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Mutation in triangulated categories and rigid Cohen-Macaulay modules. (English) Zbl 1140.18007
In the paper under review, the authors, based on the approximation theory initiated by the school of M. Auslander, introduce and study mutations of \(n\)-cluster tilting subcategories of (\(k\)-linear, Krull-Schmidt) triangulated categories (endowed with a Serre functor) as a tool for the classification of rigid objects in such a category. They show that any object in such a triangulated category has a (triangulated analogue of a) minimal left resolution of length \(n-1\) with terms in a given \(n\)-cluster tilting subcategory, hence, under some further assumptions, one can describe the indecomposable objects of the triangulated category in terms of the indecomposable objects of the subcategory. Mutations allows one to obtain new \(n\)-cluster tilting subcategories from a given one. They are closely related to another notion introduced by the authors, namely that of AR \((n+2)\)-angles, a generalization of Auslander-Reiten triangles.
As an application, they obtain a complete classification of rigid maximal Cohen-Macaulay modules over \(R=k[[\)monomials of degree \(3]] \subset k[[x_1,x_2,x_3]]=S\) and over \(R=k[[\)monomials of degree \(2]] \subset k[[x_1,x_2,x_3,x_4]]=S\). In both cases, \(R\) is a Gorenstein local ring with only an isolated singularity.
Beside the above mentioned category theoretical considerations, the proof of these computational results uses a theorem of O. Iyama [Adv. Math. 210, No. 1, 22–50 (2007; Zbl 1115.16005), Adv. Math. 210, No. 1, 51–82 (2007; Zbl 1115.16006)], and a theorem of V. G. Kac [Invent. Math. 56, 57–92 (1980; Zbl 0427.17001)]. The theorem of Iyama asserts that if \(R=S^G\), where \(S=k[[x_1,\dots ,x_d]]\) and \(G\) is a finite subgroup of \(\text{SL}(d)\) (which is equivalent to \(R\) being Gorenstein) such that \(\sigma -1\in \text{GL}(d)\), \(\forall \sigma \in G\setminus \{1\}\) (which is equivalent to \(R\) having only an isolated singularity), then the subcategory of the stable category \(\underline{\text{CM}}(R)\) of maximal Cohen-Macaulay \(R\)-modules whose objects are direct sums of indecomposable factors of the \(R\)-module \(S\) is a \((d-1)\)-cluster tilting subcategory of \(\underline{\text{CM}}(R)\).

MSC:
18E30 Derived categories, triangulated categories (MSC2010)
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
13C14 Cohen-Macaulay modules
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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