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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Hom into an injective makes projective-resolution rows exact

Statement

Let $\dots \to P_{1} \to P_{0} \to M \to 0$ be a projective resolution and $N \to I^{∙}$ an injective resolution. For every $q \geq 0$ , the augmented row $0 ⟶ Hom (M, I^{q}) ⟶ Hom (P_{0}, I^{q}) ⟶ Hom (P_{1}, I^{q}) ⟶ \dots$ is exact. These augmentations commute with postcomposition by $d_{I}$ , so their edge complex is $Hom (M, I^{∙})$ 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 $I^{q}$ is injective, so $Hom (-, I^{q})$ is exact. Applying it to the augmented projective resolution gives the displayed exact row. Naturality of Hom shows that these augmentations commute with postcomposition by $d_{I}$ .

2.1step 1.1algebra∎

The objects in horizontal degree $- 1$ are $Hom (M, I^{q})$ , with vertical differential given by postcomposition by $d_{I}$ . They therefore form $Hom (M, I^{∙})$ and give the asserted natural edge map to the total complex.

Depends on

Used by

Dependency tree · two levels

7 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 (standard reference, not scraped)