TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-29

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An object is projective exactly when Hom out of it is exact

Statement

Let $P$ and $I$ be objects of an abelian category.

$P$ is projective if and only if the functor $A (P, -)$ is exact.
$I$ is injective if and only if the functor $A (-, I)$ is exact.

Facts & Assumptions

Given: An abelian category and objects $P$ and $I$ in it.

[L1]

Hom is left exact in each variable (Hom is left exact in each variable).

[L2]

Projective objects are exactly those for which Hom out of them sends every short exact sequence to a short exact sequence (Projective object, Projective object characterisations).

[L3]

Injective objects are exactly those for which Hom into them sends every short exact sequence to a short exact sequence (Injective object, Injective object characterisations).

[L4]

An exact functor is one that is both left exact and right exact (Exact functor between abelian categories).

Proof

technique · direct

1.1L1L2L4

By [L1], the functor $A (P, -)$ is always left exact. Therefore, by [L4], it is exact exactly when it is also right exact on every short exact sequence. But [L2] says that extra right-end surjectivity is exactly the projective lifting property.

2.1L1L3L4∎

The same argument for the contravariant Hom functor uses [L1], [L3], and [L4]: $A (-, I)$ is always left exact, and exactness is equivalent to the additional surjectivity that characterizes injectivity.

Depends on

Used by

Projective Hom pairing descends and is graded sesquilinear Theorem

Dependency tree · two levels

16 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

Pavel Etingof et al., Tensor Categories, Section 1.6 (standard reference, not scraped)