
Mathematics 2012
τtilting theoryAbstract: The aim of this paper is to introduce tautilting theory, which completes (classical) tilting theory from the viewpoint of mutation. It is wellknown in tilting theory that an almost complete tilting module for any finite dimensional algebra over a field k is a direct summand of exactly 1 or 2 tilting modules. An important property in cluster tilting theory is that an almost complete clustertilting object in a 2CY triangulated category is a direct summand of exactly 2 clustertilting objects. Reformulated for path algebras kQ, this says that an almost complete support tilting modules has exactly two complements. We generalize (support) tilting modules to what we call (support) tautilting modules, and show that an almost support tautilting module has exactly two complements for any finite dimensional algebra. For a finite dimensional kalgebra A, we establish bijections between functorially finite torsion classes in mod A, support tautilting modules and twoterm silting complexes in Kb(proj A). Moreover these objects correspond bijectively to clustertilting objects in C if A is a 2CY tilted algebra associated with a 2CY triangulated category C. As an application, we show that the property of having two complements holds also for twoterm silting complexes in Kb(proj A).
