PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-05

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Positive left derived functors are effaceable by projectives

Statement

Assume the Axiom of Dependent Choice.

Let $A$ be an abelian category with enough projectives, let $P$ be supplied projective resolution data on all objects of $A$ . Let $F : A \to B$ be an additive right exact functor. Then the homological delta functor $(L_{n}^{P} F)$ is effaceable in positive degrees by projectives.

Facts & Assumptions

Given: An object $A \in A$ and an integer $n > 0$ .

[L1]

Enough projectives gives an epimorphism $q : Q ↠ A$ with $Q$ projective (A category with enough projectives and with enough injectives).

[L2]

Positive left derived functors vanish on projective objects (Positive left derived functors vanish on projective objects).

[L3]

Effaceability in positive degrees means killing the induced map from some projective epimorphism (Effaceable homological delta functor in positive degrees).

Proof

technique · direct

1.1L1L2givenconstruct

By [L1], choose a projective epimorphism $q : Q ↠ A$ . The datum $P$ is defined on all of $A$ , so $L_{n}^{P} F (Q)$ is defined; since $Q$ is projective and $n > 0$ , [L2] gives $L_{n}^{P} F (Q) = 0$ .

2.1L3step 1.1∎

The induced map $L_{n}^{P} F (q) : L_{n}^{P} F (Q) \to L_{n}^{P} F (A)$ is therefore the zero map. By [L3], this is exactly the required positive-degree effacement of $L_{n}^{P} F (A)$ by a projective object.

Depends on

Used by

FALSE: effaceability means every positive value is zero False statement
Derived functors are universal delta functors Theorem

Dependency tree · two levels

15 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, Chapter 2 `Derived Functors` (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)