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 $R$-linear category for a commutative ring $R$. 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 $$X_1 \oplus X_2 \oplus \cdots \oplus X_r \cong Y_1 \oplus Y_2 \oplus \cdots \oplus Y_s$$ where the $$X_i$$ and $$Y_j$$ are all indecomposable, then $$r=s$$, and there exists a permutation $$\pi$$ such that $$X_{\pi(i)} \cong Y_i$$ for all $i$.

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 $R$-algebra, where $R$ 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.