Prime types and geometric completeness

TL;DR

This paper reinterprets Hilbert's Nullstellensatz and related algebraic concepts within a logical framework, connecting algebraic geometry, model theory, and universal algebra to provide new insights into prime ideals and geometric saturation.

Contribution

It introduces a logical perspective on prime and radical ideals, linking geometric saturation with model-theoretic properties like positive model-completeness, and applies these ideas to group-based algebras.

Findings

Hilbert's Nullstellensatz understood as geometric saturation in logic

Model-completeness related to geometric properties in algebraic theories

Application to group-based algebras for functional field expansions

Abstract

The geometric form of Hilbert's Nullstellensatz may be understood as a property of "geometric saturation" in algebraically closed fields. We conceptualise this property in the language of first order logic, following previous approaches and borrowing ideas from classical model theory, universal algebra and positive logic. This framework contains a logical equivalent of the algebraic theory of prime and radical ideals, as well as the basics of an "affine algebraic geometry" in quasivarieties. Hilbert's theorem may then be construed as a model-theoretical property, weaker than and equivalent in certain cases to positive model-completeness, and this enables us to geometrically reinterpret model-completeness itself. The three notions coincide in the theories of (pure) fields and we apply our results to group-based algebras, which supply a way of dealing with certain functional field expansions.