LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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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A morphism has a comparison extension between the supplied injective resolutions

Statement

Assume the Axiom of Dependent Choice.

Let $I$ be a supplied injective resolution datum on a class $D$ in an abelian category. For every morphism $u : A \to B$ with $A, B \in D$ , there exists a coaugmentation-preserving cochain map ${\tilde{u}}^{∙} : I^{∙} (A) \to I^{∙} (B)$ extending $u$ .

Facts & Assumptions

Given: A morphism $u : A \to B$ with $A, B \in D$ .

[L1]

The datum $I$ supplies specific injective resolutions $A \to I^{∙} (A)$ and $B \to I^{∙} (B)$ (Supplied injective resolution data).

[L2]

Assuming Dependent Choice, injective comparison maps exist for morphisms between chosen injective resolutions (Injective comparison maps exist).

Proof

technique · direct

1.1

By [L1], the objects $A$ and $B$ come with chosen injective resolutions.

L1given

2.1

Apply [L2] to $u : A \to B$ and the resolutions from step 1.1. The resulting coaugmentation-preserving cochain map ${\tilde{u}}^{∙} : I^{∙} (A) \to I^{∙} (B)$ is the required comparison extension.

L2step 1.1construct∎

Depends on

Used by

Dependency tree · two levels

6 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

Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed. (standard reference, not scraped)