Lefschetz trace formula for open adic spaces

TL;DR

This paper extends the Lefschetz trace formula to certain non-proper adic spaces over non-archimedean fields, providing new fixed point and trace formulas applicable to formal schemes and Rapoport-Zink towers.

Contribution

It introduces a Lefschetz trace formula for separated smooth adic spaces not necessarily proper, under specific boundary conditions, and applies it to formal schemes and Rapoport-Zink towers.

Findings

Established a fixed point formula for non-proper adic spaces.

Derived a trace formula applicable to Rapoport-Zink towers.

Generalized Fujiwara's trace formula for contracting morphisms.

Abstract

In this article, we discuss the Lefschetz trace formula for an adic space which is separated smooth of finite type but not necessarily proper over an algebraically closed non-archimedean field. Under a certain condition on the absence of set-theoretical fixed points on the boundary, we obtain a fixed point formula. As an application, we can establish a trace formula for some formal schemes, which is applicable to the Rapoport-Zink tower for GSp(4). A partial generalization of Fujiwara's trace formula for contracting morphisms is also given.