# Homogeneous spaces, algebraic $K$-theory and cohomological dimension of fields

## TL;DR

This paper characterizes the cohomological dimension of perfect fields using properties of homogeneous spaces and Milnor K-theory, providing a new criterion linking algebraic geometry and field theory.

## Contribution

It establishes a new equivalence between cohomological dimension bounds and norm images in Milnor K-theory for homogeneous spaces over fields.

## Key findings

- Cohomological dimension at most q+1 characterized by norm images in Milnor K-theory.
- Extension of results to imperfect fields.
- Provides a criterion connecting algebraic groups, K-theory, and field cohomology.

## Abstract

Let $q$ be a non-negative integer. We prove that a perfect field $K$ has cohomological dimension at most $q+1$ if, and only if, for any finite extension $L$ of $K$ and for any homogeneous space $Z$ under a smooth linear connected algebraic group over $L$, the $q$-th Milnor $K$-theory group of $L$ is spanned by the images of the norms coming from finite extensions of $L$ over which $Z$ has a rational point. We also prove a variant of this result for imperfect fields.