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

Consider a bounded prism (A, I) and a bounded quasi-l. c. i algebra R over A. In this paper, for any prism S/A with a surjection S R such that Lₒ/₀ is a p-completely flat module over S, we establish an equivalence of categories between rational Hodge-Tate crystals on (R/A) _Δ and topologically nilpotent integrable connections on the Hodge--Tate cohomology ring Δₑ/ₒ. As an application, for a non-zero divisor a A, we introduce the concept of a-smallness for a rational Hodge-Tate prismatic crystal on (R/A) _Δ. Finally, we focus on some special algebras R over O ₂䂹 (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.