Suppose $\mathcal C$ is a preadditive, Karoubi category with a zero object. What further assumptions on $\mathcal C$ are required to ensure that the endomorphism ring of an indecomposable object is a local?


By an object $X$ being indecomposable, I mean that in any biproduct decomposition $X \cong X\_1 \oplus X\_2$, one has $X\_i \cong 0$ for some $i=1,2$.


Thanks for your help.



endomorphism rings of indecomposable objects
Hi benjamin.


You're asking if $\mathcal{C}$ is a so called Krull-Schmidt-category. There are several sufficient conditions know from ring theory if $\mathcal{C}$ is a module category. For example the argument George mentioned gives an affirmative answer if $\mathcal{C}$ is the category of finitely generated (left)modules over an artinian ring.
You get another characterisation by considering semiperfect rings. A ring is semiperfect iff the category of finitely generated projective (left)modules is Krull-Schmidt. Examples of semiperfect rings include all left or right artinian ring as well as all finitely generated $R$-algebras where $R$ is a complete local ring or a discrete valuation ring.


This connection between semiperfect rings and Krull-Schmidt-categories is in fact somewhat stronger: One can show that $\mathcal{C}$ is Krull-Schmidt iff $End\_\mathcal{C}(X)$ is semiperfect for all objects $X\in\mathcal{C}$.


