Krull–Schmidt category

In category theory, 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.^[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]

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.

Notes

↑ This is the classical case, see for example Krause (2012), Corollary 3.3.3.
↑ A finite $R$ -algebra is an $R$ -algebra which is finitely generated as an $R$ -module.
↑ Reiner (2003), Section 6, Exercises 5 and 6, p. 88.
↑ Atiyah (1956), Theorem 2.

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. ISBN 0-19-852673-3.
Claus Michael Ringel (1984) Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics 1099, Springer-Verlag, 1984.

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