False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-05‡ rests on unproved material (inherited)

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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.
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‡ 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.

FALSE: effaceability means every positive value is zero

Statement

Assume the Axiom of Dependent Choice.

False. If a delta functor is effaceable in positive degrees, then all of its positive-degree values are zero.

Facts & Assumptions

Given: The right exact functor $F = (-) \otimes_{Z} Z / n$ on abelian groups, where $n > 1$ is an integer, together with supplied projective resolution data on all abelian groups.

[L1]

Effaceability only says that a suitable induced map is zero (Effaceable homological delta functor in positive degrees, Effaceable cohomological delta functor in positive degrees).

[L2]

Positive left derived functors of a right exact functor on a category with enough projectives are effaceable (Positive left derived functors are effaceable by projectives).

[L3]

Replacing supplied projective resolution data gives naturally isomorphic left derived functors (Two supplied projective resolution data define naturally isomorphic left derived functors).

Refutation

technique · direct

1.1

Every abelian group is a quotient of a free abelian group, so $A b$ has enough projectives. Hence [L2] makes the positive left derived functors of $F$ effaceable in the sense of [L1].

L1L2given

1.2

Let $Q$ be supplied projective resolution data obtained from the given datum by using $0 \to Z \overset{n}{\to} Z \to Z / n \to 0$ at the object $Z / n$ . Applying $F$ to this resolution gives a deleted complex whose differential $n : Z / n \to Z / n$ is zero. Therefore $L_{1}^{Q} F (Z / n) ≅ \ker (Z / n \overset{0}{\to} Z / n) ≅ Z / n \neq 0.$ By [L3], $L_{1}^{P} F (Z / n) ≅ L_{1}^{Q} F (Z / n)$ for the original supplied datum $P$ , so its first derived value is nonzero as well.

L3givenalgebra

2.1

Thus this homological delta functor is effaceable in positive degrees but has a nonzero positive-degree value, refuting the statement.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

Alexandre Grothendieck, Some aspects of homological algebra (Barr translation) (standard reference, not scraped)
The Stacks Project, Section 12.12: Cohomological delta-functors (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Sections 2.4 and 2.5 (standard reference, not scraped)