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A bounded above acyclic complex of injective objects is contractible when its cycle monomorphisms split

Statement

Let $C_{∙}$ be a bounded-above acyclic chain complex of injective objects in an abelian category. If every cycle inclusion $i_{n} : Z_{n} (C) ↪ C_{n}$ admits a retraction $r_{n} : C_{n} \to Z_{n} (C)$ , then $C_{∙}$ is contractible.

Facts & Assumptions

Given: A bounded-above acyclic complex $C_{∙}$ and retractions $r_{n} : C_{n} \to Z_{n} (C)$ with $r_{n} i_{n} = 1$ .

[L1]

Boundaries and cycles are defined degreewise in a chain complex (Cycle and boundary subobjects of a complex).

[L2]

Acyclic means exact at every degree (Exactness of a complex at a degree and acyclic complexes).

[L3]

The previous theorem treats the dual split criterion on the epimorphism side (A bounded below acyclic complex of projective objects is contractible when its cycle epimorphisms split).

[L4]

The stated proof uses the chosen retractions. Injectivity of the terms $C_{n}$ alone does not provide retractions onto $Z_{n} (C)$ ; the extension property in Injective object would provide such a retraction if the target $Z_{n} (C)$ were injective.

[L5]

Opposite abelian categories are abelian (The opposite of an abelian category is abelian).

Proof

technique · direct

1.1

Because $i_{n}$ splits, each exact sequence $0 \to Z_{n} (C) \overset{i_{n}}{\to} C_{n} \to Z_{n - 1} (C) \to 0$ decomposes $C_{n}$ as $C_{n} ≅ Z_{n} (C) \oplus Z_{n - 1} (C),$ with the differential again equal to projection onto the second factor followed by inclusion into $C_{n - 1}$ .

L1L2givenalgebra

2.1

Step 1.1 is the same compatible splitting pattern used in [L3], so the same contraction argument applies and $C_{∙}$ is contractible. The proof depends on the assumed retractions, as [L4] emphasizes; [L5] explains the formal duality with the epimorphism-side criterion.

L3L4L5step 1.1algebra∎

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Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)