Abstract
We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real (Formula presented.) -theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of (Formula presented.) is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of (Formula presented.) and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2-torsion group.
| Original language | English (US) |
|---|---|
| Article number | e12333 |
| Journal | Journal of Topology |
| Volume | 17 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jun 2024 |
Funding
We would like to thank Elden Elmanto, David Gepner, Tyler Lawson, Akhil Mathew, and Bertrand To\u00EBn for helpful conversations about the subject matter of this paper over the years. We further thank Sven van Nigtevecht and the anonymous referee for their remarks improving the\u00A0exposition. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the first and third authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester. The authors would also like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program \u201CHomotopy harnessing higher structures\u201D when work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1. We moreover thank the Hausdorff Research Institute for Mathematics for its hospitality, as the completion of this project took place during the program \u201CSpectral Methods in Algebra, Geometry, and Topology.\u201D The first author was supported by NSF Grants DMS-2120005 and DMS-2102010, and by a Simons Fellowship. The second author was supported by the NWO grant VI.Vidi.193.111. The third author was supported by NSF Grants DMS-1812122 and DMS-2304797, and by a Simons\u00A0Fellowship. The first author was supported by NSF Grants DMS\u20102120005 and DMS\u20102102010, and by a Simons Fellowship. The second author was supported by the NWO grant VI.Vidi.193.111. The third author was supported by NSF Grants DMS\u20101812122 and DMS\u20102304797, and by a Simons Fellowship. This material is based upon work supported by the National Science Foundation under Grant No. DMS\u20101440140, while the first and third authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester. The authors would also like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program \u201CHomotopy harnessing higher structures\u201D when work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1. We moreover thank the Hausdorff Research Institute for Mathematics for its hospitality, as the completion of this project took place during the program \u201CSpectral Methods in Algebra, Geometry, and Topology.\u201D
ASJC Scopus subject areas
- Geometry and Topology