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In [[category theory]], a '''Krull–Schmidt category''' is a generalization of categories in which the [[Krull–Schmidt theorem]] holds. They arise in the study of the [[representation theory]] of finite-dimensional [[algebra over a field|algebra]]s.
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== Definition ==
Let ''k'' be a field. A category [[enriched category|enriched]] over finite-dimensional ''k''-[[vector space]]s is a Krull–Schmidt category if all idempotents split. In other words, if <math>e \in \text{End}(X)</math> satisfies <math>e^2 = e</math>, then there exists an object ''Y'' and morphisms <math>\mu \colon Y \to X</math> and <math>\rho \colon X \to Y</math> such that <math>\mu \rho = e</math> and <math>\rho\mu = 1_Y</math>. If <math>\text{End}(X)</math> is a [[local ring]] whenever ''X'' is indecomposable, i.e., not isomorphic to the [[coproduct]] of two nonzero objects, then the condition is satisfied and the category is Krull–Schmidt.
 
To every Krull–Schmidt category ''K'', one associates an [[Auslander–Reiten quiver]].
 
== Properties ==
One has the analogue of the Krull–Schmidt theorem in Krull–Schmidt categories. Namely, given isomorphisms <math>X_1 \oplus X_2 \oplus \cdots \oplus X_r \cong Y_1 \oplus Y_2 \oplus \cdots \oplus Y_s</math> where the <math>X_i</math> and <math>Y_j</math> are indecomposable, then <math>r=s</math>, and there exists a permutation <math>\pi</math> such that <math>X_{\pi(i)} \cong Y_i</math> for all ''i''.
 
== See also ==
* [[Quiver (mathematics)|Quiver]]
* [[Karoubi envelope]]
 
== References ==
* Claus Michael Ringel, ''Tame Algebras and Integral Quadratic Forms'', Lecture Notes in Mathematics '''1099''', Springer-Verlag, 1984.
 
{{DEFAULTSORT:Krull-Schmidt category}}
[[Category:Category theory]]
[[Category:Representation theory]]

Latest revision as of 12:56, 10 June 2014

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