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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Global dimension of tiled orders over a discrete valuation ring


Author: Vasanti A. Jategaonkar
Journal: Trans. Amer. Math. Soc. 196 (1974), 313-330
MSC: Primary 16A18
MathSciNet review: 0349729
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Abstract: Let R be a discrete valuation ring with maximal ideal $ \mathfrak{m}$ and the quotient field K. Let $ \Lambda = ({\mathfrak{m}^{{\lambda _{ij}}}}) \subseteq {M_n}(K)$ be a tiled R-order, where $ {\lambda _{ij}} \in {\mathbf{Z}}$ and $ {\lambda _{ii}} = 0$ for $ 1 \leq i \leq n$. The following results are proved. Theorem 1. There are, up to conjugation, only finitely many tiled R-orders in $ {M_n}(K)$ of finite global dimension. Theorem 2. Tiled R-orders in $ {M_n}(K)$ of finite global dimension satisfy DCC. Theorem 3. Let $ \Lambda \subseteq {M_n}(R)$ and let $ \Gamma $ be obtained from $ \Lambda $ by replacing the entries above the main diagonal by arbitrary entries from R. If $ \Gamma $ is a ring and if gl $ \dim \;\Lambda < \infty $, then gl $ \dim \;\Gamma < \infty $. Theorem 4. Let $ \Lambda $ be a tiled R-order in $ {M_4}(K)$. Then gl $ \dim \;\Lambda < \infty $ if and only if $ \Lambda $ is conjugate to a triangular tiled R-order of finite global dimension or is conjugate to the tiled R-order $ \Gamma = ({\mathfrak{m}^{{\lambda _{ij}}}}) \subseteq {M_4}(R)$, where $ {\gamma _{ii}} = {\gamma _{1i}} = 0$ for all i, and $ {\gamma _{ij}} = 1$ otherwise.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0349729-3
Keywords: Orders, tiled orders, discrete valuation ring, Dedekind domain, global dimension
Article copyright: © Copyright 1974 American Mathematical Society