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 CalabiâYau type property. If đ \mathcal{C} is such a category, we say that đ \mathcal{C} is CalabiâYau with dim ⥠đ †1 \dim\mathcal{C}\leq 1 . We extend the notion of CalabiâYau 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 CM ÂŻ R \mathop{\underline{\textup{CM}}}R of CohenâMacaulay modules. We explain the connection between CalabiâYau reduction of CM ÂŻ R \mathop{\underline{\textup{CM}}}R and both partial crepant resolutions and â \mathbb{Q} -factorial terminalizations of Spec ⥠R \operatorname{Spec}R , and we show under quite general assumptions that CalabiâYau reductions exist. In the remainder of the paper we focus on complete local c âą A n cA_{n} singularities R. By using a purely algebraic argument based on CalabiâYau reduction of CM ÂŻ R \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, 2443â2484], [H. Dao and C. Huneke, Vanishing of Ext, cluster tilting and finite global dimension of endomorphism rings, Amer. J. Math. 135 2013, 2, 561â578], 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 = â k=\mathbb{C} we obtain many autoequivalences of the derived category of the â \mathbb{Q} -factorial terminalizations of Spec ⥠R \operatorname{Spec}R .