Homological Dimensions of Gentle Algebras via Geometric Models

Abstract

Let $A=kQ/I$ be a finite dimensional basic algebra over an algebraically closed field $k$ which is a gentle algebra with the marked ribbon surface $(\mathcal{S}_A,\mathcal{M}_A,Γ_A)$. It is known that $\mathcal{S}_A$ can be divided into some elementary polygons $\{Δ_i\mid 1\le i\le d\}$ by $Γ_A$ which has exactly one side in the boundary of $\mathcal{S}_A$. Let $\mathfrak{C}(Δ_i)$ be the number of sides of $Δ_i$ belonging to $Γ_A$ if the unmarked boundary component of $\mathcal{S}_A$ is not a side of $Δ_i$; otherwise, $\mathfrak{C}(Δ_i)=\infty$, and let $\mathsf{f}\text{-}Δ$ be the set of all non-$\infty$-elementary polygons and $\mathcal{F}_A$ (respectively, ${\mathsf{f}\text{-}\mathcal{F}}_A$) the set of all forbidden threads (respectively, of finite length). Then we have \begin{enumerate} \item[{\rm (1)}] The global dimension of $A=\max\limits_{1\leq i\leq d}{\mathfrak{C}(Δ_i)}-1 =\max\limits_{\mathitΠ\in\mathcal{F}_A} l(\mathitΠ)$, where $l(\mathitΠ)$ is the length of $\mathitΠ$. \item[{\rm (2)}] The left and right self-injective dimensions of $A=$ \begin{center} $\begin{cases} 0,\ \mbox{\text{if {\it Q} is either a point or an oriented cycle with full relations};}\\ \max\limits_{Δ_i\in{\mathsf{f}\text{-}Δ}}\big\{1, {\mathfrak{C}(Δ_i)}-1 \big\}= \max\limits_{\mathitΠ\in{\mathsf{f}\text{-}\mathcal{F}}_A} l(\mathitΠ),\ \mbox{\text{otherwise}.} \end{cases}$ \end{center} \end{enumerate} As a consequence, we get that the finiteness of the global dimension of gentle algebras is invariant under AG-equivalence. In addition, we get that the number of indecomposable non-projective Gorenstein projective modules over gentle algebras is also invariant under AG-equivalence.