Krull–Schmidt category

In category theory, a branch of mathematics, a Krull–Schmidt category is a generalization of categories in which the Krull–Schmidt theorem holds. They arise, for example, in the study of finite-dimensional modules over an algebra.

Definition

Let C be an additive category, or more generally an additive -linear category for a commutative ring . We call C a Krull–Schmidt category provided that every object decomposes into a finite direct sum of objects having local endomorphism rings. Equivalently, C has split idempotents and the endomorphism ring of every object is semiperfect.

Properties

One has the analogue of the Krull–Schmidt theorem in Krull–Schmidt categories:
An object is called indecomposable if it is not isomorphic to a direct sum of two nonzero objects. In a Krull–Schmidt category we have that

an object is indecomposable if and only if its endomorphism ring is local.
every object is isomorphic to a finite direct sum of indecomposable objects.
if where the and are all indecomposable, then, and there exists a permutation such that for all.

One can define the Auslander–Reiten quiver of a Krull–Schmidt category.

Examples

An abelian category in which every object has finite length. This includes as a special case the category of finite-dimensional modules over an algebra.
The category of finitely-generated modules over a finite -algebra, where is a commutative Noetherian complete local ring.
The category of coherent sheaves on a complete variety over an algebraically-closed field.

A non-example

The category of finitely-generated projective modules over the integers has split idempotents, and every module is isomorphic to a finite direct sum of copies of the regular module, the number being given by the rank. Thus the category has unique decomposition into indecomposables, but is not Krull-Schmidt since the regular module does not have a local endomorphism ring.