A Degree Formula for Equivariant Cohomology Rings

TL;DR

This paper extends the understanding of the degree invariant in equivariant cohomology rings, providing an additive formula that links algebraic and geometric aspects of these invariants.

Contribution

It introduces a new additivity formula for the degree of equivariant cohomology rings, connecting algebraic invariants with geometric data.

Findings

Derived an explicit additivity formula for the degree of equivariant cohomology rings.

Linked algebraic invariants to geometric and group-theoretic data.

Demonstrated the algebraic and geometric nature of the degree invariant.

Abstract

This paper generalizes a result of Lynn on the "degree" of an equivariant cohomology ring $H^*_G(X)$. The degree of a graded module is a certain coefficient of its Poincar\'{e} series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree: $$deg(H^*_G(X)) = \sum_{[A,c] \in \mathcal{Q'}_{max}(G,X)}\frac{1}{|W_G(A,c)|} \deg(H^*_{C_G(A,c)}(c)).$$ We also show how this formula relates to the additivity formula from commutative algebra, demonstrating both the algebraic and geometric character of the degree invariant.