Reduction of triangulated categories and maximal modification algebrasfor cAn singularities

In this paper we define and study triangulated categories in which the Hom-spaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi–Yau type property. If <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒞</m:mi> </m:math> C is such a category, we say that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒞</m:mi> </m:math> C is Calabi–Yau with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>dim</m:mo> <m:mo>⁡</m:mo> <m:mi>𝒞</m:mi> </m:mrow> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> ≤ 1 . We extend the notion of Calabi–Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder> <m:mtext>CM</m:mtext> <m:mo>¯</m:mo> </m:munder> <m:mi>R</m:mi> </m:mrow> </m:math> {{{CM}}}R of Cohen–Macaulay modules. We explain the connection between Calabi–Yau reduction of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder> <m:mtext>CM</m:mtext> <m:mo>¯</m:mo> </m:munder> <m:mi>R</m:mi> </m:mrow> </m:math> {{{CM}}}R and both partial crepant resolutions and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℚ</m:mi> </m:math> Q -factorial terminalizations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Spec</m:mi> <m:mo>⁡</m:mo> <m:mi>R</m:mi> </m:mrow> </m:math> SpecR , and we show under quite general assumptions that Calabi–Yau reductions exist. In the remainder of the paper we focus on complete local <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>c</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:math> cAₙ singularities R . By using a purely algebraic argument based on Calabi–Yau reduction of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder> <m:mtext>CM</m:mtext> <m:mo>¯</m:mo> </m:munder> <m:mi>R</m:mi> </m:mrow> </m:math> {{{CM}}}R , we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [I. Burban, O. Iyama, B. Keller and I. Reiten, Cluster tilting for one-dimensional hypersurface singularities, Adv. Math. 217 2008, 6, 2443–2484], [H. Dao and C. Huneke, Vanishing of Ext, cluster tilting and finite global dimension of endomorphism rings, Amer. J. Math. 135 2013, 2, 561–578], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mi>ℂ</m:mi> </m:mrow> </m:math> k=C we obtain many autoequivalences of the derived category of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℚ</m:mi> </m:math> Q -factorial terminalizations of