Tiled orders of finite global dimension

Abstract:

We define a projective link between maximal ideals, with respect to which an idealizer preserves being of finite global dimension. Let $D$ be a local Dedekind domain with the quotient ring $K$. We show that for $2 \leq n \leq 5$, every tiled $D$-order of finite global dimension in ${(K)_n}$ is obtained by iterating idealizers w.r.t. projective links from a hereditary order. For $n \geq 6$, we give a tiled $D$-order in ${(K)_n}$ without this property, which is also a counterexample to Tarsy’s conjecture, saying that the maximum finite global dimension of such an order is $n - 1$.