Adelic Descent for Equivariant Elliptic Cohomology

TL;DR

This paper introduces a new adelic descent approach to define and analyze rationalized G-equivariant elliptic cohomology, connecting it with singular cohomology and K-theory, and comparing it with cyclic homology theories.

Contribution

It develops a novel adelic descent framework for equivariant elliptic cohomology and provides new descriptions and comparison results with other homology theories.

Findings

Defined k-rationalized G-equivariant elliptic cohomology via adelic descent

Provided adelic descriptions of rationalized G-equivariant singular cohomology and K-theory

Established comparison results with periodic cyclic homology theories

Abstract

We define $k$-rationalized $G$-equivariant elliptic cohomology, for a field of characteristic zero $k$ and a compact Lie group $G$, via adelic descent. We also give adelic descriptions of rationalized $G$-equivariant singular cohomology and K-theory. This completes a program first proposed by Ro\c{s}u. These descriptions are then used to obtain comparison results with periodic cyclic homology theories defined via derived algebraic geometry.