Krull–Schmidt category

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.

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 ${\displaystyle X_{1}\oplus X_{2}\oplus \cdots \oplus X_{r}\cong Y_{1}\oplus Y_{2}\oplus \cdots \oplus Y_{s}}$ where the ${\displaystyle X_{i}}$ and ${\displaystyle Y_{j}}$ are all indecomposable, then ${\displaystyle r=s}$, and there exists a permutation ${\displaystyle \pi }$ such that ${\displaystyle X_{\pi (i)}\cong Y_{i}}$ for all i.

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

• An abelian category in which every object has finite length.^[1] This includes as a special case the category of finite-dimensional modules over an algebra.
• The category of finitely-generated modules over a finite^[2] R-algebra, where R is a commutative Noetherian complete local ring.^[3]
• The category of coherent sheaves on a complete variety over an algebraically-closed field.^[4]

• Quiver
• Karoubi envelope