Prisms and Tambara functors I: Twisted powers, transversality, and the perfect sandwich

Abstract

We construct a faithful and conservative functor from prisms to $C_{p^\infty}$-Tambara functors; in appropriate situations, this gives an algebraic description of $\underline\pi_0{\mathrm{TC}^-}$. We also present two integral variants using the generalized $n$-series of Devalapurkar-Misterka. The construction is based on the "twisted $I$-adic" or "$(p^\bullet)_q$" filtration, and is closely related to $q$-divided powers. To verify the axioms, we introduce a new technique for constructing Tambara functors, inspired by transversal prisms. We apply this to give a conceptual construction of Molokov's de Rham-Witt comparison map, and generalize it to a triangle sandwiching prismatic theory between theories built from Witt vectors and adjunction of $p$-power roots.