CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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