Points in algebraic geometry

TL;DR

This paper provides algebraic geometric descriptions of fibre functors on sheaf categories across various topologies in algebraic geometry, with applications to exactness of direct images.

Contribution

It offers scheme-theoretic characterizations of fibre functors for multiple topologies, extending the understanding of their structure and applications in algebraic geometry.

Findings

Describes fibre functors for Zariski, Nisnevich, étale, and other topologies.

Shows direct image along closed immersion is exact for all topologies except qfh.

Provides methods applicable to other sheaf categories.

Abstract

We give scheme-theoretic descriptions of the category of fibre functors on the categories of sheaves associated to the Zariski, Nisnevich, \'etale, rh, cdh, ldh, eh, qfh, and h topologies on the category of separated schemes of finite type over a separated noetherian base. Combined with a theorem of Deligne on the existence of enough points, this provides an algebro-geometric description of a conservative family of fibre functors on these categories of sheaves. As an example of an application we show direct image along a closed immersion is exact for all these topologies except qfh. The methods are transportable to other categories of sheaves as well.