DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Homotopically injective bounded below complex

Definition

A cochain complex $I$ is homotopically injective, or K-injective, if ${Hom}_{K} (A, I [r]) = 0$ for every acyclic cochain complex $A$ and every integer $r$ . Equivalently, $\underset{}{Hom} (A, I)$ is acyclic. This uses the Hom complex and shifted Hom identification fixed in Homotopically projective bounded above complex. A bounded-below K-injective complex additionally has $I^{n} = 0$ for all sufficiently negative $n$ . The K-injective property itself neither assumes boundedness nor means termwise injectivity.

Depends on

Homotopically projective bounded above complex

Used by

fs-an-unbounded-total-derived-functor-exists-from-enough-injectives-alone.md False statement
Morphisms into a homotopically injective complex need no roof Proposition
A bounded below complex of injectives is homotopically injective Theorem

Dependency tree · two levels

4 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

6.5.1 and 13.1 (K-injective definition); cochain convention (standard reference, not scraped)