Tilting-completion for gentle algebras

TL;DR

This work studies tilting completion questions for gentle algebras using a surface-model framework. By interpreting indecomposable string modules as zigzag arcs on marked surfaces and employing a cutting-surface induction, the authors prove that any almost-tilting module over a gentle algebra is partial-tilting and that such modules have at most $2n$ complements, where $n$ is the algebra’s rank. They further construct explicit counterexamples showing that for $n\ge 3$ and $1\le m\le n-2$, pre-tilting modules can fail to be partial-tilting, thereby demonstrating sharp boundary behavior in this class. The results yield a positive answer to the almost-tilting completion question in the gentle setting (for the adapted Happel conjectures) and provide a versatile geometric reduction method with potential applications beyond gentle algebras.

Abstract

It is demonstrated that any almost-tilting module over a gentle algebra is indeed partial-tilting, meaning it can be completed as a tilting module. Furthermore, such a module has at most $2n$ possible complements, thereby confirming a (modified) conjecture of Happel for the case of gentle algebras. Additionally, for any $n\geq 3$ and $1\leq m \leq n-2$, there always exists a (connected) gentle algebra with rank $n$ and a pre-tilting module of rank $m$ which is not partial-tilting. The tool we use is the surface model associated with the module category of a gentle algebra. The main technique is an induction process involving surface cuts, which is hoped to be beneficial for other applications as well.

Figures (25)

• Figure 1: The left picture is a marked surface with a simple coordinate $\Delta^*$, where the arcs in $\Delta^*$ are the $\color{red}{\bullet}$-arcs. The right picture shows the gentle algebra associated with it, where the dotted lines represent the (quadratic) relations in the algebra.
• Figure 2: Two types of polygons formed by arcs in a simple coordinate and boundary segments, where a zigzag arc $\alpha$ passes through the polygon along an oriented intersection ${\bf{a}}_{i+1}$.
• Figure 3: For a zigzag arc $\alpha$ with endpoint $p$ which intersects $\ell^*_t$, the weight $w_p(\alpha)$ of $\alpha$ at $p$ equals $n-t$. The weight $\omega(\mathfrak{p})$ of an oriented intersection $\mathfrak{p}$ from $\alpha$ to $\beta$ is defined as $w_p(\alpha)-w_p(\beta)$, which equals $\omega$.
• Figure 4: The top picture depicts a simple $\circ$-arc $\gamma$ on $(\mathcal{S},\mathcal{M})$ cuts an $\color{red}{\bullet}$-arc $\ell^*$ into several segments $\varsigma_i$, with endpoints $\wp_{i-1}$ and $\wp_i$, which naturally induce segments $\overline{\varsigma}_i$ on the surface $\mathcal{S}_\gamma$, whose endpoints are $\wp'_{i-1}/\wp"_{i-1}$ and $\wp'_{i}/\wp"_{i}$, that is the red crossed points in the middle picture. Then after smoothing these segments with the line segments $\wp'_{i-1}q'/\wp"_{i-1}q"$ and $\wp'_iq'/\wp"_iq"$, we get the $\color{red}{\bullet}$-arcs $\ell^*_{i}$ on $(\mathcal{S}_\gamma,\mathcal{M}_\gamma)$. The final simple coordinate $\Delta^*_\gamma$ is obtained from $\Delta^*$ after replacing each $\ell^*$ by arcs $\ell^*_{i}$.
• Figure 5: An example of the cutting surface.

Theorems & Definitions (49)

• Theorem 1: Theorem \ref{['thm:almost-tilting is partial-tilting']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Definition 2.5
• Definition 2.6
• Example 2.7
• Definition 2.8
• Definition 2.9