-tilting theory

Abstract

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

Introduction to τ-tilting theory

“…Corollary 20 (9). The exchange quiver of sτ-tiltΛ coincides with the Hasse quiver of the partially ordered set sτ-tiltΛ.…”

“…Theorem 14 (9). The following conditions are equivalent for a τ-rigid Λ-module T. In addition to τ-rigid and support τ-tilting modules, it is also convenient to consider associated pairs of modules.…”

“…We denote by τ-rigidΛ the set of isomorphism classes of basic τ-rigid pairs of Λ. Theorem 15 (9). ð−Þ † gives bijections τ-rigidΛ↔τ-rigid Λ op and sτ-tiltΛ↔sτ-tilt Λ op such that ð−Þ † † = id.…”

“…Theorem 17 (9). Let ðX; 0Þ and ðY ; 0Þ be two complements of an almost complete support τ-tilting pair ðU; PÞ.…”

“…This is accomplished by introducing τ-tilting modules, or more precisely, support τ-tilting pairs (9). Such a property is of central interest when dealing with categorification of cluster algebras.…”