Computing global dimension of endomorphism rings via ladders

Abstract

This paper deals with computing the global dimension of endomorphism rings of maximal Cohen–Macaulay (=MCM) modules over commutative rings. Several examples are computed. In particular, we determine the global spectra, that is, the sets of all possible finite global dimensions of endomorphism rings of MCM-modules, of the curve singularities of type An for all n, Dn for n≤13 and E6,7,8 and compute the global dimensions of Leuschke's normalization chains for all ADE curves, as announced in [12]. Moreover, we determine the centre of an endomorphism ring of a MCM-module over any curve singularity of finite MCM-type.In general, we describe a method for the computation of the global dimension of an endomorphism ring EndRM, where R is a Henselian local ring, using add(M)-approximations. When M≠0 is a MCM-module over R and R is Henselian local of Krull dimension ≤2 with a canonical module and of finite MCM-type, we use Auslander–Reiten theory and Iyama's ladder method to explicitly construct these approximations.