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Transactions of the American Mathematical Society

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

 

 

Global dimension of tiled orders over commutative noetherian domains


Author: Vasanti A. Jategaonkar
Journal: Trans. Amer. Math. Soc. 190 (1974), 357-374
MSC: Primary 16A18
DOI: https://doi.org/10.1090/S0002-9947-1974-0354754-2
MathSciNet review: 0354754
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Abstract: Let R be a commutative noetherian domain and $ \Lambda = ({\Lambda _{ij}}) \subseteq {M_n}(R)$ be a tiled R-order. The main result of this paper is the following

Theorem. Let gl $ \dim R = d < \infty $ and $ \Lambda $ a triangular tiled R-order (i.e., $ {\Lambda _{ij}} = R$ whenever $ i \leq j$). Then the following three conditions are equivalent:

(1) gl $ \dim \Lambda < \infty $;

(2) $ {\Lambda _{i,i - 1}} = R$ or gl $ \dim \;(R/{\Lambda _{i,i - 1}}) < \infty $, whenever $ 2 \leq i \leq n$;

(3) gl $ \dim \Lambda \leq d(n - 1)$.

If $ d = 1$ or 2 then the upper bound in the above theorem is best possible. We give a sufficient condition for an arbitrary tiled R-order $ \Lambda $ to be of finite global dimension.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0354754-2
Keywords: Orders, tiled orders over commutative noetherian domains, global dimension, regular local rings
Article copyright: © Copyright 1974 American Mathematical Society