The formalism of Grothendieck's six operations in p-adic cohomologies

TL;DR

This paper develops a formal framework for Grothendieck's six operations within p-adic cohomology, specifically constructing and verifying the properties of overholonomy categories over schemes in mixed characteristic.

Contribution

It introduces the category of overholonomy type in p-adic cohomology and proves it satisfies the formalism of Grothendieck's six operations, advancing the theoretical foundation.

Findings

Constructed the overholonomy category over schemes in mixed characteristic.

Verified the six operations formalism holds in this new setting.

Provides a foundation for further p-adic cohomological studies.

Abstract

Let $\mathcal{V}$ be a complete discrete valued ring of mixed characteristic $(0,p)$, $K$ its field of fractions, $k$ its residue field which is supposed to be perfect. Let $X$ be a separated $k$-scheme of finite type and $Y$ be an open subscheme of $X$. We construct the category $F\text{-}D ^\mathrm{b}_\mathrm{ovhol} (\mathcal{D} ^\dag_{(Y,X)/K})$ of overholonomy type over $(Y,X)/K$. We check that these categories satisfy a formalism of Grothendieck's six operations.