Cartier modules and cyclotomic spectra

Abstract

We construct and study a t-structure on p-typical cyclotomic spectra and explain how to recover crystalline cohomology of smooth schemes over perfect fields using this t-structure. Our main tool is a new approach to p-typical cyclotomic spectra via objects we call p-typical topological Cartier modules. Using these, we prove that the heart of the cyclotomic t-structure is the full subcategory of derived V -complete objects in the abelian category of p-typical Cartier modules.We say that a p-typicalThe existence and uniqueness of such a t-structure is a formal consequence of the fact that (CycSp p ) 0 is presentable and is closed under colimits and extensions in CycSp p . The difficult part of the theorem is the identification of the heart.Recall Bökstedt's theorem, which says that π * THH(F p ) = F p [b], a polynomial ring on a degree 2 generator. More generally, using the vanishing of the cotangent complex, one deduces that π * THH(k) = k[b] for any perfect ring k. Our interest in the cyclotomic t-structure was piqued by the discovery of the next result. 1 Unless otherwise specified, all quotients M/V n are computed in the derived sense (and hence are given as the cofiber of M V n −− → M in the derived category) as are all limits.Corollary 10. If X is a bounded below p-typical cyclotomic spectrum, then the natural S 1 -equivariant map TR(X) → X induces a p-adic equivalence TR(X) tS 1 → X tS 1 .Proof. By Theorem 9, the counit map TR(X)/TR(X) hCp → X is an equivalence. In particular, we have a cofiber sequence TR(X) hCp → TR(X) → X.Applying (−) tS 1 , we obtain a cofiber sequence p-adically equivalent to[NS17, Lemma II.4.2]. But, (TR(X) hCp ) tCp ≃ 0 by the Tate orbit lemma [NS17, Lemma I.2.1].2 In early talks on this project, we called these topological Dieudonné modules. The cyclotomic t-structureIn this section we define the cyclotomic t-structure for integral and p-typical cyclotomic spectra in their genuine and non-genuine flavors. With some difficulty, one can prove some basic facts about truncations in

On -local

“…Suppose is -complete with trivialized Frobenius. Then, as in [AN21, Remark 2.5] and [NS18, Corollary II.4.7], we obtain a natural equivalence For this, we also use the identification for , , see [NS18, Lemma II.4.1].…”

“…We reduce by dévissage to the case where is concentrated in degree , and . The completion of is given by where the maps are given by ; see [AN21, Proposition 3.22].…”

“…Now since is discrete, the Verschiebung is simply given by a map of abelian groups, and the -action is trivial. We can (as in the proof of [AN21, Lemma 3.25]) form a -indexed diagram such that the transition maps in the -direction are given by and in the -direction are given by the canonical projections. By the above, the completion of is given by the cofiber of the map .…”

“…By dévissage, we can reduce to the case where is an -module (in topological Cartier modules). Let be the associated cyclotomic spectrum, so is the derived -completion of by the correspondence between bounded-below -complete Cartier modules and bounded-below cyclotomic spectra; see [AN21, Theorem 3.21]. Then the proof of Proposition 5.2 shows that the cyclotomic Frobenius is -truncated, since .…”

“…We write for the -category of spectra and for the sphere spectrum. We use the theory of cyclotomic spectra in the form developed in [NS18], as well as the theory of topological Cartier modules developed in [AN21]; we write for the -category of cyclotomic spectra. We write for -typical .…”