A finite global Azumaya theorem in additive categories

Abstract

Let C {\mathbf {C}} be an additive category such that idempotent endomorphisms have kernels, C C a class of objects of C {\mathbf {C}} having Dedekind domains as endomorphism rings, and assume that if X X and Y Y are quasi-isomorphic objects of C C then Hom ( X , Y ) {\operatorname {Hom}}(X,Y) is a torsion-free module over the endomorphism ring of X X . A ⊕ B = C 1 ⊕ ⋯ ⊕ C n A \oplus B = {C_1} \oplus \cdots \oplus {C_n} with each C i {C_i} in C C , then A = A 1 ⊕ ⋯ ⊕ A m A = {A_1} \oplus \cdots \oplus {A_m} , where each A j {A_j} is locally in C C , and End ( A j ) ≃ End ( C i ) {\operatorname {End}}({A_j}) \simeq {\operatorname {End}}({C_i}) for some i i . The proof includes a characterization of tiled orders. Moreover, there is a "local" uniqueness for finite direct sums of objects of C C .