Krull–Schmidt category

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 algebras.

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Definition

Let k be a field. A category enriched over finite-dimensional k-vector spaces is a Krull–Schmidt category if all idempotents split. In other words, if e \in \text{End}(X) satisfies e^2 = e, then there exists an object Y and morphisms \mu \colon Y \to X and \rho \colon X \to Y such that \mu \rho = e and \rho\mu = 1_Y. If \text{End}(X) 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 a Auslander–Reiten quiver.

Properties

One has the analogue of the Krull–Schmidt theorem in Krull–Schmidt categories. Namely, given isomorphisms 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 indecomposable, then r=s, and there exists a permutation \pi such that X_{\pi(i)} \cong Y_i for all i.

See also

References