论文标题
通过几何模型的温和代数的同源维度
Homological Dimensions of Gentle Algebras via Geometric Models
论文作者
论文摘要
令$ a = kq/i $是一个有限的尺寸基本代数，在代数封闭的字段$ k $上，这是一个温和的代数，带有标记的色带表面$（\ Mathcal {s} _a，\ Mathcal {m Mathcal {m} _a，_a，γ_A）$。众所周知，可以将$ \ Mathcal {s} _a $分为某些基本多边形$ \ {Δ_i\ mid 1 \ le i \ le i \ le i \ le d \} $ by $γ_a$，它在$ \ mathcal {s} _a _a $的边界上完全具有一个面。如果如果$ \ mathcal {s} _a $的未标记边界组件不是$δ_i$的一面，则令$ \ mathfrak {c}（Δ_i）$为属于$γ_A$的$δ_i$的侧面数；否则，$ \ mathfrak {c}（δ_i）= \ infty $，然后让$ \ m athsf {f} \ text { - }δ$是所有非non- $ \ undty $ elementary polygons and $ \ \ \ \ \ \ \ \ \ m imparcal {f} _a $（分别分别$ {\ mathsf {f} \ text { - } \ Mathcal {f}} _ a $）所有禁止线程的集合（分别为有限长度）。那么我们就有\ begin { l（\mathitπ）$，其中$ l（\mathitπ）$是$ \mathitπ$的长度。 \ item [{\ rm（2）}] $ a = $ a = $ a = $ \ begin {center} $ \ begin {cases} 0，\ \ \ mbox {\ text {f text {if {\ it q}是一个点或一个方向循环，具有完全关系};} \ \ \ \ \\ \\\\\\\\\\\\\\\\\\ rm（2）}] \ max \ limits_ {Δ_i\ in {\ Mathsf {f} \ text { - }δ}}}}}}}} \ big \ {1，{\ mathfrak {c}（Δ_i）} - 1 \ big big \} = \ max \ limits _ {\Mathitπ\ in {\ MathSf {f} \ text { - } \ Mathcal {f}} _ a} l（\mathitπ），\ \ \ \ \ \ mbox {\ mbox {\ mbox {\ fext {eref text {eref}}}}。在Ag等效的情况下，温和代数的全球维度的有限性是不变的。此外，我们发现，在柔和的代数上，不可分解的非注射性Gorenstein投影模块在Ag等效性下也不变。
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.
