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The aim of this paper is to introduce τ -tilting theory, which 'completes' (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field k is a direct summand of exactly one or two tilting modules. An important property in clustertilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras kQ, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) τ -tilting modules, and show that an almost complete support τ -tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional k-algebra Λ, we establish bijections between functorially finite torsion classes in mod Λ, support τ -tilting modules and two-term silting complexes in K b (proj Λ). Moreover, these objects correspond bijectively to cluster-tilting objects in C if Λ is a 2-CY tilted algebra associated with a 2-CY triangulated category C. As an application, we show that the property of having two complements holds also for two-term silting complexes in K b (proj Λ). Contents

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules over certain Veronese subrings.

We investigate cluster-tilting objects (and subcategories) in triangulated 2-CalabiYau and related categories. In particular, we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. For these 2-Calabi-Yau categories, we construct cluster-tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.

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