Computing global dimension of endomorphism rings via ladders

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• 作者：Brandon Doherty^a ; ^{Brandon.Doherty@unb.ca" class="auth_mail" title="E-mail the corresponding author} ; Eleonore Faber^b ; ¹ ; ^{emfaber@umich.edu" class="auth_mail" title="E-mail the corresponding author} ; Colin Ingalls^a ; ^{cingalls@unb.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16G30 ; 13C14 ; 16E10 ; 16G70 ; 14B05
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：458
• 期：Complete
• 页码：307-350
• 全文大小：728 K

文摘

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$A_n$ for all n  , D_n$D_n$ for n≤13$n\le 13$ and E_6,7,8$E_{6,7,8}$ 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 height="15" width="61" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316300084-si180.gif">${\mathrm{End}}_{R}M$, where R   is a Henselian local ring, using add(M)$\mathrm{add}(M)$-approximations. When M≠0$M\ne 0$ 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.