Rational Hodge--Tate prismatic crystals of quasi-l.c.i algebras and non-abelian $p$-adic Hodge theory
Xiaoyu Qu, Jiahong Yu
TL;DR
The paper extends p-adic non-abelian Hodge theory to quasi-l.c.i. algebras by establishing a de Rham realization that classifies rational Hodge-Tate prismatic crystals as topologically nilpotent integrable connections on Hodge–Tate cohomology. It introduces a-smallness and a robust cosimplicial-stratification framework, enabling the translation between prismatic crystals and Higgs/MIC data in a broad non-noetherian setting, including pre-smooth and geometric valuation rings. The results generalize prior classifications (e.g. Tian, Ogus, Min–Wang) to non-topologically finitely generated algebras and connect with Sen theory via geometric Sen operators on valuation rings. Collectively, the work yields new insights into p-adic non-abelian Hodge theory and expands the applicability of prismatic methods to a wide class of algebras.
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 $(R/A)_Δ$ and topologically nilpotent integrable connections on the Hodge--Tate cohomology ring $\overlineΔ_{R/S}$. 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 $(R/A)_Δ$. 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.