A Finite Global Azumaya Theorem in Additive Categories

Abstract

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

Noncommutative motives of separable algebras

“…Recall from [17, §III Prop. 4.1] that a commutative k-algebra A is separable if and only if it is isomorphic to a finite product of finite separable field extensions of k. Thanks to [17, §III Thm. 1.4(1) and Prop.…”

“…It is possible to prove Lemma 11.9(ii) without invoking the results of Arnold [4] (i.e. Theorem 11.5).…”

The max-plus algebra of exponent matrices of tiled orders

“…These include homological aspects [14,15,10,11,17,19,20,26,28], representation theory [27,32,33,34,39], structure [12,23,37,36,38], K-theory [18,22] and others. In addition, tiled orders turned out to be useful to prove Krull-Remak-Schmidt-Azumaya type theorems in additive categories [3] and, more recently, a strong connection between cluster categories and Cohen-Macaulay representation theory of some tiled orders was established in [4].…”

The cone of quasi-semimetrics and exponent matrices of tiled orders

“…Order of "tiled form" appeared as essential ingredients in the study of large classes of orders in algebras, gaining special significance in the theory of orders of finite representation type and in the investigation of global dimension. Since then tiled orders (also called Schurian orders and monomial orders) were studied from various points of view, and, apart from their structural, representation theoretic and homological applications, they turned out to be useful to additive categories [1] and, more recently, to the connection between Cohen-Macaulay representation theory and cluster categories [4] (see also the bibliography in [11]).…”

“…, where πO is the maximal ideal of O and (α ij ) ∈ M n (Z) 1 . Actually, for Λ to be multiplicatively closed it is necessary and sufficient that the triangle inequalities (2) α ij + α jk α ik , hold for any i, j and k, whereas for it to be unital, it is necessary that α ii 0 for all i.…”