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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.

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