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 $X_1 \oplus X_2 \oplus \cdots \oplus X_r \cong Y_1 \oplus Y_2 \oplus \cdots \oplus Y_s$ where the $X_i$ and $Y_j$ are all indecomposable, then $r=s$, and there exists a permutation $\pi$ such that $X_ \cong Y_i$ 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.Atiyah (1956), Theorem 2.

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.

See also

* Quiver
* Karoubi envelope

Notes

References

* Michael Atiyah (1956) ''On the Krull-Schmidt theorem with application to sheaves'
Bull. Soc. Math. France 84
307–317.
* Henning Krause
Krull-Remak-Schmidt categories and projective covers
May 2012.
* Irving Reiner (2003) ''Maximal orders. Corrected reprint of the 1975 original. With a foreword by M. J. Taylor.'' London Mathematical Society Monographs. New Series, 28. The Clarendon Press, Oxford University Press, Oxford. .
* Claus Michael Ringel (1984) ''Tame Algebras and Integral Quadratic Forms'', Lecture Notes in Mathematics 1099, Springer-Verlag, 1984.

{{DEFAULTSORT:Krull-Schmidt category
Category theory
Representation theory