Stabilization of Betti tables (Q2453707)

Let \(\mathbf{k}\) be a field and let \(I\subseteqq R=\mathbf{k}[x_1,\ldots,x_n]\) be an ideal. The graded Betti numbers of a homogeneous ideal \(I\) are given by \(\beta_{i,j}(I)=\text{dim}_\mathbf{k}\text{Tor}_i(\mathbf{k},I)\). The author organize this data into the Betti table of \(I\) displaying \(\beta_{i,i+j}(R/I)\) in \(i\)th column and \(j\)th row. An equigenerated ideal \(I\) is an ideal of \(\mathbf{k}[x_1,\ldots,x_n]\) if \(I\) generated by homogenous generators of the same degree. Using techniques similar by \textit{K. Borna} [Osaka J. Math. 46, No. 4, 1047--1058 (2009; Zbl 1183.13016)], \textit{S. D. Cutkosky} et al. [Compos. Math. 118, No. 3, 243--261 (1999; Zbl 0974.13015)] and \textit{T. Römer} [Ill. J. Math. 45, No. 4, 1361--1376 (2001; Zbl 1094.13525)] the author proves a sharper result on the asymptotic of Betti tables of powers \(I^d\). Theorem. Let \(I=(f_0,f_1,\ldots,f_k)\subseteqq\mathbf{k}[x_1,\ldots,x_n]=R\) be an equigenerated ideal in degree \(r\). Then there exists a \(D\) such that, for all \(d>D\), we have \(\beta_{i,j+rd}(I^d)\neq0 \Leftrightarrow \beta_{i,j+rD}(I^D)\neq0\).