A tiled order having large global dimension
For a discrete valuation domain \(R\) with quotient field \(K\), and any even integer \(N\geq 8\), a tiled \(R\)-order \(\Lambda_N\) in \(M_N(K)\) of global dimension \(2N-8\) is constructed. If \(P_1,\dots,P_N\) are the indecomposable projective right \(\Lambda_N\)-modules, the authors first prove that all the syzygies of \(\text{rad }P_{N-1}\) and \(\text{rad }P_N\) decompose into irreducible \(\Lambda_N\)-representations. Therefore, the projective dimension of these modules is easy to compute. In this way, a family of \(\Lambda_N\)-modules with known projective dimensions is obtained. The other direct summands of \(\text{Rad }\Lambda_N\) are then related to these modules by certain short exact sequences, and thus the global dimension of \(\Lambda_N\) is evaluated.
scientific article; zbMATH DE number 1038258