Isocrystals and de Rham-Witt connections
Rubén Muñoz--Bertrand
TL;DR
This work establishes a bridge between (overconvergent) $F$-isocrystals and Frobenius structures on integrable de Rham–Witt connections. By formulating integrable $\mathcal{F}$-connections on sheaves of $d$gas and translating them into right differential graded modules, it provides a unified framework that works over weak formal schemes and relies on global Frobenius lifts without invoking Grothendieck topologies. Central results include explicit functors $\acute E$ and $\acute E^{\dagger}$ that yield equivalences between convergent/overconvergent $F$-isocrystals and Frobenius-equipped $W\Omega$-connections on locally finite projective modules, supported by de Rham–Witt matrices and pseudovaluation techniques to control overconvergence. The paper thereby clarifies coefficients for rigid and crystalline cohomology in a global, Zariski-topology setting and shows that isocrystal categories can be studied via de Rham–Witt connections with Frobenius structures. These insights enable a Frobenius-compatible, topology-friendly description of $p$-adic cohomology theories tied to rigid and crystalline frameworks.
Abstract
We introduce the notion of integrable connections for a sheaf of differential graded algebras on a topological space. We then describe them in the finite locally projective setting, when the sheaf is either the de Rham complex of a formal or a weakly formal scheme, or for the convergent or the overconvergent de Rham-Witt complex on a smooth scheme over a perfect field of positive characteristic. This enables us to give a new description of convergent and overconvergent isocrystals with a Frobenius structure.