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Reduction of triangulated categories and maximal modification algebras for \(cA_n\) singularities. (English) Zbl 1428.18012
In this paper, several geometrical properties of partial resolutions of \(\mathrm{Spec}\,R\) are studied using new algebraic tools, among others CY reductions. Let \(R\) be a \(d\)-dimensional equi-codimensional Cohen-Macaulay ring with a canonical module \(\omega_R\) and \(\mathcal C\) be an \(n\)-Calabi-Yau (CY) \(R\)-linear triangulated category of dimension at most one, i.e. the Hom spaces have Krull dimension at most one. For any modifying object \(M\) of \(\mathcal C\), the authors construct a new \(n\)-CY triangulated category \({\mathcal C}_M\) (the Calabi-Yau reduction of \(\mathcal C\)) with dimension at most one. When \(\mathcal C\) is Krull-Schmidt and \(M\) is basic, then there is a bijection between basic modifying objects with summand \(M\) is \(\mathcal C\) and basic modifying objects in \({\mathcal C}_M\).
In the next section, these results are applied to the \(D_{sg}(R)=\underline{CM}\; R\), the stable category of maximal Cohen-Macaulay modules, for \(R\) a commutative equi-codimensional ring with \(\dim\; R=d\), this category is shown to be \((d-1)\)-CY triangulated category with dimension at most one. Moreover, if \(M\) is a modifying generator then \(\underline{CM}\;\mathrm{End}(M)\) is equivalent to \((\underline{CM}\; R)_M\) as triangulated categories. Consider the ring \(R= S/(fg)\), where \(S\) is the formal power series ring in two variables and \(f,g\in S\), then \(\underline{CM}\; R\) is a \(2\)-CY triangulated category with dimension at most one. It is shown that \(S/(f)\) is a modifying object in \(\underline{CM}\; R\) and \((\underline{CM}\; R)_{S/(f)}\) is equivalent to \(\underline{CM} (S/(f))\times \underline{CM}(S/(g))\). The authors conjecture that if \(R\) is a three dimensional Gorenstein normal domain over \(\mathbb C\) with rational singularities, there exists a CY reduction of \(\underline{CM}\; R\) with dimension cero and without non-zero rigid objects. The conjecture is verify for such rings with \({\mathbb Q}\)-factorial terminalization \(Y\) of \(\mathrm{Spec}\, R\) derived equivalent to some ring \(\Lambda\).
In Section 4, a method to detect when a set of MM generator are all using mutation of modifying modules, introduced in [the authors, Invent. Math. 197, No. 3, 521–586 (2014; Zbl 1308.14007)], is obtained. This method extends previous result in the case of 2-CY triangulated categories by T. Adachi et al. [Compos. Math. 150, No. 3, 415–452 (2014; Zbl 1330.16004)]
In the last Section, the authors classify all maximal modifying (generators) modules for complete local \(cA_n\) singularities using their techniques of reductions and mutation modification considered in the previous sections.

18E10 Abelian categories, Grothendieck categories
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13D09 Derived categories and commutative rings
Full Text: DOI arXiv
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