Local constancy of intersection numbers

Mihatsch, Andreas

doi:10.2140/ant.2022.16.505

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Local constancy of intersection numbers

Andreas Mihatsch

Vol. 16 (2022), No. 2, 505–519

DOI: 10.2140/ant.2022.16.505

Abstract

We prove that, in certain situations, intersection numbers on formal schemes that come in profinite families vary locally constantly in the parameter. To this end, we define the product $S \times M$ of a profinite set $S$ with a locally noetherian formal scheme $M$ and study intersections thereon. Our application is to the arithmetic fundamental lemma of W. Zhang where the result helps to overcome a restriction in its recent proof. Namely, it allows to spread out the validity of the AFL identity from an open to the whole set of regular semisimple elements.

Keywords

formal scheme, intersection theory, profinite set, arithmetic fundamental lemma

Mathematical Subject Classification

Primary: 11G18, 14C17

Milestones

Received: 30 March 2021

Accepted: 13 June 2021

Published: 27 April 2022

Authors

Andreas Mihatsch
Mathematical Institute Bonn University Endenicher Allee 60 53115 Bonn Germany

Vol. 16, No. 2, 2022