Classical tilting and $\tau$-tilting theory via duplicated algebras

Abstract

$\tau$-tilting theory can be thought of as a generalization of the classical tilting theory which allows mutations at any indecomposable summand of a support $\tau$-tilting pair. Indeed, for any algebra $\Lambda$ its tilting modules $\text{tilt}\,\Lambda$ form a subposet of the support $\tau$-tilting poset $\text{s}\tau-\text{tilt}\,\Lambda$. We show that conversely the $\tau$-tilting theory of an algebra $\Lambda$ can be naturally identified with the classical tilting theory of its duplicated algebra $\bar\Lambda$ by establishing a poset isomorphism $\text{s}\tau-\text{tilt}\,\Lambda\cong \text{tilt}\,\bar\Lambda$. As a result, $\tau$-tilting theory may be considered to be a special case of tilting theory. This extends the results of Assem-Br\"ustle-Schiffler-Todorov in the case of hereditary algebras. We also show that the product $\text{s}\tau-\text{tilt}\,\Lambda\times \text{s}\tau-\text{tilt}\,\Lambda$ embeds into the support $\tau$-tilting poset of its duplicated algebra $\text{s}\tau-\text{tilt}\,\bar\Lambda$ as a collection of Bongartz intervals. As an application we obtain a similar inclusion on the level of maximal green sequences.