Corrigendum: On the Gorenstein locus of simplicial affine semigroup rings [Comm. Algebra 50 (2022), no. 9, 4032–4039.]

Abstract

The Cohen-Macaulay condition is added to the statements of Proposition 3.3 and Corollary 3.4. We also correct some small typos.

Keywords:

• Apéry set
• Gorenstein rings
• simplicial affine semigroup

2020 Mathematics Subject Classification:

• 13H10
• 20M25
• 05E40

This article refers to:

On the Gorenstein locus of simplicial affine semigroup rings

Raheleh Jafari^a, Abdoljavad Taherizadeh^b, Marjan Yaghmaei^b

^aMosaheb Institute of Mathematics, Kharazmi University,Tehran, Iran

^bFaculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran

CONTACT Raheleh Jafari [email protected] Mosaheb Institute of Mathematics, Kharazmi University, Tehran, Iran.

In proposition 3.3, for the implication (2) $\Rightarrow$ (1), we use Theorem 3.1 where we need $R_{(\mathfrak{p})}$ to be Cohen-Macaulay. Thus, we need the Cohen-Macaulay assumption in Proposition 3.3 and Corollary 3.4, as well.

Proposition 3.3

Let $n\ge 0$ be the dimension of the non-Gorenestein locus of $\mathbb{K}[S]$. If $\mathbb{K}[S]$ is Cohen-Macaulay, then the following statements are equivalent for an integer $k\ge 0$.

1. $n\le k$.

2. For any face F of S such that $E\cap F=\{{\mathbf{a}}_{i_1},\dots ,{\mathbf{a}}_{i_{k+1}}\}$ has k + 1 elements, there exists $\mathbf{m}\in \text{Ap}(S,E)$ and ${\lambda }_1,\dots ,{\lambda }_{k+1}\in \mathbb{N}$ such that $w{⪯}_S\mathbf{m}+{\displaystyle \sum _{j=1}^{k+1}{\lambda }_j}{a}_{i_j}$ for all $\mathbf{w}\in \text{Ap}(S,E)$.

Corollary 3.4

If $\mathbb{K}[S]$ is Cohen-Macaulay, then $\mathbb{K}[S]$ is Gorenstein on the punctured spectrum precisely when, for any , there e $1\le i\le d$ xist ${\mathbf{m}}_i\in \text{Ap}(S,E)$ and ${\lambda }_i\in \mathbb{N}$ such that $w{⪯}_S{\mathbf{m}}_i+{\lambda }_i{a}_i$ for all $\mathbf{w}\in \text{Ap}(S,E)$.

On the 3rd line of the preliminaries section, the phrase $\mathbb{N}{\mathbf{a}}_1+\dot{c}+\mathbb{N}{\mathbf{a}}_{i_d}$ should be $\mathbb{N}{\mathbf{a}}_1+\cdots +\mathbb{N}{\mathbf{a}}_{i_d}$.

In the proof of Corollary 3.4, “Proposition 3.3” should be “Corollary 3.3”.

On the 3rd line of the proof of Corollary 3.5, the correct equation is$$〈{\mathfrak{n}}_i,{\mathbf{w}}_i〉=\max \{〈{\mathfrak{n}}_i,\mathbf{w}〉;\mathbf{w}\in \text{Ap}(S,E)\}.$$