A study of a quadratic almost complete intersection ideal and its linked Gorenstein ideal
Rachel Diethorn, Sema Güntürkün, Alexis Hardesty, Pinar Mete, Liana Şega, Aleksandra Sobieska, Oana Veliche
TL;DR
This work analyzes the almost complete intersection $R$ in ${\sf k}[x_1,\dots,x_n]$ defined by $I=(x_1^2,\dots,x_n^2,(x_1+\dots+x_n)^2)$ and its Gorenstein link $G=J:I$, with $J=(x_1^2,\dots,x_n^2)$. It computes graded Betti numbers of $R$ (and of the linked ring $A={\sf Q}/G$) for odd $n$ and extends even-$n$ results, proving ${\sf Q}/I$ is level with socle dimension equal to the Catalan number and describing generators and the reverse-lex initial ideal of $G$. The paper then provides explicit Betti tables for $R$ and $A$ over ${\sf Q}$, introducing sequences $\{\rho_k(n)\}$ and $\{\gamma_i(n)\}$ to express the Betti numbers in closed forms, and uses liftability to ${\sf Q}$ and reduction to intermediate rings to derive these results. Finally, it gives a detailed presentation of the generators and initial ideal of $G$ and proves that the Gorenstein quotient $A$ satisfies the Strong Lefschetz Property in characteristic $0$ via the Macaulay inverse system and Hessian criteria. These results offer precise homological data for ideals generated by $n+1$ general quadrics and lay groundwork for broader investigations in the aci framework.
Abstract
We examine the ideal $I=(x_1^2, \dots, x_n^2, (x_1+\dots+x_n)^2)$ in the polynomial ring $Q=k[x_1, \dots, x_n]$, where $k$ is a field of characteristic zero or greater than $n$. We also study the Gorenstein ideal $G$ linked to $I$ via the complete intersection ideal $(x_1^2, \dots, x_n^2)$. We compute the Betti numbers of $I$ and $G$ over $Q$ when $n$ is odd and extend known computations when $n$ is even. A consequence is that the socle of $Q/I$ is generated in a single degree (thus $Q/I$ is level) and its dimension is a Catalan number. We also describe the generators and the initial ideal with respect to reverse lexicographic order for the Gorenstein ideal $G$.