DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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Projective resolutions in an abelian category

Definition

Let $A$ be an object of an abelian category. A projective resolution of $A$ is an augmented chain complex $\dots \to P_{2} \to P_{1} \to P_{0} \overset{ε}{\to} A \to 0$ such that every $P_{n}$ is projective and the augmented complex is exact at every displayed term.

Thus a projective resolution is an exact way of recovering $A$ from projective objects arranged in homological degrees.

Depends on

Used by

Under the Axiom of Choice, every module admits a projective resolution Corollary
An F-acyclic resolution Definition
Augmentation-preserving maps of projective resolutions Definition
Deleted resolutions Definition
Projective dimension of an object Definition
Supplied projective resolution data Definition
Syzygies and cosyzygies relative to a chosen resolution Definition
The first-quadrant tensor double complex of two projective resolutions Definition
The length of a resolution Definition
Tor from a projective resolution of the left module Definition
Tor from a projective resolution of the right module Definition
An explicit projective resolution of the vertex module S₂ for A₂ Example
Extending a partial comparison map by one degree Lemma
The degree-zero horseshoe lift Lemma
The graded horseshoe lemma for finite graded projective resolutions Lemma
The Khovanov-Seidel grid resolutions of the vertex modules Lemma
A projective object has a length-zero projective resolution Proposition
A chosen chain of projective epimorphisms gives a projective resolution Theorem
Finite homological dimension of the finite graded Khovanov-Seidel module category Theorem
Projective comparison maps exist Theorem
The diagonal Koszul complex is a finite free resolution of R Theorem
The horseshoe lemma for projective resolutions Theorem
The two-sided bar complex is a projective Aᵉ-resolution Theorem
The zero-th left derived functor of a right exact functor recovers the functor Theorem
Tor one of a cyclic abelian group detects n-torsion Theorem

Dependency tree · two levels

8 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)