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

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A chosen chain of projective epimorphisms gives a projective resolution

Statement

Let $A$ be an object of an abelian category. Suppose one has chosen an epimorphism $P_{0} ↠ A$ with $P_{0}$ projective and, for each $n \geq 0$ , an epimorphism $P_{n + 1} ↠ K_{n}$ from a projective object onto the current kernel $K_{n}$ of the previous displayed map. Then composing each chosen epimorphism with its kernel inclusion produces an augmented complex $\dots \to P_{2} \to P_{1} \to P_{0} \to A \to 0$ that is a projective resolution of $A$ .

Facts & Assumptions

Given: An object $A$ of an abelian category, together with a chosen projective epimorphism onto $A$ and a chosen projective epimorphism onto each successive kernel.

[L1]

A chosen projective epimorphism onto the current kernel extends a partial resolution by one exact step (One-step extension of a partial projective resolution).

[L2]

A projective resolution is an exact augmented complex of projective objects (Projective resolutions in an abelian category).

Proof

technique · direct

1.1

Start with the chosen epimorphism $P_{0} ↠ A$ . Applying [L1] to the chosen epimorphism $P_{1} ↠ K_{0}$ makes $P_{1} \to P_{0} \to A \to 0$ exact, and repeating the same step with the chosen epimorphism onto each later kernel produces an augmented exact complex $\dots \to P_{2} \to P_{1} \to P_{0} \to A \to 0$ whose terms are all projective.

L1givenconstruct

2.1

By [L2], the complex assembled in step 1.1 is a projective resolution of $A$ , including the case $A = 0$ when the chosen initial epimorphism may be $0 ↠ 0$ .

L2step 1.1∎

Depends on

Used by

FALSE: every abelian category has enough projectives and enough injectives False statement
FALSE: objectwise projective-resolution choices uniquely determine a resolution functor False statement

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)
The Stacks Project, Section 12.28: Projectives (standard reference, not scraped)