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 projective resolution

Statement

Let $P_{n} \to P_{n - 1} \to \dots \to P_{0} \overset{ε}{\to} A \to 0$ be an augmented chain complex that is exact at every displayed term except possibly at $P_{n}$ . Let $K_{n}$ be the kernel of the previous displayed map, so $K_{0} = \ker (ε)$ and $K_{n} = \ker (P_{n} \to P_{n - 1})$ for $n \geq 1$ .

If $q : P_{n + 1} ↠ K_{n}$ is an epimorphism from a projective object $P_{n + 1}$ , then composing $q$ with the kernel inclusion $K_{n} ↪ P_{n}$ extends the complex by one term and makes it exact at $P_{n}$ .

Facts & Assumptions

Given: The displayed partial augmented complex and a chosen epimorphism $q : P_{n + 1} ↠ K_{n}$ with $P_{n + 1}$ projective.

[L1]

Exactness at a degree means that the image of the incoming differential is the kernel subobject of the outgoing differential (Exactness of a complex at a degree and acyclic complexes).

[L2]

An augmented chain complex records the extra map to the resolved object (Augmented chain complexes over an object).

[L3]

Projective objects are the allowable terms in a projective resolution (Projective object).

Proof

technique · direct

1.1

Let $i_{n} : K_{n} ↪ P_{n}$ be the kernel inclusion, and define the new differential by $d_{n + 1} : = i_{n} q : P_{n + 1} \to P_{n} .$ Because $i_{n}$ lands in the kernel of the previous displayed map, the composite of the new differential with that previous map is zero, so the extended row is again an augmented chain complex in the sense of [L2].

givenL2construct

2.1

The image of $d_{n + 1}$ is the image of $i_{n} q$ , which is exactly $K_{n}$ because $q$ is epic. By [L1], this is precisely the exactness condition at $P_{n}$ . The new term is projective by the given hypothesis and [L3].

L1L3step 1.1∎

Depends on

Used by

Dependency tree · two levels

8 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)