Reconstructing function fields from Milnor K-theory

Cadoret, Anna; Pirutka, Alena

doi:10.2140/ant.2021.15.2261

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Reconstructing function fields from Milnor K-theory

Anna Cadoret and Alena Pirutka

Vol. 15 (2021), No. 9, 2261–2288

DOI: 10.2140/ant.2021.15.2261

Abstract

Let $F$ be a finitely generated regular field extension of transcendence degree $\geq 2$ over a perfect field $k$ . We show that the multiplicative group $F^{\times} ∕ k^{\times}$ endowed with the equivalence relation induced by algebraic dependence on $F$ over $k$ determines the isomorphism class of $F$ in a functorial way. As a special case of this result, we obtain that the isomorphism class of the graded Milnor K-ring $K_{*}^{M} (F)$ determines the isomorphism class of $F$ , when $k$ is algebraically closed or finite.

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Keywords

Milnor K-theory, function fields, reconstruction

Mathematical Subject Classification

Primary: 11R58, 14C35, 19D45

Milestones

Received: 11 April 2020

Revised: 11 January 2021

Accepted: 17 February 2021

Published: 23 December 2021

Authors

Anna Cadoret
Institut de Mathématiques de Jussieu–Paris Rive Gauche Sorbonne Université Paris France

Alena Pirutka
Courant Institute of Mathematical Sciences New York University New York, NY United States	National Research University Higher School of Economics Moscow, Russia

Vol. 15, No. 9, 2021