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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A finite global Azumaya theorem in additive categories

Author: David M. Arnold
Journal: Proc. Amer. Math. Soc. 91 (1984), 25-30
MSC: Primary 18E05; Secondary 16A32
MathSciNet review: 735557
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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$.

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Keywords: Additive categories, Azumaya theorem, uniqueness of direct sum decompositions, orders in simple algebras
Article copyright: © Copyright 1984 American Mathematical Society

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