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Adelic Descent for Equivariant Elliptic Cohomology

Paolo Tomasini

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 Roşu. These descriptions are then used to obtain comparison results with periodic cyclic homology theories defined via derived algebraic geometry.

Adelic Descent for Equivariant Elliptic Cohomology

Abstract

We define -rationalized -equivariant elliptic cohomology, for a field of characteristic zero and a compact Lie group , via adelic descent. We also give adelic descriptions of rationalized -equivariant singular cohomology and K-theory. This completes a program first proposed by Roşu. These descriptions are then used to obtain comparison results with periodic cyclic homology theories defined via derived algebraic geometry.
Paper Structure (18 sections, 27 theorems, 155 equations)

This paper contains 18 sections, 27 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complexified Equivariant Elliptic Cohomology
  4. Adelic descent
  5. Shifted tangent bundles and loop spaces
  6. The Adelic Decomposition of Equivariant Elliptic Cohomology
  7. The rank one case
  8. The higher rank case
  9. Extension to compact Lie groups
  10. Cochain-level variant
  11. The Adelic Decomposition for Equivariant Cohomology and K-theory
  12. Equivariant cohomology
  13. Equivariant K-theory
  14. Geometric presentations for Equivariant Cohomology and K-theory
  15. Equivariant K-theory
  16. ...and 3 more sections

Key Result

Theorem A

Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$ acted on by a reductive group $G$. There is an equivalence of $\mathbb{Z}_2$-periodic coherent sheaves on the GIT adjiont quotient $G//G$ which is natural in $X$ with respect to $G$-equivariant maps. Here, the left-hand side is periodic cyclic homology and the right-hand side is $G^\mathrm{an}$-equivariant K-theory of the analytificati

Theorems & Definitions (77)

  • Theorem A: Theorem \ref{['thm:HPKG']}
  • Theorem B: Theorem \ref{['thm:HPlHG']}
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Theorem 3.1 in Groech
  • Theorem 2.4: Beilinson BeilAd
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 67 more