Real syntomic cohomology

TL;DR

This paper develops a new version of syntomic cohomology called Real syntomic cohomology for ring spectra with involution, extending and refining existing theories, and computes it for specific topological theories.

Contribution

It introduces Real syntomic cohomology for ring spectra with involution, extending prior syntomic theories and providing explicit computations for topological K-theory and modular forms.

Findings

Extended syntomic cohomology to include involutions

Refined existing syntomic theories by Bhatt--Morrow--Scholze and others

Computed Real syntomic cohomology for topological K-theory and modular forms

Abstract

We introduce a theory of syntomic cohomology for ring spectra with involution, which we call Real syntomic cohomology. We show that our construction extends the theory of syntomic cohomology for rings with involution due to Park. Our construction also refines syntomic cohomology as developed by Bhatt--Morrow--Scholze, Morin, Bhatt--Lurie, and Hahn--Raksit--Wilson. We compute the Real syntomic cohomology of Real topological K-theory and topological modular forms with level structure.