Some Foundational Results In Adic Geometry

TL;DR

This paper develops foundational aspects of adic geometry, including Proj construction, lci closed immersions, and an etale six functor formalism, filling gaps in the existing literature.

Contribution

It introduces new foundational results in adic geometry, particularly in the context of locally noetherian analytic adic spaces, with categorical and formalism developments.

Findings

Developed the Proj construction for adic spaces

Established a theory of lci closed immersions in adic geometry

Formulated an etale six functor formalism in the analytic setting

Abstract

In this paper, we record some foundational results on adic geometry that seem to be missing in the existing literature. Namely, we develop the Proj construction and a theory of lci closed immersions in the context of locally noetherian analytic adic spaces. In the context of rigid-analytic spaces, these topics have previously been considered in [GL21] and [Con07]. We also develop an etale six functor formalism in the analytic geometry and give a categorical description of lisse and constructible sheaves. All results of this paper are probably well-known to the experts.