P-adic Simpson correpondence via prismatic crystals (2201.08030v4)

Abstract: Let $\frakX$ be a smooth $p$-adic formal scheme over $\calO_K$ with adic generic fiber $X$. We obtain a global equivalence between the category $\Vect((\frakX){\Prism},\overline\calO{\Prism}[\frac{1}{p}])$ of rational Hodge--Tate crystals on the absolute prismatic site $(\frakX){\Prism}$ and the category $\HIG^{\nil}(X)$ of enhanced Higgs bundles on $X$. Along the way, we construct an inverse Simpson functor from $\HIG^{\nil}_(X)$ to the category $\Vect(X_{\proet},\widehat\calO_X)$ of generalised representations on $X$, which turns out to be fully faithful.