nLab
schemes are sober

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

h

Algebra

Contents

Statement

Proposition

The topological space underlying a scheme is a sober topological space. In particular for affine schemes: The Zariski topology on the prime spectrum of a commutative ring is sober.

For proof see this prop at Zariski topology.

Remark

If instead of the prime spectrum of a commutative ring one considers only the topological subspace of maximal ideals in prop. 1, as in algebraic geometry before the introduction of schemes by Alexander Grothendieck, then one does not get a sober space. But if the ring is a Jacobson ring (in that every prime ideal in the ring is an intersection of maximal ideals), then the soberification of the topological subspace of maximal ideals inside the prime ideals is the full prime spectrum.

References

Revised on April 25, 2017 17:16:58 by Urs Schreiber (92.218.150.85)