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In this paper we define and study triangulated categories in which theHom-spaces have Krull dimension at most one over some base ring (hence theyhave a natural 2-step filtration), and each factor of the filtration satisfiessome Calabi-Yau type property. If C is such a category, we say that C isCalabi-Yau with dimension at most one. We extend the notion of Calabi-Yaureduction to this setting, and prove general results which are an analogue ofknown results in cluster theory.  Such categories appear naturally in the setting of Gorenstein singularitiesin dimension three as the stable categories of Cohen-Macaulay modules. Weexplain the connection between Calabi-Yau reduction of $\underline{\rm CM} R$and both partial crepant resolutions and Q-factorial terminalizations of SpecR, and we show under quite general assumptions that Calabi-Yau reductionsexist.  In the remainder of the paper we focus on complete local cA_n singularitiesR. By using a purely algebraic argument based on Calabi-Yau reduction of$\underline{\rm CM} R$, we give a complete classification of maximal modifyingmodules in terms of the symmetric group, generalizing and strengthening resultsin [BIKR] and [DH], where we do not need any restriction on the ground field.We also describe the mutation of modifying modules at an arbitrary (notnecessarily indecomposable) direct summand. As a corollary when the field isthe complex numbers, we obtain many autoequivalences of the derived category ofthe Q-factorial terminalizations of Spec R.

Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities