DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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.

Injective resolutions in an abelian category

Definition

Let $A$ be an object of an abelian category. An injective resolution of $A$ is a coaugmented cochain complex $0 \to A \overset{η}{\to} I^{0} \to I^{1} \to I^{2} \to \dots$ such that every $I^{n}$ is injective and the coaugmented complex is exact at every displayed term.

Thus the object $A$ sits as the initial term of an exact cochain complex of injective objects.

Depends on

Used by

Every Grothendieck category has enough injectives, and every object admits an injective resolution Corollary
Every module admits an injective resolution Corollary
Deleted resolutions Definition
Syzygies and cosyzygies relative to a chosen resolution Definition
The length of a resolution Definition
A chosen chain of injective embeddings gives an injective resolution Theorem
Injective comparison maps exist Theorem
The horseshoe lemma for injective resolutions Theorem

Dependency tree · two levels

9 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)