From the homotopy category of projective modules over gentle algebras to poset representations
Germán Benitez, Gustavo Costa
TL;DR
The paper develops triangulated structures for Bondarenko's poset-based matrix category and links these to the homotopy/derived categories of gentle algebras. It defines a new triangulated quotient $\mathcal{K}(\mathscr{Y},\Bbbk)$ of Bondarenko's category using $\mathcal{K}$-matrices, with $\mathcal{K}$-standard triangles forming the distinguished triangles. It then constructs a functor $\mathbf{F}$ that embeds the bounded homotopy category $\mathbf{K}^b(\mathrm{proj}\,A)$ of a gentle algebra $A=\Bbbk Q/I$ into $\mathcal{K}(\mathscr{Y}(A),\Bbbk)$ by encoding complexes on a poset $\mathscr{Y}(A)$, and shows this induces a triangulated embedding of $\mathbf{D}^b(A)$ when $A$ has finite global dimension. Together, these results connect derived-category techniques with poset representations and provide a framework for transferring triangulated structure across these settings.
Abstract
In [BCP24], the authors describe a triangulated structure of a quotient of a certain category of representations of posets, nowadays known as the Bondarenko's category. This category was essential in [BM03] for classify all indecomposable objects of the derived category of gentle algebras. In view of this connection with the derived category, which possess a triangulated structure. In this paper, we identify another triangulated structure for Bondarenko's category, allowing us to utilize the functor presented in [BM03]. This functor will establishes a connection between the triangulated structure of the homotopy category of gentle algebras and the new triangulated structure of a quotient of a certain Bondarenko's category.