The $S^1$-Equivariant Cohomology of Spaces of Long Exact Sequences

TL;DR

This paper studies the $S^1$-equivariant cohomology of moduli spaces of exact sequences, providing new obstructions and generalizations of classical equations through cohomological computations.

Contribution

It introduces a novel interpretation of chain complexes as equivariant maps and computes their cohomology to derive obstructions and generalize existing equations.

Findings

Computed the cohomology of moduli spaces of exact sequences.

Derived obstructions to certain equivariant maps.

Generalized Herzog-Kühl equations.

Abstract

Let $S$ denote the graded polynomial ring $\C[x_1,...,x_m]$. We interpret a chain complex of free $S$-modules having finite length homology modules as an $S^1$-equivariant map $\C^m\sm\{0\} \to X$, where $X$ is a moduli space of exact sequences. By computing the cohomology of such spaces $X$ we obtain obstructions to such maps, including a slight generalization of the Herzog-K\"uhl equations.