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 $D_{2p} = \mu_p \rtimes C_2$. 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2019
- DOI:
- 10.48550/arXiv.1909.03920
- arXiv:
- arXiv:1909.03920
- Bibcode:
- 2019arXiv190903920Q
- Keywords:
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- Mathematics - Algebraic Topology;
- Mathematics - K-Theory and Homology;
- 19D55;
- 55P42;
- 55P43;
- 55P91;
- 18D05;
- 16E40;
- 13D03
- E-Print:
- 110 pages