On the equivalence of two theories of real cyclotomic spectra

Abstract

We give a new formula for real topological cyclic homology that refines the fiber sequence formula discovered by Nikolaus and Scholze for topological cyclic homology to one involving genuine C 2 -spectra. To accomplish this, we give a new definition of the ∞-category of real cyclotomic spectra that replaces the usage of genuinely equivariant dihedral spectra with the parametrized Tate construction. We then define an ∞-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. As an application, we compute the real topological cyclic homology of perfect Fp-algebras for all primes p. Contents 1. Introduction 1.1. Motivation 1.2. Main results 1.3. Conventions, notation, and prerequisites 1.4. Acknowledgments 2. Two theories of real p-cyclotomic spectra 2.1. Borel real p-cyclotomic spectra 2.2. Genuine real p-cyclotomic spectra 2.3. C 2 -symmetric monoidal structures 3. Comparison of the theories 3.1. The dihedral Tate orbit lemma 3.2. The comparison at finite level 3.3. Exchanging a lax equalizer for an equalizer 3.4. Proof of the main theorem 3.5. Variant: O(2)-spectra at the prime p 4. Extension to integral theories 4.1. Borel real cyclotomic spectra 4.2. F-genuine real cyclotomic spectra 4.3. Integral comparison 5. Real cyclotomic structure on THR of C 2 -E ∞ -algebras 6. TCR of the constant mod p Mackey functor 6.1. Computation for p odd 6.2. Computation for p = 2 6.3. Extension to perfect F p -algebras Appendix A. On the geometric fixed points of real topological cyclic homology