Overconvergent Witt Vectors

TL;DR

This paper introduces overconvergent Witt vectors as a subring of Witt vectors for finitely generated algebras over a field of characteristic p, establishing their sheaf property and laying groundwork for a new cohomology theory.

Contribution

It defines overconvergent Witt vectors and proves they form an étale sheaf on schemes, enabling new approaches in p-adic cohomology theories.

Findings

Overconvergent Witt vectors form an étale sheaf.

The subring is well-behaved on schemes of finite type.

Foundation laid for overconvergent de Rham-Witt complex.

Abstract

Let A be a finitely generated algebra over a field K of characteristic p >0. We introduce a subring of the ring of Witt vectors W(A). We call it the ring of overconvergent Witt vectors. We prove that on a scheme X of finite type over K the overconvergent Witt vectors are an \'etale sheaf. In a forthcoming paper (Annales ENS) we define an overconvergent de Rham-Witt complex on a smooth scheme X over a perfect field K whose hypercohomology is the rigid cohomology of X in the sense of Berthelot.