Hodge-Tate stacks and non-abelian $p$-adic Hodge theory of v-perfect complexes on rigid spaces

TL;DR

This paper develops a new functorial framework connecting perfect complexes on Hodge-Tate stacks to those on the v-site of the generic fiber, advancing non-abelian p-adic Hodge theory for rigid spaces.

Contribution

It constructs a fully faithful functor linking perfect complexes on Hodge-Tate stacks to the v-site, and introduces a derived p-adic Simpson functor for rigid spaces.

Findings

Established a fully faithful functor between categories of perfect complexes.

Described perfect complexes on Hodge-Tate stacks via Higgs and Higgs-Sen modules.

Developed a derived p-adic Simpson correspondence.

Abstract

Let $X$ be a quasi-compact quasi-separated $p$-adic formal scheme that is smooth either over a perfectoid $\mathbb{Z}_p$-algebra or over some ring of integers of a $p$-adic field. We construct a fully faithful functor from perfect complexes on the Hodge-Tate stack of $X$ up to isogeny to perfect complexes on the v-site of the generic fibre of $X$. Moreover, we describe perfect complexes on the Hodge-Tate stack in terms of certain derived categories of Higgs, resp. Higgs-Sen modules. This leads to a derived $p$-adic Simpson functor.