On the parametrized Tate construction and two theories of real $p$-cyclotomic spectra

Abstract

We give a new formula for $p$-typical real topological cyclic homology that refines the fiber sequence formula discovered by Nikolaus and Scholze for $p$-typical topological cyclic homology to one involving genuine $C_2$-spectra. To accomplish this, we give a new definition of the $\infty$-category of real $p$-cyclotomic spectra that replaces the usage of genuinely equivariant dihedral spectra with the parametrized Tate construction $(-{)}^{t_{C_2}{\mu }_p}$ associated to the dihedral group $D_{2p}={\mu }_p⋊C_2$. We then define a $p$-typical and $\infty$-categorical version of Høgenhaven's $O(2)$-orthogonal cyclotomic spectra, construct a forgetful functor relating the two theories, and show that this functor restricts to an equivalence between full subcategories of appropriately bounded below objects.

arXiv comment110 pages

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• Real motivic and $C_2$-equivariant Mahowald invariants
• On the geometric fixed-points of real topological cyclic homology
• The Picard group in equivariant homotopy theory via stable module categories
• Real topological Hochschild homology via the norm and Real Witt vectors
• Tate blueshift and vanishing for Real oriented cohomology
• Integral Models for Spaces via the Higher Frobenius
• On the parametrized Tate construction
• Recollements and stratification
• Dualizable objects in stratified categories and the 1-dimensional bordism hypothesis for recollements