CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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${Hom}_{Z} (-, Z)$ need not preserve surjections on the right

Statement refuted

The contravariant functor ${Hom}_{Z} (-, Z)$ need not carry a short exact sequence to a short exact sequence: it is left exact but need not be right exact.

Facts & Assumptions

Given: The short exact sequence $0 \to Z \overset{\times 2}{\to} Z \to Z / 2 Z \to 0$ .

[F1]

Precomposition induces the contravariant maps on Hom groups (The abelian group ${Hom}_{R} (M, N)$ and maps induced by pre- and postcomposition).

[L1]

Contravariant Hom is left exact (Covariant and contravariant $Hom$ are left exact).

Counterexample

technique · direct

1.1

Every homomorphism $Z / 2 Z \to Z$ is zero, because the image of its generator would be killed by two; and evaluation at $1$ identifies ${Hom}_{Z} (Z, Z)$ with $Z$ .

algebra

1.2

Under this identification, precomposition with multiplication by two is the map $Z \overset{\times 2}{\to} Z$ , because $(f \circ (\times 2)) (1) = 2 f (1)$ .

F1algebra

2.1

Applying the functor therefore produces $0 \to 0 \to Z \overset{\times 2}{\to} Z,$ which is exact as [L1] predicts but whose final map is not surjective. Thus the functor is not right exact.

step 1.1step 1.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)

Hom⁡Z(−,Z)HomZ​(−,Z) need not preserve surjections on the right