Reduction of triangulated categories and maximal modification algebras for \(cA_n\) singularities.

*(English)*Zbl 1428.18012In 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.

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.

Reviewer: Blas Torrecillas (Almeria)

##### MSC:

18E10 | Abelian categories, Grothendieck categories |

13D07 | Homological functors on modules of commutative rings (Tor, Ext, etc.) |

13D09 | Derived categories and commutative rings |

##### Keywords:

CY-reductions; maximal Cohen-Macaulay modules; singularities category; modifying modules; mutation; rational singularities##### References:

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