$τ $-tilting theory. (March 2014) - Adachi, Takahide; Iyama, Osamu; Reiten, Idun
Abstract The aim of this paper is to introduce <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh4wz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\tau $]]> </alternatives>-tilting theory, which 'completes' (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh6k8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$k$]]> </alternatives> is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh6qb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$kQ$]]> </alternatives>, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh51h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\tau $]]> </alternatives>-tilting modules, and show that an almost complete support <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh59n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\tau $]]> </alternatives>-tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh411" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$k$]]> </alternatives>-algebra <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh70z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\Lambda $]]> </alternatives>, we establish bijections between functorially finite torsion classes in <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh3pc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$ \mathsf{mod} \hspace{0.167em} \Lambda $]]> </alternatives>, support <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh6bn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\tau $]]> </alternatives>-tilting modules and two-term silting complexes in <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh39p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$]]> </alternatives>. Moreover, these objects correspond bijectively to cluster-tilting objects in <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh542" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$ \mathcal{C} $]]> </alternatives> if <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh3g8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\Lambda $]]> </alternatives> is a 2-CY tilted algebra associated with a 2-CY triangulated category <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh6gq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$ \mathcal{C} $]]> </alternatives>. As an application, we show that the property of having two complements holds also for two-term silting complexes in <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh5qv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$]]> </alternatives>.