$\tau $-tilting theory. (March 2014)

• Record Type: Journal Article
• Authors: Adachi, Takahide; Iyama, Osamu; Reiten, Idun
• Is Part Of: Compositio mathematica. Volume 150:Number 3(2014:May)
• Page Start: 415
• Page End: 452
• Publication Date: 2014-03
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X13007422
• Languages: English
• ISSNs: 0010-437X

Abstract

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