Full faithfulness without Frobenius structure and partially overcoherent 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 $X$, $Y^{\prime }$ 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 $F\text{-}{\mathrm{Isoc}}^{†}(Y,X/K)\to F\text{-}{\mathrm{Isoc}}^{†}(Y,Y/K)$ is fully faithfull. Suppose now $Y$ smooth. We construct the category of partially overcoherent isocrystals over $(Y,X)$ denoted by ${\mathrm{Isoc}}^{††}(Y,X/K)$ whose objects are some particular arithmetic $\D$-modules. Furthermore, we check the equivalence of categories $\sp_{(Y,X),+} : \mathrm{Isoc} ^† (Y,X/K) \cong \mathrm{Isoc} ^{††} (Y,X/K)$.

References in corpus (1)

• Relative log convergent cohomology and relative rigid cohomology I

Cited by in corpus (1)

• Purity for overconvergence