ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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The idempotent completion of the matrix category gives the finitely generated projective modules

Example

For a commutative ring $R$ , the matrix functor extends to a fully faithful functor $\overline{K} : Kar ({M a t}_{R}) ⟶ R - M o d$ whose essential image is exactly the finitely generated projective $R$ -modules.

Facts & Assumptions

Given: A commutative ring $R$ .

[L1]

The matrix category is equivalent to the finitely generated free $R$ -modules (The matrix category is fully faithful in modules and, with chosen bases, equivalent to finite free modules).

[L2]

The Karoubi envelope adjoins splitting objects for idempotents (The idempotent completion of a preadditive category, The idempotent completion is idempotent complete and its inclusion is fully faithful and universal).

[L3]

Finite free modules are projective without any choice principle, and projective means having the lifting property against surjections (Free modules are projective, with the exact choice boundary, Projective modules and the lifting property).

[L4]

A functor is fully faithful when it induces a bijection on every hom-set, and its essential image consists of the objects isomorphic to objects in its image (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors).

Verification

technique · direct

1.1

For an object $(n, e)$ , define its splitting module $P_{e} : = im (e : R^{n} \to R^{n})$ . [L1, L2, construct] By [L1], $e$ is an idempotent endomorphism of $R^{n}$ . Let $i_{e} : P_{e} ↣ R^{n}$ be inclusion, and let $p_{e} : R^{n} ↠ P_{e}$ be $e$ with codomain restricted to its image. Then $i_{e} p_{e} = e$ and $p_{e} i_{e} = 1_{P_{e}}$ , so these maps give an actual splitting in $R - M o d$ .

L1L2algebra

2.1

The module $P_{e}$ is finitely generated and projective. [L3, step 1.1] Since $R^{n}$ is projective by [L3], composing any lifting problem for $P_{e}$ with $p_{e}$ and then restricting the resulting lift along $i_{e}$ proves that $P_{e}$ is projective. It is finitely generated because the images under $p_{e}$ of the standard generators of $R^{n}$ generate it.

L3step 1.1

2.2

The object assignment $(n, e) \mapsto P_{e}$ and restriction on morphisms define a functor $\overline{K}$ . [L1, L2, step 1.1, construct] If $u : (n, e) \to (m, f)$ is a Karoubi morphism, [L2] gives $f u e = u$ , equivalently $f u = u = u e$ . Hence $u$ maps $P_{e}$ into $P_{f}$ ; define $\overline{K} (u) : = u ∣_{P_{e}} : P_{e} \to P_{f}$ . Restrictions preserve identities and composition, so this defines a functor.

L1L2step 1.1

3.1

The functor $\overline{K}$ is fully faithful. [L1, L2, L4, step 1.1, step 2.2, algebra] For objects $(n, e)$ and $(m, f)$ , the map on hom-sets in step 2.2 is bijective. Indeed, for any homomorphism $h : P_{e} \to P_{f}$ , the composite $u : = i_{f} h p_{e} : R^{n} ⟶ R^{m}$ corresponds by [L1] to a matrix and satisfies $f u e = u$ , so it is a Karoubi morphism whose restriction is $h$ . Conversely, $u e = u$ forces every Karoubi morphism to equal $i_{f} (u ∣_{P_{e}}) p_{e}$ , proving uniqueness. Thus $\overline{K}$ is fully faithful by [L4].

L1L2L4step 1.1step 2.2algebra

3.2

Every finitely generated projective $R$ -module lies in the essential image of $\overline{K}$ . [L2, L3, step 1.1, step 2.2] Indeed, let $P$ be a finitely generated projective module. A finite generating set gives a surjection $q : R^{n} \to P$ . Projectivity from [L3] supplies a section $s : P \to R^{n}$ with $q s = 1_{P}$ . Then $e : = s q$ is idempotent, and $q$ restricts to an isomorphism $P_{e} \to P$ with inverse $s$ . Hence every finitely generated projective module lies in the essential image of $\overline{K}$ .

L2L3step 1.1step 2.2

4.1

Step 2.1 and step 3.2 identify the essential image, and step 3.1 proves full faithfulness; hence the stated claim holds.

L4step 2.1step 3.1step 3.2∎

Depends on

Used by

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Sources

Dixy Msapato, The Karoubi envelope and weak idempotent completion of an extriangulated category, Section 2.1 (standard reference, not scraped)
Kiran S. Kedlaya, Solid modules over an ordinary ring, Example 1.2.6 (standard reference, not scraped)