False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-05

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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

1.1L1L2given

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].

1.2L3givenalgebra

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.

2.1step 1.1step 1.2∎

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

Depends on

Used by

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Dependency tree · two levels

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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)