Rational Hodge--Tate prismatic crystals of quasi-l.c.i algebras and non-abelian $p$-adic Hodge theory

Abstract

Consider a bounded prism $(A,I)$ and a bounded quasi-l.c.i algebra $R$ over $\overline{A}$. In this paper, for any prism $S/A$ with a surjection $S\to R$ such that $\widehat{\mathbb L}_{\overline{S}/\overline{A}}$ is a $p$-completely flat module over $\overline{S}$, we establish an equivalence of categories between rational Hodge-Tate crystals on $(R/A)_{\Delta}$ and topologically nilpotent integrable connections on the Hodge--Tate cohomology ring $\overline{\Delta}_{R/S}$. As an application, for a non-zero divisor $a\in \overline{A}$, we introduce the concept of $a$-smallness for a rational Hodge-Tate prismatic crystal on $(R/A)_{\Delta}$. Finally, we focus on some special algebras $R$ over $\mathcal O_{\mathbb C_p}$ (or generally, the ring of integers of an algebraic closed and complete non-archimedean field) including all $p$-completely smooth algebras, $p$-complete algebras with semi-stable reductions and geometric valuation rings. By using our equivalence, we analyze the restriction functor from the category of $a$-small rational Hodge-Tate prismatic crystals to the category of $v$-vector bundles. This yields some new results in $p$-adic non-abelian Hodge Theory.