TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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A bounded below acyclic complex of projective objects is contractible when its cycle epimorphisms split

Statement

Let $C_{∙}$ be a bounded-below acyclic chain complex of projective objects in an abelian category. For each $n$ , let $π_{n} : C_{n} \to Z_{n - 1} (C)$ be the canonical epimorphism characterized by $d_{n} = i_{n - 1} π_{n},$ where $i_{n - 1} : Z_{n - 1} (C) ↪ C_{n - 1}$ is the cycle inclusion. If every $π_{n}$ admits a section $σ_{n}$ , then $C_{∙}$ is contractible.

Facts & Assumptions

Given: A bounded-below acyclic complex $C_{∙}$ and splittings $σ_{n} : Z_{n - 1} (C) \to C_{n}$ with $π_{n} σ_{n} = 1$ .

[L1]

Boundaries and cycles are the image of $d_{n + 1}$ and kernel of $d_{n}$ (Cycle and boundary subobjects of a complex).

[L2]

Acyclic means exact at every degree, so $B_{n - 1} (C) = Z_{n - 1} (C)$ (Exactness of a complex at a degree and acyclic complexes).

[L3]

A bounded-below complex has only finitely many nonzero terms below each degree (Bounded, bounded below, and bounded above complexes).

[L4]

The split-exact criterion of the previous lemma yields contractibility (A degreewise split exact complex with compatible splittings is contractible).

[L5]

The stated proof uses the chosen sections. Projectivity of the terms $C_{n}$ alone does not provide sections of $C_{n} ↠ Z_{n - 1} (C)$ ; the lifting property in Projective object would provide such a section if the target $Z_{n - 1} (C)$ were projective.

Proof

technique · direct

1.1L1L2givenalgebra

By [L2], each short exact sequence $0 \to Z_{n} (C) \to C_{n} \overset{π_{n}}{\to} Z_{n - 1} (C) \to 0$ is exact, and the section $σ_{n}$ splits it. Therefore $C_{n} ≅ Z_{n} (C) \oplus Z_{n - 1} (C)$ for every $n$ , with differential equal to projection onto the second summand followed by the cycle inclusion.

2.1L3L4L5step 1.1algebra∎

Step 1.1 is exactly the compatible decomposition required by [L4], so $C_{∙}$ is contractible. As [L5] emphasizes, the contraction comes from the chosen sections rather than from projectivity of the terms alone.

Depends on

Used by

Dependency tree · two levels

13 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)