Global dimension of tiled orders over commutative noetherian domains

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.