On the Hyperhomology of the Small Gobelin in Codimension 2

Abstract

Given a zero-dimensional Gorenstein algebra $\mathbb{B}$ and two syzygies between two elements $f_1,f_2\in\mathbb{B}$, one constructs a double complex of $\mathbb{B}$-modules, ${\cal G}_\mathbb{B},$ called the small Gobelin. We describe an inductive procedure to construct the even and odd hyperhomologies of this complex. For high degrees, the difference $\dim \mathbb{H}_{j+2}({\cal G}_\mathbb{B}) - \dim\mathbb{H}_j({\cal G}_\mathbb{B})$ is constant, but possibly with a different value for even and odd degrees. We describe two flags of ideals in $\mathbb{B}$ which codify the above differences of dimension. The motivation to study this double complex comes from understanding the tangency condition between a vector field and a complete intersection, and invariants constructed in the zero locus of the vector field $\hbox{Spec}(\mathbb{B})$.