-tilting theory

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 cluster-tilting 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 0.167em Λ , support τ -tilting modules and two-term silting complexes in K ᵇ ( proj 0.167em 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 0.167em Λ ) .