Injective dimension of quaternion orders

Abstract:

Tarsy has shown that if $R$ is a discrete valuation ring with quotient field $K$, and $\Sigma$ is a quaternion $K$-algebra, then the finitistic global dimension of any $R$-order in $\Sigma$ is 1. In this paper we allow $R$ to be any regular local ring of dimension $n$ and study the $R$-free orders $\Lambda$ in $\Sigma$. First we show that the finitistic global dimension of $\Lambda$ is $n$. Our main result concerns the injective dimension of $\Lambda$ (considered as either a left or a right $\Lambda$-module). Let $\mathfrak {M}$ denote the maximal ideal of $R$. Then the injective dimension of $\Lambda$ is $n$, unless $\Lambda /\mathfrak {M}\Lambda$ is a commutative local ring whose socle is not principal. In this case, the injective dimension of $\Lambda$ is $\infty$.