Computing global dimension of endomorphism rings via ladders

TL;DR

This paper develops methods to compute the global dimension of endomorphism rings of MCM-modules over certain singularities, providing explicit calculations for various ADE curve singularities and their normalization chains.

Contribution

It introduces a new approach using ladders and approximations to determine global dimensions of endomorphism rings over Henselian local rings with finite MCM-type.

Findings

Computed global spectra for $A_n$, $D_n$, and $E_{6,7,8}$ singularities.

Determined centers of endomorphism rings over finite MCM-type curve singularities.

Explicitly constructed approximations using Auslander--Reiten theory and Iyama's ladder method.

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 $A_n$ for all $n$, $D_n$ for $n \leq 13$ and $E_{6,7,8}$ and compute the global dimensions of Leuschke's normalization chains for all ADE curves, as announced in [Dao-Faber-Ingalls]. 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 $\mathrm{End}_R M$, where $R$ is a Henselian local ring, using $\mathrm{add}(M)$-approximations. When $M\neq 0$ is a MCM-module over $R$ and $R$ is Henselian local of Krull dimension $\leq 2$ with a canonical module and of finite MCM-type, we use Auslander--Reiten theory and Iyama's ladder method to explicitly construct these approximations.