On the Gorensteiness of string algebras

TL;DR

The paper develops a combinatorial criterion for Gorensteinness of string algebras by introducing ELIS, effective left-intersecting sequences, and related cosyzygy analysis. It obtains a precise formula for the self-injective dimension: if A=\mathds{k}\mathcal{Q}/\mathcal{I} is not self-injective, then inj.dim(A)=\sup_{v\in \mathcal{Q}_0} \mathfrak{l}(\mathrm{ELIS}(P(v)))+1, and A is Gorenstein exactly when all ELIS(P(v)) have finite length. The method hinges on explicit descriptions of the first cosyzygies of indecomposable projectives and their decomposition into directed string modules, governed by vertex types and effective intersections. The results unify and extend known cases such as gentle algebras being Gorenstein and provide practical, combinatorial criteria for determining Gorensteinness in string algebras.

Abstract

In this paper, we give a description of the self-injective dimension of string algebras and obtain a necessary and sufficient condition for a string algebra to be Gorenstein.

Figures (11)

• Figure 2.1: The vertex $v_0$ in the bound quiver of string algebra.
• Figure 3.1: The vertex $v_0$ of Type $(2^{\mathrm{in}}, 2^{\mathrm{out}})$, where $a_1c_1\in\mathcal{I}$ and $b_1d_1\in\mathcal{I}$
• Figure 3.2: A maximal path with respect to $p$
• Figure 3.3: Left and right-distances
• Figure 3.4: If $\alpha$ exists, that is, $\mathfrak{l}(\wp')>0$, then $r_2=r_{\widehat{p}}^{\mathrm{u}}(w_{t_1})$ effectively left-intersects with $r_1$; otherwise, $r_2=r_{\widehat{p}}^{\mathrm{u}}(\mathfrak{s}(a_{t_1+1}))$ (it is possible for $r_{\widehat{p}}^{\mathrm{u}}(w_{t_1})=r_{\widehat{p}}^{\mathrm{u}}(\mathfrak{s}(a_{t_1+1}))$)

Theorems & Definitions (32)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Example 2.4
• Definition 3.1
• Proposition 3.2
• proof
• Lemma 3.3