#### Vol. 16, No. 2, 2022

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Local constancy of intersection numbers

### Andreas Mihatsch

Vol. 16 (2022), No. 2, 505–519
##### 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 $S×M$ 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.

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##### Keywords
formal scheme, intersection theory, profinite set, arithmetic fundamental lemma
##### Mathematical Subject Classification
Primary: 11G18, 14C17
##### Milestones
Accepted: 13 June 2021
Published: 27 April 2022
##### Authors
 Andreas Mihatsch Mathematical Institute Bonn University Endenicher Allee 60 53115 Bonn Germany