Global dimension of tiled orders over a discrete valuation ring

Jategaonkar, Vasanti A.

doi:10.1090/S0002-9947-1974-0349729-3

Global dimension of tiled orders over a discrete valuation ring
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Trans. Amer. Math. Soc. 196 (1974), 313-330 Request permission

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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Global dimension of tiled orders over a DVR

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