$p$-adic Simpson correspondence via prismatic crystals

Abstract

Let $\mathfrak{X}$ be a smooth $p$-adic formal scheme over ${\mathcal{O}}_K$ with adic generic fibre $X$. We obtain a global equivalence between the category $\mathrm{Vect}((\mathfrak{X}{)}_{△△},{\overline{\mathcal{O}}}_{△△}[\frac{1}{p}])$ of rational Hodge–Tate crystals on the absolute prismatic site $(\mathfrak{X}{)}_{△△}$ and the category ${\mathrm{HIG}}_{\ast }^{\mathrm{nil}}(X)$ of enhanced Higgs bundles on $X$. Along the way, we construct an inverse Simpson functor from ${\mathrm{HIG}}_{\ast }^{\mathrm{nil}}(X)$ to the category $\mathrm{Vect}(X_{\text{proeˊt}},{\hat{\mathcal{O}}}_X)$ of generalised representations on $X$, which turns out to be fully faithful.