Un th\'eor\`eme de l'application ouverte sur les corps valu\'es alg\'ebriquement clos

TL;DR

This paper proves that for algebraically closed valued fields, universally open morphisms induce open maps on rational points when considering the topology from the absolute value, extending understanding of topological properties in algebraic geometry.

Contribution

It establishes that universally open morphisms between finite type schemes over algebraically closed valued fields induce open maps on K-rational points with respect to the absolute value topology.

Findings

Induced maps are open for algebraically closed valued fields.

Extends topological understanding of morphisms in algebraic geometry.

Bridges algebraic properties with topological openness.

Abstract

Let K be an algebraically closed valued field, and let f:X--->Y be a universally open morphism of K-schemes of finite type. We show that the induced map on K-rational points is open for the topologies deduced from the absolute value of K. ------ Soit K un corps valu\'e alg\'ebriquement clos, et soit f:X--->Y un morphisme universellement ouvert de K-sch\'emas de type fini. On montre que l'application de X(K) dans Y(K) induite par f est ouverte pour les topologies d\'eduites de la valeur absolue sur K.