Dirac geometry II: coherent cohomology

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Dirac rings are commutative algebras in the symmetric monoidal category of Z-graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger ∞-category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to MU and Fp in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves.
OriginalsprogEngelsk
Artikelnummere27
TidsskriftForum of Mathematics, Sigma
Vol/bind12
Antal sider93
DOI
StatusUdgivet - 2024

Bibliografisk note

Funding Information:
Both authors were supported by the Danish National Research Foundation through the Copenhagen Center for Geometry and Topology (DNRF151). Lars Hesselholt received support from JSPS Grant-in-Aid for Scientific Research number 21K03161 and from a grant from the Institute for Advanced Study School of Mathematics, and Piotr Pstrągowski received support from NSF Grant #DMS-1926686 and Deutsche Forschungsgemeinschaft #EXC-2047/1 – 390685813.

Funding Information:
It is a pleasure to thank Maxime Ramzi for carefully reading a first draft of Appendix 1 and for several helpful suggestions. The second author also would like to thank Kazuhiro Fujiwara and Nagoya University, and Takeshi Saito and the University of Tokyo, for their support and hospitality during his visit to Japan, where part of this paper was written. Finally, we thank an anonymous referee for helpful comments. Both authors were supported by the Danish National Research Foundation through the Copenhagen Center for Geometry and Topology (DNRF151). Lars Hesselholt received support from JSPS Grant-in-Aid for Scientific Research number 21K03161 and from a grant from the Institute for Advanced Study School of Mathematics, and Piotr Pstragowski received support from NSF Grant #DMS-1926686 and Deutsche Forschungsgemeinschaft #EXC-2047/1 – 390685813.

Publisher Copyright:
© The Author(s), 2024.

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