Cartan–Eilenberg resolution

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.

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 an 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

• Quiver
• Karoubi envelope

References

• Claus Michael Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics 1099, Springer-Verlag, 1984.