Reduction of triangulated categories and maximal modification algebras for cAn singularities

Abstract

Abstract 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 \mathcal{C} is such a category, we say that \mathcal{C} is Calabi–Yau with \dim\mathcal{C}\leq 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 \mathop{\underline{\textup{CM}}}R of Cohen–Macaulay modules. We explain the connection between Calabi–Yau reduction of \mathop{\underline{\textup{CM}}}R and both partial crepant resolutions and \mathbb{Q} -factorial terminalizations of \operatorname{Spec}R , and we show under quite general assumptions that Calabi–Yau reductions exist. In the remainder of the paper we focus on complete local cA_{n} singularities R. By using a purely algebraic argument based on Calabi–Yau reduction of \mathop{\underline{\textup{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 k=\mathbb{C} we obtain many autoequivalences of the derived category of the \mathbb{Q} -factorial terminalizations of \operatorname{Spec}R .