On equivariant affineness in elliptic cohomology
Abstract
We investigate equivariant affineness in spectral algebraic geometry: the phenomenon where ∞-categories of quasi-coherent sheaves of a spectral stack are equivalent to ∞-categories of modules over an equivariant ring spectrum. In the nonequivariant case, this is known as 0-affineness, and has been studied by Gaitsgory, Mathew--Meier, amongst others. Allowing for equivariant phenomena opens up a much wider class of spectral stacks to be studied through the lens of spectra. The spectral stacks at the heart of our analysis are oriented elliptic curves, their associated Hilbert schemes, and related constructions. When working over the universal base, so the moduli stack of oriented elliptic curves, we show that these spectral stacks are U(1)-, U(n)-, and SU(n)-equivariantly affine, with associated G-ring spectrum Gepner--Meier's equivariant topological modular forms TMFG. Our results also encompass the groups Sp(n) as well as all finite products of such groups. Our most general result is a relative statement over an arbitrary base stack. These results can be viewed as an integral and global refinement of Greenlees' algebraic model for modules over rational U(1)-equivariant elliptic cohomology.
Related
- Equivariant elliptic cohomology via loop spaces
- The ℤ/2-equivariant Seiberg--Witten Floer cohomology of the $(2, 1)$-cable is an unknot detector
- Local coefficients for genuine equivariant cohomology I
- Formal Neighborhood Equivariant Cohomology and Kirwan Surjectivity at Singular Values
- Bounded cohomology property on Jacobian elliptic surfaces with simplicial Mori cones
BibTeX
@misc{davies2026equivariant,
title = {{On equivariant affineness in elliptic cohomology}},
author = {Jack Morgan Davies and Sil Linskens and Leonard Tokic},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39583},
eprint = {2610.39583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39583},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On equivariant affineness in elliptic cohomology
AU - Jack Morgan Davies
AU - Sil Linskens
AU - Leonard Tokic
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39583
UR - https://arxiv.org/abs/2610.39583
AB - We investigate equivariant affineness in spectral algebraic geometry: the phenomenon where ∞-categories of quasi-coherent sheaves of a spectral stack are equivalent to ∞-categories of modules over an equivariant ring spectrum. In the nonequivariant case, this is known as 0-affineness, and has been studied by Gaitsgory, Mathew--Meier, amongst others. Allowing for equivariant phenomena opens up a much wider class of spectral stacks to be studied through the lens of spectra. The spectral stacks at the heart of our analysis are oriented elliptic curves, their associated Hilbert schemes, and related constructions. When working over the universal base, so the moduli stack of oriented elliptic curves, we show that these spectral stacks are U(1)-, U(n)-, and SU(n)-equivariantly affine, with associated G-ring spectrum Gepner--Meier's equivariant topological modular forms TMF_G. Our results also encompass the groups Sp(n) as well as all finite products of such groups. Our most general result is a relative statement over an arbitrary base stack. These results can be viewed as an integral and global refinement of Greenlees' algebraic model for modules over rational U(1)-equivariant elliptic cohomology.
ER -