On the periodicity of the first Betti number of the semigroup rings under translations (Q2860851)

Let \(k\) be a field of characteristic \(0\). Consider the monomial curve \(\Gamma_{\underline{a}}:=\{(t^{a_1}, \dots , t^{a_n}):t \in k\} \subset \mathbb{A}^n_k\) associated to a vector \(\underline{a}=(a_1, \dots , a_n)\in \mathbb{N}^n_0\) with \(a_1<\cdots < a_n\). The defining ideal \(P_{\underline{a}}\) of \(\Gamma_{\underline{a}}\) is the kernel of the \(k\)-algebra homomorphism \(\varphi\) defined by \(R:=k[x_1, \dots , x_n] \to k[t], x_i \mapsto t^{a_i}\) for every \(i=1, 2, \dots , n\). J. Herzog and H. Srinivasan conjectured that the Betti numbers NEWLINE\[NEWLINE \beta_i(P_{\underline{a}+j})=\dim_{k}\mathrm{Tor}^R_{i} (P_{\underline{a}+j},k) NEWLINE\]NEWLINE with \(\underline{a}+j=(a_1+j, a_2+j, \dots , a_n+j)\), are eventually periodic in \(j\) for every \(j \in \mathbb{N}\) with period \(a_n-a_1\).NEWLINENEWLINEThe purpose of this paper is to prove that, for the special case \(\underline{a}=(1,1+a, 1+a+b,1+a+b+c) \in \mathbb{N}^4\), the following statements hold: {\parindent=6mm \begin{itemize}\item[(1)] If either \(a\) divides \(b+c\) or \(c\) divides \(a+b\), and \(j\) divides \(a+b+c\), then \(\beta_0(P_{\underline{a}+j})=3\). \item[(2)] Assume either \(c\) divides \(a+b\) and \(j=(a+b+c)n+(a+b)m\), or \(a\) divides \(b+c\) and \(j=(a+b+c)n+(b+c)m\) for some \(n,m \in \mathbb{N}\); then \(\beta_0(P_{\underline{a}+j})=4\). NEWLINENEWLINE\end{itemize}} Moreover the author consider the semigroup ring \(k[\Gamma_{\underline{a}}]\) associated with the monomial curve \(\Gamma_{\underline{a}}\) as the image of the homomorphism \(\varphi\). He takes a vector \(\underline{a}=(a,b,c) \in \mathbb{N}^3\) and sets \(j=(a+b+c)n+r\) for some nonnegative integers \(n,r\), and either \(a\) dividing \(b+c\) or \(c\) dividing \(a+b\), in such a way that NEWLINE\[NEWLINE \underline{a}+j=(j,a+j,a+b+j,a+b+c+j) \in \mathbb{N}^4. NEWLINE\]NEWLINE For \(j \geq (a+b+c)^3\) the author proves that the semigroup rings \(k[\Gamma_{\underline{a}+j}]\) are complete intersection if and only if \(j\) divides \(a+b+c\), and this complete intersection appears in the family of such semigroup rings eventually with period \(a+b+c\).NEWLINENEWLINEThe conjecture by Herzog and Srinivasan was recently proven by \textit{T. Vu} in his paper [J. Algebra 418, 66--90 (2014; Zbl 1317.13037)].