Perfectoid Nullstellensatz: Results and counterexamples

TL;DR

This paper investigates the perfectoid Nullstellensatz, providing conditions for its validity and demonstrating its failure over certain p-adic fields like _p.

Contribution

It establishes necessary and sufficient conditions for the perfectoid Nullstellensatz and shows counterexamples over specific p-adic fields.

Findings

Nullstellensatz does not hold for _p and similar fields

Provides criteria for when the Nullstellensatz holds in perfectoid settings

Identifies limitations of algebraic geometry principles in p-adic perfectoid contexts

Abstract

We give necessary conditions and we give sufficient conditions for perfectoid Nullstellensatz to hold. As a consequence, we prove that perfectoid Nullstellensatz does not hold for $\mathbb{C}_p$ and other natural p-adic fields.