The prismatization of $p$-adic formal schemes

Abstract

In this note, we introduce and study the Cartier-Witt stack WCartX attached to a padic formal scheme X as well as some variants. In particular, we reinterpret the notion of prismatic crystals on X and their cohomology in terms of quasicoherent sheaf theory on WCartX in favorable situations. Contents 1. Introduction 2. Animated prisms 3. Absolute prismatization 4. Revisiting the prismatic logarithm 5. The relative prismatization 6. Comparison with prismatic cohomology 7. Derived relative prismatization 8. Derived absolute prismatization 9. Examples: The Hodge-Tate stack of some regular schemes 10. Some questions on the Hodge-Tate stack and regularity Appendix A. Animated δ-rings ReferencesThis document is a postscript to [2]. In particular, the introduction below should be read in conjunction with that of [2]. Moreover, this document is a preliminary version, and we hope to revisit and expand on the exposition in the future. In particular, the definition and basic properties of the key object of this study in this paper -the prismatization WCart X of a bounded p-adic formal scheme X -rely critically on the notion of mapping spaces provided by derived p-adic formal geometry, and some of our arguments rely on a working theory of derived formal δ-schemes; neither of these theories has been systematically documented in the literature yet as far as we know.References to [2] in this paper have the form APC.x.y.z.

P-adic Simpson correpondence via prismatic crystals

“…Remark 1.7. Similar results have been obtained independently in works of Bhatt-Lurie [BL22a], [BL22b] by using the Hodge-Tate stack WCart HT X when the base field K is absolutely unramified. For example, Theorem 1.6 may give some evidence for [BL22b, Conjecture 10.1].…”

“…Remark 1.17. In [BL22a], [BL22b], Bhatt and Lurie can confirm Conjecture 1.16 when K is absolutely unramified.…”

Hodge--Tate crystals on the logarithmic prismatic sites of semi-stable formal schemes

“…Finally, it is worth pointing out that the usual absolute prismatic theory was recently independently studied by Drinfeld [Dri20] and Bhatt-Lurie [BL22a], [BL22b] in a stacky way. Indeed, one can relate a p-adic formal scheme X to a certain stack, which is called the prismatization of X, such that studying prismatic theory on (X) ∆ amounts to studying coherent theory on the corresponding stack.…”

“…So one may ask whether we can recover and generalise all known results mentioned above by using this stack. In particular, we hope that one can get an integral p-adic non-abelian Hodge theory in this way (for example, see [BL22b,Remark 9.2] when X is equipped with a δ-structure over W(k)).…”

v-vector bundles on p-adic fields and Sen theory via the Hodge-Tate stack

“…A new perspective via the Hodge-Tate stack. The starting point of this article is the idea that the recent introduction of ring stacks in p-adic Hodge theory by Drinfeld [Dri20] and Bhatt-Lurie [BL22a], [BL22b] will help finding a more canonical formulation of such results and thereby shed new light on the (as yet still mostly conjectural) p-adic Simpson correspondence. The goal of this paper is to realize this concretely in the case X = Spa(K).…”