On the Hyperhomology of the Small Gobelin in Codimension 2

TL;DR

This paper investigates the hyperhomology of the small Gobelin complex associated with zero-dimensional Gorenstein algebras, providing an inductive method to compute it and revealing stable dimension differences in high degrees related to geometric invariants.

Contribution

It introduces an inductive procedure for calculating hyperhomologies of the small Gobelin complex and links these to geometric invariants of vector fields and complete intersections.

Findings

Hyperhomology differences stabilize in high degrees.

Two ideal flags encode the dimension differences.

Application to tangency conditions in algebraic geometry.

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 $\hbox{Spec}(\mathbb{B})$.