Sur la p-dimension des corps

TL;DR

This paper establishes a formula relating the p-cohomological dimension of a field to the dimension of a local ring and the p-rank of its residue field, extending previous results to more general cases.

Contribution

It generalizes the p-dimension formula for fields associated with excellent henselian local rings, using algebraization techniques and building on Kato's one-dimensional case.

Findings

Proves the formula cd_p(K) = dim(A) + p-rank(k) under specific conditions.

Extends the formula to cases without restrictions on the residue field and the field K.

Uses algebraization techniques to relate local ring properties to field cohomology.

Abstract

Let A be an excellent integral henselian local noetherian ring, k its residue field of characteristic p>0 and K its fraction field. Using an algebraization technique introduced by the first named author, and the one-dimension case already proved by Kazuya KATO, we prove the following formula: cd_p(K) = dim(A) + p-rank(k), if k is separably closed and K of characteristic zero. A similar statement is valid without those assumptions on k and K.