Full faithfulness without Frobenius structure and partially overcoherent isocrystals

TL;DR

This paper extends Kedlaya's full faithfulness results for overconvergent isocrystals to cases without smoothness assumptions and introduces partially overcoherent isocrystals, establishing an equivalence with existing categories.

Contribution

It generalizes full faithfulness of isocrystal functors beyond smooth cases and constructs a new category of partially overcoherent isocrystals with an established equivalence.

Findings

Extended Kedlaya's full faithfulness to non-smooth cases.

Constructed the category of partially overcoherent isocrystals.

Proved the equivalence of categories with existing overconvergent isocrystals.

Abstract

Let $K$ be a mixed characteristic complete discrete valuation field with perfect residue field $k$. Let $X$ be a variety over $k$, $Y$ be an open of $X$, $Y'$ be an open of $Y$ dense in $X$. We extend Kedlaya's full faithfulness as follows (we do not suppose $Y$ to be smooth): the canonical functor $F\text{-}\mathrm{Isoc} ^{\dag} (Y,X/K) \to F\text{-}\mathrm{Isoc} ^{\dag} (Y,Y/K) $ is fully faithfull. Suppose now $Y$ smooth. We construct the category of partially overcoherent isocrystals over $(Y,X)$ denoted by $\mathrm{Isoc} ^{\dag\dag} (Y,X/K) $ whose objects are some particular arithmetic $\D$-modules. Furthermore, we check the equivalence of categories $\sp_{(Y,X),+} : \mathrm{Isoc} ^{\dag} (Y,X/K) \cong \mathrm{Isoc} ^{\dag\dag} (Y,X/K)$.