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.
Contents
1 Definition
2 Properties
3 Examples
3.1 A non-example
4 See also
5 Notes
6 References
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 X1⊕X2⊕⋯⊕Xr≅Y1⊕Y2⊕⋯⊕Ysdisplaystyle X_1oplus X_2oplus cdots oplus X_rcong Y_1oplus Y_2oplus cdots oplus Y_s
where the Xidisplaystyle X_i
and Yjdisplaystyle Y_j
are all indecomposable, then r=sdisplaystyle r=s
, and there exists a permutation πdisplaystyle pi
such that Xπ(i)≅Yidisplaystyle 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
^ 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. .mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
ISBN 0-19-852673-3. - Claus Michael Ringel (1984) Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics 1099, Springer-Verlag, 1984.
