Projective resolutions and the global dimension of subhereditary orders (Q1262374)

Let \(\Lambda\) be an order over a complete d.v.r. such that there exists a hereditary overorder \(\Gamma\) with Rad \(\Gamma\) \(\subset \Lambda\). For an indecomposable \(\Lambda\)-representation M with minimal projective resolution P, it is shown that the reduction \(\bar P\) of P modulo Rad \(\Gamma\) (replacing the \(P_ i\) by \(\bar P_ i:=P_ i/(Rad \Gamma)P_ i)\) decomposes into complexes \(\bar P_ j\), where \(P_ j\) is a minimal projective resolution of some irreducible \(\Gamma\)-representation \(G_ j\), and a minimal projective resolution of \(\bar M.\) From this fact, a number of corollaries is derived. In particular, a simple proof of the result in \textit{K. W. Roggenkamp}'s paper [Math. Z. 160, 63-67 (1978; Zbl 0383.16002)] is obtained, and some interesting estimates for gld \(\Lambda\), especially for Bäckström or generalized Bäckström orders \(\Lambda\), are established.