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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The degree-zero injective construction of Ext is Hom

Statement

Assume the Axiom of Dependent Choice, and let $I$ be supplied injective resolution data on the objects under consideration. For $0 \to N \overset{ι}{\to} I^{0} \overset{d^{0}}{\to} I^{1} \to \dots$ , $f \mapsto ι f$ gives a natural isomorphism $Hom (M, N) ≅ {Ext}_{I}^{0} (M, N)$ .

Facts & Assumptions

Given: The displayed injective resolution and an object $M$ .

Proof

technique · direct

1.1givenconstruct

Exactness gives $\ker d^{0} = im ι$ ; thus $ι f$ is a degree-zero cocycle, and $f \mapsto ι f$ is injective because $ι$ is monic.

2.1step 1.1algebra∎

If $d^{0} g = 0$ , then $g$ factors uniquely through $ι$ . Since there are no negative-degree coboundaries, this is the required $H^{0}$ identification. Precomposition makes it natural in $M$ . For $b : N \to N^{'}$ , choose a comparison extension between the supplied resolutions; its degree-zero square with the coaugmentations commutes, so the identification is natural in $N$ , and the independence lemma makes this map independent of the chosen extension.

Depends on

Used by

Ext zero as Hom in both constructions Example

Dependency tree · two levels

17 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)