A finite global Azumaya theorem in additive categories

Abstract:

Let ${\mathbf {C}}$ be an additive category such that idempotent endomorphisms have kernels, $C$ a class of objects of ${\mathbf {C}}$ having Dedekind domains as endomorphism rings, and assume that if $X$ and $Y$ are quasi-isomorphic objects of $C$ then ${\operatorname {Hom}}(X,Y)$ is a torsion-free module over the endomorphism ring of $X$. $A \oplus B = {C_1} \oplus \cdots \oplus {C_n}$ with each ${C_i}$ in $C$, then $A = {A_1} \oplus \cdots \oplus {A_m}$, where each ${A_j}$ is locally in $C$, and ${\operatorname {End}}({A_j}) \simeq {\operatorname {End}}({C_i})$ for some $i$. The proof includes a characterization of tiled orders. Moreover, there is a "local" uniqueness for finite direct sums of objects of $C$.