Global dimension of tiled orders over a discrete valuation ring

Vasanti A. Jategaonkar
1974 Transactions of the American Mathematical Society  
Let R be a discrete valuation ring with maximal ideal m and the quotient field K. Let A = (m ") Ç M (K) be a tiled /{-order, where X.. e Z and X.. =0 for 1 s i s,7». The following results are proved. Theorem 1. There are, up to conjugation, only finitely many tiled R-orders in M"(X) of finite global dimension. Theorem 2. Tiled R-orders in M (K) of finite global dimension satisfy DCC. Theorem 3. Let A CM (/?) and let T be obtained from A by replacing the entries above the main diagonal by
more » ... diagonal by arbitrary entries from R. If r is a ring and if gl dim A < oo, then gl dim r < oo. Theorem 4. Let A be a tiled R-order in M.(K). Then gl dim A < oo if and only if A is conjugate to a triangular tiled R-order of finite global dimension or is conjugate to the tiled R-order T ■ (r^iftC M¿R), where 7-= 7j.b0 for all i, and Yy-l otherwise. Introduction. This paper is a continuation of the author's previous paper, Global dimension of tiled orders over commutative noetherian domains [7], Throughout this paper R will denote a discrete valuation ring (DVR) with maximal ideal m, generated by r, and the quotient field K. In this paper we will use notations and terminologies of [7]. Let A be a tiled R-order in Mn(K), i.e., an R-order in Al (K) containing n orthogonal idempotents. If a tiled R-order A in M (K) contains the usual system e.., 1 < i < n, of « orthogonal idempotents,
doi:10.1090/s0002-9947-1974-0349729-3 fatcat:zpyyeds275fp5cod7hvzyl7w2q