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 is the classical case, see for example Krause (2012), Corollary 3.3.3. This includes as a special case the category of finite-dimensional modules over an algebra. The category of finitely-generated modules over a finiteA finite -algebra is an -algebra which is finitely generated as an -module. -algebra, where is a commutative Noetherian complete local ring.Reiner (2003), Section 6, Exercises 5 and 6, p. 88. 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. Bulletin de la Société Mathématique de France, Volume 84 (1956), pp. 307-317. doi:10.24033/bsmf.1475. https://www.numdam.org/articles/10.24033/bsmf.1475/ 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.