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Comparing tempered and equivariant elliptic cohomology

Jack Morgan Davies

TL;DR

The paper constructs and compares two sophisticated equivariant cohomology theories—tempered cohomology and equivariant elliptic cohomology—within derived algebraic geometry. It achieves a natural, functorial equivalence by expressing both theories as Kan extensions of geometric abelian data and comparing their Hom-stacks, using the P^\infty-torsion and preoriented structures. The main results establish that, on finite abelian groups, Ell and Temp agree, and that this agreement extends to all compact Lie groups that factor as a torus times a finite group, yielding a dualisability result for G-fixed points of TMF. The work clarifies naturality and coherence of the two theories, enabling transfers of results between them and offering a path toward a global, structured framework with norms and level structures. Together, these contributions provide a coherent bridge between algebro-geometric and topological constructions of equivariant cohomology.

Abstract

Lurie and Gepner--Meier each define equivariant cohomology theories, namely \emph{tempered cohomology} and \emph{equivariant elliptic cohomology}, respectively, using derived algebraic geometry. We construct a natural equivalence between these theories where they overlap. Moreover, we emphasise the naturality and coherence of both these equivariant theories as well as our comparison. To demonstrate the use of this comparison, we show that the $G$-fixed points of equivariant topological modular forms is dualisable as a $\mathrm{TMF}$-module for all compact Lie groups $G$ that decompose as a product of a torus and a finite group by formally reducing to an argument of Gepner--Meier.

Comparing tempered and equivariant elliptic cohomology

TL;DR

The paper constructs and compares two sophisticated equivariant cohomology theories—tempered cohomology and equivariant elliptic cohomology—within derived algebraic geometry. It achieves a natural, functorial equivalence by expressing both theories as Kan extensions of geometric abelian data and comparing their Hom-stacks, using the P^\infty-torsion and preoriented structures. The main results establish that, on finite abelian groups, Ell and Temp agree, and that this agreement extends to all compact Lie groups that factor as a torus times a finite group, yielding a dualisability result for G-fixed points of TMF. The work clarifies naturality and coherence of the two theories, enabling transfers of results between them and offering a path toward a global, structured framework with norms and level structures. Together, these contributions provide a coherent bridge between algebro-geometric and topological constructions of equivariant cohomology.

Abstract

Lurie and Gepner--Meier each define equivariant cohomology theories, namely \emph{tempered cohomology} and \emph{equivariant elliptic cohomology}, respectively, using derived algebraic geometry. We construct a natural equivalence between these theories where they overlap. Moreover, we emphasise the naturality and coherence of both these equivariant theories as well as our comparison. To demonstrate the use of this comparison, we show that the -fixed points of equivariant topological modular forms is dualisable as a -module for all compact Lie groups that decompose as a product of a torus and a finite group by formally reducing to an argument of Gepner--Meier.
Paper Structure (6 sections, 18 theorems, 88 equations)

This paper contains 6 sections, 18 theorems, 88 equations.

Table of Contents

  1. Functoriality of equivariant elliptic cohomology
  2. Functoriality of tempered cohomology
  3. Comparing definitions of preorientations
  4. Comparing homomorphism stacks
  5. Proof of \ref{['maintheoremalg', 'maintheoremgeo']}
  6. An application to equivariant topological modular forms

Key Result

Theorem A

There is a natural equivalence of functors induced by the inclusion ${\mathscr{F}}[\mathbf{P}^\infty]\to {\mathscr{F}}$.

Theorems & Definitions (54)

  • Theorem A
  • Theorem B
  • Theorem C
  • Conjecture 1
  • Definition 1.1
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • Definition 1.6
  • ...and 44 more