(2019-05-15) Sulyma, Yuri John Fraser; Blumberg, Andrew J.; Freed, Daniel S; Ben-Zvi, David D; Hill, Michael A

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We study two invariants of topological Hochschild homology coming from equivariant homotopy theory: its RO(C [subscript p superscript n])-graded homotopy Mackey functors, and the regular slice filtration. In the case of RO(C [subscript p superscript n])-graded homotopy, we explain how to relate Angeltveit-Gerhardt's work to the gold elements, and in cases of interest give canonical identifications of the relevant groups in terms of the kernels of the Fontaine maps θ̃ [subscript r]. This is then used as input for studying the slice filtration on THH. When R is a torsionfree perfectoid ring, we show that the C [subscript p] -regular slice spectral sequence of THH(R; Z [subscript p]) collapses at E².

Let R be a connective ring spectrum and let M be an R-bimodule. In this paper
we prove several results that relate the K-theory of R⋉M and T[superscript M, subscript R] to a “topological Witt vectors” construction W(R; M), where R ⋉ M is the square-zero extension of R by M and T [superscript M, subscript R] is the tensor algebra on M. Our main results include a desciption
of the Taylor tower of K(R ⋉ (−)) and the derived functor of K̃(TR(−)) on the category
of R-bimodules in terms of the Taylor tower of W(R;−). W(R;−) has an easily described Taylor tower, given explicitly by Lindenstrauss and McCarthy in [17]. Our main results serve as generalizations of the results for discrete rings in [17, 18] and also extend the computations by Hesselholt and Madsen [15] showing that π₀(TR(R; p)) is isomorphic to the p-typical Witt vectors over R when R a commutative ring.