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 ⊕ X 2 ⊕ ⋯ ⊕ X r ≅ Y 1 ⊕ Y 2 ⊕ ⋯ ⊕ Y s {\displaystyle 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 {\displaystyle X_{i}} and Y j {\displaystyle Y_{j}} are all indecomposable, then r = s {\displaystyle r=s} , and there exists a permutation π {\displaystyle \pi } such that X π ( i ) ≅ Y i {\displaystyle 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.

See also

  • Quiver
  • Karoubi envelope

Notes

  1. ^ This is the classical case, see for example Krause (2012), Corollary 3.3.3.
  2. ^ A finite R-algebra is an R-algebra which is finitely generated as an R-module.
  3. ^ Reiner (2003), Section 6, Exercises 5 and 6, p. 88.
  4. ^ 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.