tilting theory

The aim of this paper is to introduce $\tau $ -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) $\tau $ -tilting modules, and show that an almost complete support $\tau $ -tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional $k$ -algebra $\Lambda $ , we establish bijections between functorially finite torsion classes in $ \mathsf{mod} \hspace{0.167em} \Lambda $ , support $\tau $ -tilting modules and two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$ . Moreover, these objects correspond bijectively to cluster-tilting objects in $ \mathcal{C} $ if $\Lambda $ is a 2-CY tilted algebra associated with a 2-CY triangulated category $ \mathcal{C} $ . As an application, we show that the property of having two complements holds also for two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$ .