PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-05‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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

1.1

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 $L_{n}^{P} F (Q) = 0$ .

L1L2givenconstruct

2.1

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.

L3step 1.1∎

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)