-tilting theory

<jats:title>Abstract</jats:title><jats:p>The aim of this paper is to introduce <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline3" /><jats:tex-math>$\tau $</jats:tex-math></jats:alternatives></jats:inline-formula>-tilting theory, which ‘completes’ (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline4" /><jats:tex-math>$k$</jats:tex-math></jats:alternatives></jats:inline-formula> is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline5" /><jats:tex-math>$kQ$</jats:tex-math></jats:alternatives></jats:inline-formula>, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline6" /><jats:tex-math>$\tau $</jats:tex-math></jats:alternatives></jats:inline-formula>-tilting modules, and show that an almost complete support <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline7" /><jats:tex-math>$\tau $</jats:tex-math></jats:alternatives></jats:inline-formula>-tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline8" /><jats:tex-math>$k$</jats:tex-math></jats:alternatives></jats:inline-formula>-algebra <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline9" /><jats:tex-math>$\Lambda $</jats:tex-math></jats:alternatives></jats:inline-formula>, we establish bijections between functorially finite torsion classes in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline10" /><jats:tex-math>$ \mathsf{mod} \hspace{0.167em} \Lambda $</jats:tex-math></jats:alternatives></jats:inline-formula>, support <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline11" /><jats:tex-math>$\tau $</jats:tex-math></jats:alternatives></jats:inline-formula>-tilting modules and two-term silting complexes in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline12" /><jats:tex-math>${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$</jats:tex-math></jats:alternatives></jats:inline-formula>. Moreover, these objects correspond bijectively to cluster-tilting objects in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline13" /><jats:tex-math>$ \mathcal{C} $</jats:tex-math></jats:alternatives></jats:inline-formula> if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline14" /><jats:tex-math>$\Lambda $</jats:tex-math></jats:alternatives></jats:inline-formula> is a 2-CY tilted algebra associated with a 2-CY triangulated category <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline15" /><jats:tex-math>$ \mathcal{C} $</jats:tex-math></jats:alternatives></jats:inline-formula>. As an application, we show that the property of having two complements holds also for two-term silting complexes in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007422_inline16" /><jats:tex-math>${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>