On the Witt vector Frobenius

TL;DR

This paper investigates the properties of the Frobenius map on Witt vectors over arbitrary rings, providing conditions for its surjectivity and exploring stability under extensions, with implications for a generalized almost purity theorem.

Contribution

It offers new criteria for Frobenius surjectivity on Witt vectors and links these to integral extensions and a generalized almost purity theorem.

Findings

Conditions equivalent to Frobenius surjectivity established

Surjectivity stability under certain integral extensions demonstrated

Connections to a generalized Faltings's almost purity theorem made

Abstract

We study the kernel and cokernel of the Frobenius map on the $p$-typical Witt vectors of a commutative ring, not necessarily of characteristic $p$. We give some equivalent conditions to surjectivity of the Frobenus map on both finite and infinite length Witt vectors; the former condition turns out to be stable under certain integral extensions, a fact which relates closely to a generalization of Faltings's almost purity theorem.