LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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.

One-step extension of a partial injective resolution

Statement

Let $0 \to A \overset{η}{\to} I^{0} \to I^{1} \to \dots \to I^{n}$ be a coaugmented cochain complex that is exact at every displayed term except possibly at $I^{n}$ . Let $C^{n}$ be the cokernel of the previous displayed map, so $C^{0} = coker (η)$ and $C^{n} = coker (I^{n - 1} \to I^{n})$ for $n \geq 1$ .

If $j : C^{n} ↣ I^{n + 1}$ is a monomorphism into an injective object $I^{n + 1}$ , then composing the quotient map $I^{n} ↠ C^{n}$ with $j$ extends the complex by one term and makes it exact at $I^{n}$ .

Facts & Assumptions

Given: The displayed partial coaugmented complex and a chosen monomorphism $j : C^{n} ↣ I^{n + 1}$ with $I^{n + 1}$ injective.

[L1]

Exactness at a degree means that the image of the incoming map equals the kernel of the outgoing map (Exactness of a complex at a degree and acyclic complexes).

[L2]

A coaugmented cochain complex records the extra map from the resolved object (Coaugmented cochain complexes under an object).

[L3]

Injective objects are the allowable terms in an injective resolution (Injective object).

Proof

technique · direct

1.1

Let $π_{n} : I^{n} ↠ C^{n}$ be the cokernel map and define the new differential by $d^{n} : = j π_{n} : I^{n} \to I^{n + 1} .$ Since $π_{n}$ kills the image of the previous displayed map, the composite of that previous map with $d^{n}$ is zero, so the extended row is again a coaugmented cochain complex in the sense of [L2].

givenL2construct

2.1

Because $j$ is monic, the kernel of $d^{n} = j π_{n}$ is the kernel of $π_{n}$ , namely the image of the previous displayed map. By [L1], this is exactly the required exactness at $I^{n}$ . The new term is injective by the given hypothesis and [L3].

L1L3step 1.1∎

Depends on

Used by

Dependency tree · two levels

9 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)