Equivariant cohomology and the super reciprocal plane of a hyperplane arrangement

TL;DR

This paper explores the algebraic structure of rings related to the reciprocal plane of hyperplane arrangements, providing presentations, interpretations, and applications in equivariant cohomology calculations.

Contribution

It offers a new presentation and interpretation of rings associated with reciprocal planes, extending their use in equivariant cohomology computations for arbitrary finite groups.

Findings

Presented generators and relations for the rings.

Connected these rings to superscheme interpretations.

Applied to compute localized equivariant cohomology coefficients.

Abstract

In this paper, we investigate certain graded-commutative rings which are related to the reciprocal plane compactification of the coordinate ring of a complement of a hyperplane arrangement. We give a presentation of these rings by generators and defining relations. This presentation was used by Holler and I. Kriz to calculate the $\mathbb{Z}$-graded coefficients of localizations of ordinary $RO((\mathbb{Z}/p)^n)$-graded equivariant cohomology at a given set of representation spheres, and also more recently by the author in a generalization to the case of an arbitrary finite group. We also give an interpretation of these rings in terms of superschemes, which can be used to further illuminate their structure.