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

$τ$-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.

💡 Research Summary

The paper establishes a precise and comprehensive bridge between classical tilting theory and τ‑tilting theory by means of duplicated algebras. For any finite‑dimensional algebra Λ over an algebraically closed field, the authors consider its duplicated algebra (\bar\Lambda), defined as the 2 × 2 triangular matrix algebra (\begin{pmatrix}\Lambda&0\ D\Lambda&\Lambda\end{pmatrix}) where (D\Lambda=\operatorname{Hom}_k(\Lambda,k)) is the standard dual bimodule. This construction yields two copies of Λ inside (\bar\Lambda): the “top” copy (denoted simply Λ) and the “bottom” copy (denoted Λ·). Lemma 2.4 describes the indecomposable projective and injective (\bar\Lambda)‑modules, showing that the top projectives coincide with those of Λ, while the bottom projectives are injective‑projective and fit into short exact sequences involving the dual bimodule.

The core of the work is the explicit functor‑like map (\bar F\colon \operatorname{s}\tau\text{-tilt},\Lambda\to\operatorname{tilt},\bar\Lambda). For a Λ‑module M, one starts with a minimal projective presentation (P_1\xrightarrow{f}P_0\to M). Embedding (P_1) into a projective‑injective (\bar\Lambda)‑module via an injective envelope (g), the authors take the cokernel of (\begin{pmatrix}f&g\end{pmatrix}) and discard any projective‑injective summand. The remaining summand is defined to be (\bar F(M)). Lemma 3.2 proves that this construction yields a minimal projective resolution of (\bar F(M)) in (\operatorname{mod}\bar\Lambda) and that the assignment respects the τ‑rigid condition. Consequently, Theorem 1.1 shows a poset isomorphism \