Global Azumaya theorems in additive categories

Arnold, David M.; Hunter, Roger; Richman, Fred

doi:10.1016/0022-4049(80)90026-2

Cited by 33 publications

(12 citation statements)

References 7 publications

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“…Applications of Theorem I and Corollary H include the applications listed in [AHR1,§6], with appropriate and obvious modifications, the finite rank case of [KOl,Theorem 8], and a generalization of [AR1,Theorem 6.11].…”

Section: Assume That R Is a Dedekind Domain With Quotient Field K Amentioning

confidence: 99%

“…Theorem I is an example of a finite Azumaya theorem, a deduction of properties of Gi © ■ • • © Cn from conditions on the endomorphism rings of the C^s [AHR1,WW1]. In this case, each direct sum decomposition has a refinement into a direct sum of indécomposables.…”

mentioning

confidence: 99%

“…Special cases of Theorem I and Corollary LI are proved in [AHR1,Theorems A and D], wherein G is assumed to be a class of objects of C such that E(X) is a principal ideal domain for each X in C. Results of [AHR1] are used to reduce to the homogeneous case (the G¿'s are pairwise quasi-isomorphic) which, in turn, is a consequence of a ring theoretic theorem. This approach is in contrast to [AHR1] which uses additive category techniques for the homogeneous case.…”

mentioning

confidence: 99%

“…This approach is in contrast to [AHR1] which uses additive category techniques for the homogeneous case.…”

mentioning

confidence: 99%

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A Finite Global Azumaya Theorem in Additive Categories

Arnold¹

1984

Proceedings of the American Mathematical Society

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Abstract.Let C be an additive category such that idempotent endomorphisms have kernels, C a class of objects of C having Dedekind domains as endomorphism rings, and assume that if X and Y are quasi-isomorphic objects of C then Hom(X, Y") is a torsion-free module over the endomorphism ring of X. lîA®B = Ci®---®Cn with each C¿ in C, then A = A\ ©• ■ •©Am, where each Aj is locally in C, and End(A,) ~ End(C¿) 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.Let' C be an additive category such that idempotent endomorphisms of objects of C have kernels and let G be a class of objects of C with Dedekind domains as endomorphism rings. Two objects A and B of G are quasi-isomorphic if there are maps / G Hom(A, P) and g G Hom(P, A) with fg ?* 0. The class G has the torsionfree-hom condition if whenever A and B are quasi-isomorphic objects of G then Hom(A, B) is a torsion-free £'(A)-module, where E(A) is the endomorphism ring of A. If A and X are objects of C such that E(A) and E(X) are integral domains and if P is a prime ideal of E(A) then A is isomorphic to X at P if there are maps

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Section: Assume That R Is a Dedekind Domain With Quotient Field K Amentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…This approach is in contrast to [AHR1] which uses additive category techniques for the homogeneous case.…”

mentioning

confidence: 99%

See 2 more Smart Citations

A Finite Global Azumaya Theorem in Additive Categories

Arnold¹

1984

Proceedings of the American Mathematical Society

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“…To see this we first establish the following Proof. Suppose that, for each prime/), there is a group in C that is isomorphic to K at p. By Lemma 4.3, there is a fixed group G in C that is isomorphic to K at each p. By [AHR,Lemma 3.5] K is isomorphic to G. G Corollary 4.5. Let C be the class of groups G(A) where A is a doubly incomparable n-tuple of subgroups of Q.…”

mentioning

confidence: 99%