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

### Diego Izquierdo

Max-Planck-Institut für Mathematik, Bonn, Germany; Institut Polytechnique de Paris, Palaiseau, France### Giancarlo Lucchini Arteche

Universidad de Chile, Santiago, Chile

## 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.

## Cite this article

Diego Izquierdo, Giancarlo Lucchini Arteche, Homogeneous spaces, algebraic $K$-theory and cohomological dimension of fields. J. Eur. Math. Soc. 24 (2022), no. 6, pp. 2169–2189

DOI 10.4171/JEMS/1129