Classical tilting and $τ$-tilting theory via duplicated algebras

Abstract

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