Article contents
$\tau $-tilting theory
Published online by Cambridge University Press: 03 December 2013
Abstract
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
${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$.
Keywords
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s) 2013
References







- 257
- Cited by