Hodge--Tate crystals on the logarithmic prismatic sites of semi-stable formal schemes

Abstract

Let $\calO_K$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with a perfect residue field. In this paper, for a semi-stable $p$-adic formal scheme $\frakX$ over $\calO_K$ with rigid generic fibre $X$ and canonical log structure $\calM_{\frakX} = \calO_{\frakX}\cap\calO_X^{\times}$, we study Hodge--Tate crystals over the absolute logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$. As an application, we give an equivalence between the category of rational Hodge--Tate crystals on the absolute logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$ and the category of enhanced log Higgs bundles over $\frakX$, which leads to an inverse Simpson functor from the latter to the category of generalised representations on $X_{\proet}$.

We show that for semi-stable small R, its associated prismatic site of perfect prisms is equal to that of perfect log prisms. Comments welcome!