2015/08/25 by Brandon Doherty, Doherty, Brandon, Eleonore Faber +3
Mathematics · #13C14 #14B05 #16E10 #16G30 #16G70 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT) #math.AC #math.AG #math.RT #msc:13C14 #msc:14B05 #msc:16E10 #msc:16G30 #msc:16G70
paper · pdf · doi:10.48550/arxiv.1508.06287
v2: Fixed an error in the code of our SAGE program: some computational results changed; minor revision. v1:35 pages
arxiv created 2015/10/02 · arxiv updated 2015/10/06
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 [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 EndR M, 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.