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Computing global dimension of endomorphism rings via ladders. (English) Zbl 1431.16015

Summary: 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 [H. Dao et al., Algebr. Represent. Theory 18, No. 3, 633–664 (2015; Zbl 1327.14020)]. 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 \(\operatorname{End}_R M\), where \(R\) is a Henselian local ring, using \(\operatorname{add}(M)\)-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.

MSC:

16G30 Representations of orders, lattices, algebras over commutative rings
13C14 Cohen-Macaulay modules
16E10 Homological dimension in associative algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
14B05 Singularities in algebraic geometry

Citations:

Zbl 1327.14020
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References:

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