TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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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.

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A chosen chain of injective embeddings gives an injective resolution

Statement

Let $A$ be an object of an abelian category. Suppose one has chosen a monomorphism $A ↣ I^{0}$ into an injective object and, for each $n \geq 0$ , a monomorphism $C^{n} ↣ I^{n + 1}$ from the current cokernel $C^{n}$ of the previous displayed map into an injective object. Then composing each quotient map with its chosen embedding produces a coaugmented complex $0 \to A \to I^{0} \to I^{1} \to I^{2} \to \dots$ that is an injective resolution of $A$ .

Facts & Assumptions

Given: An object $A$ of an abelian category, together with a chosen injective embedding of $A$ and a chosen injective embedding of each successive cokernel.

[L1]

A chosen injective embedding of the current cokernel extends a partial coaugmented resolution by one exact step (One-step extension of a partial injective resolution).

[L2]

An injective resolution is an exact coaugmented complex of injective objects (Injective resolutions in an abelian category).

Proof

technique · direct

1.1L1givenconstruct

Start with the chosen monomorphism $A ↣ I^{0}$ . Applying [L1] to the chosen embedding $C^{0} ↣ I^{1}$ makes $0 \to A \to I^{0} \to I^{1}$ exact, and repeating the same step with the chosen embedding of each later cokernel produces an exact coaugmented complex $0 \to A \to I^{0} \to I^{1} \to I^{2} \to \dots$ whose terms are all injective.

2.1L2step 1.1∎

By [L2], the complex assembled in step 1.1 is an injective resolution of $A$ , including the case $A = 0$ when the chosen initial embedding may be $0 ↣ 0$ .

Depends on

Used by

Cartan-Eilenberg injective resolutions exist Theorem

Dependency tree · two levels

5 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)