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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Global dimension of tiled orders over a discrete valuation ring
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by Vasanti A. Jategaonkar
Trans. Amer. Math. Soc. 196 (1974), 313-330
DOI: https://doi.org/10.1090/S0002-9947-1974-0349729-3

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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Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 196 (1974), 313-330
  • MSC: Primary 16A18
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0349729-3
  • MathSciNet review: 0349729