Points in the fppf topology

TL;DR

This paper constructs topos-theoretical points for the fppf topology on schemes, linking algebraic and topos-theoretic concepts, and provides criteria for comparing point spaces across different topologies.

Contribution

It introduces a new method to construct points in the fppf topology using topos-theoretic and algebraic techniques, and relates these points to known algebraic structures.

Findings

Fppf-local rings have henselian localizations with algebraically closed residue fields.

Constructed points are indexed by geometric points and limit ordinals.

Criteria are provided for when different sites have equivalent point spaces.

Abstract

Using methods from commutative algebra and topos-theory, we construct topos-theoretical points for the fppf topology of a scheme. These points are indexed by both a geometric point and a limit ordinal. The resulting stalks of the structure sheaf are what we call fppf-local rings. We show that for such rings all localizations at primes are henselian with algebraically closed residue field, and relate them to AIC and TIC rings. Furthermore, we give an abstract criterion ensuring that two sites have point spaces with identical sobrification. This applies in particular to some standard Grothendieck topologies considered in algebraic geometry: Zariski, etale, syntomic, and fppf.