TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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A bounded below complex of injectives is homotopically injective

Statement

A bounded-below cochain complex $I$ of injective objects is K-injective. Assume dependent choice for the countable successive homotopy extensions, or supply those extensions as data.

Facts & Assumptions

Given: A bounded-below cochain complex $I$ of injective objects is K-injective. Assume dependent choice for the countable successive homotopy extensions, or supply those extensions as data.

[F1]

An injective object extends maps from a subobject to the ambient object (Injective object).

[F2]

K-injectivity is vanishing of Hom from every acyclic complex into every shift of the target (Homotopically injective bounded below complex).

[F3]

DC supplies a sequence extending successive choices on a nonempty set (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Proof

1.1givenalgebra

Let $A$ be acyclic and $f : A \to I$ a chain map. Choose $a$ with $I^{n} = 0$ for $n < a$ and set $h^{n} : A^{n} \to I^{n - 1}$ equal to zero for $n \leq a$ . The equation $f^{n} = d_{I} h^{n} + h^{n + 1} d_{A}$ holds below $a$ . In particular it is valid for zero complexes.

2.1F1step 1.1algebra

If the equation holds below $n$ , then $u^{n} = f^{n} - d_{I}^{n - 1} h^{n}$ vanishes on $im d_{A}^{n - 1}$ by the chain-map identity. Since $A$ is acyclic this image equals $\ker d_{A}^{n}$ , so $u^{n}$ factors through $im d_{A}^{n} ↪ A^{n + 1}$ . Injectivity of $I^{n}$ extends it to $h^{n + 1} : A^{n + 1} \to I^{n}$ . This establishes the equation in degree $n$ .

3.1F2F3step 2.1∎

Apply DC to the set of finite partial homotopies with the entire extension relation, or take the supplied successive extensions. The resulting homotopy makes $f$ zero in $K$ . Apply the same argument to every shift $I [r]$ , which remains bounded below and termwise injective. This is the K-injective condition.

Depends on

Dependency tree · 2 levels

Sources

13.18.3–13.18.8; W 10.4.8 for the equivalence