2.1step 1.1algebra∎

The objects in vertical degree $- 1$ are $Hom (P_{p}, N)$ , with horizontal differential given by precomposition by $d_{P}$ . They therefore form $Hom (P_{∙}, N)$ and give the asserted natural edge map to the total complex.

Depends on

Dependency tree · 2 levels

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2