Vol. 16, No. 2, 2022

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Local constancy of intersection numbers

Andreas Mihatsch

Vol. 16 (2022), No. 2, 505–519

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.

formal scheme, intersection theory, profinite set, arithmetic fundamental lemma
Mathematical Subject Classification
Primary: 11G18, 14C17
Received: 30 March 2021
Accepted: 13 June 2021
Published: 27 April 2022
Andreas Mihatsch
Mathematical Institute
Bonn University
Endenicher Allee 60
53115 Bonn