Homological Dimensions of Gentle Algebras via Geometric Models

Let $A=kQ\mathrm{/}I$ be a finite dimensional basic algebra over an algebraically closed field $k$ which is a gentle algebra with the marked ribbon surface $({\mathcal{S}}_A,{\mathcal{M}}_A,{\mathrm{\Gamma }}_A)$. It is known that ${\mathcal{S}}_A$ can be divided into some elementary polygons $\{{\mathrm{\Delta }}_i\mid 1\le i\le d\}$ by ${\mathrm{\Gamma }}_A$ which has exactly one side in the boundary of ${\mathcal{S}}_A$. Let $\mathfrak{C}({\mathrm{\Delta }}_i)$ be the number of sides of ${\mathrm{\Delta }}_i$ belonging to ${\mathrm{\Gamma }}_A$ if the unmarked boundary component of ${\mathcal{S}}_A$ is not a side of ${\mathrm{\Delta }}_i$; otherwise, $\mathfrak{C}({\mathrm{\Delta }}_i)=\mathrm{\infty }$, and let $\mathsf{f}\text{-}\mathrm{\Delta }$ be the set of all non-$\mathrm{\infty }$-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 $A=\underset{1\le i\le d}{\max }\mathfrak{C}({\mathrm{\Delta }}_i)-1=\underset{\Pi \in {\mathcal{F}}_A}{\max }l(\Pi )$, where $l(\Pi )$ is the length of $\Pi$. \item[{\rm (2)}] The left and right self-injective dimensions of $A=$ \begin{center} $\{\begin{array}{l}0,\text{if }\mathit{\text{Q}}\mathit{\text{is}}\mathit{\text{either}}\mathit{\text{a}}\mathit{\text{point}}\mathit{\text{or}}\mathit{\text{an}}\mathit{\text{oriented}}\mathit{\text{cycle}}\mathit{\text{with}}\mathit{\text{full}}\mathit{\text{relations}}\text{;}\\ \underset{{\mathrm{\Delta }}_i\in \mathsf{f}\text{-}\mathrm{\Delta }}{\max }\{1,\mathfrak{C}({\mathrm{\Delta }}_i)-1\}=\underset{\Pi \in {\mathsf{f}\text{-}\mathcal{F}}_A}{\max }l(\Pi ),\text{ otherwise.}\end{array}$ \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.