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On the relative Nullstellensatz in nonarchimedean geometry

Kiran S. Kedlaya, Yutaro Mikami

TL;DR

This work develops a relative Nullstellensatz for algebras topologically of finite type over a Banach Tate base ring, reducing to the hypothesis that the Nullstellensatz holds for rational localizations. It proves a relative Weierstrass preparation and Noether normalization, enabling a finite presentation after suitable localization. The results yield strong Nullstellensatz properties for key FF-geometry base rings, including $A^r_{L,E}$ and $B^{[s,r]}_{L,E}$, and connect to Jacobson--Tate theory, expanding the toolkit for nonarchimedean analytic geometry. The paper also discusses limitations, potential generalizations, and future directions, notably in perfectoid settings and in addressing remaining open questions about the range of rings for which the weak/strong Nullstellensatz hold.

Abstract

We establish a relative version of the Nullstellensatz for algebras topologically of finite type over a given Banach Tate ring $A$, under the assumption that the corresponding statement holds for rational localizations of $A$. This applies in particular to pseudoaffinoid algebras and to the coordinate rings of affinoid subspaces of a Fargues--Fontaine curve.

On the relative Nullstellensatz in nonarchimedean geometry

TL;DR

This work develops a relative Nullstellensatz for algebras topologically of finite type over a Banach Tate base ring, reducing to the hypothesis that the Nullstellensatz holds for rational localizations. It proves a relative Weierstrass preparation and Noether normalization, enabling a finite presentation after suitable localization. The results yield strong Nullstellensatz properties for key FF-geometry base rings, including and , and connect to Jacobson--Tate theory, expanding the toolkit for nonarchimedean analytic geometry. The paper also discusses limitations, potential generalizations, and future directions, notably in perfectoid settings and in addressing remaining open questions about the range of rings for which the weak/strong Nullstellensatz hold.

Abstract

We establish a relative version of the Nullstellensatz for algebras topologically of finite type over a given Banach Tate ring , under the assumption that the corresponding statement holds for rational localizations of . This applies in particular to pseudoaffinoid algebras and to the coordinate rings of affinoid subspaces of a Fargues--Fontaine curve.

Paper Structure

This paper contains 4 sections, 14 theorems, 4 equations.

Table of Contents

  1. Relative Weierstrass preparation and Noether normalization
  2. Relative Nullstellensatz
  3. An application to Fargues--Fontaine curves
  4. A final remark

Key Result

Lemma 1.1

Let $A$ be a commutative Banach Tate ring. Let $n_0\geq 0$ be an integer. Let $f(T)=\sum_{n=0}^{\infty}f_nT^n \in A\langle T \rangle$ be a series such that: Then for any $g\in A\langle T \rangle$, there exists a unique pair $(q,r)$ with $q\in A\langle T \rangle$, $r\in A[T]$, and $\deg r <n_0$ such that $g=fq+r$. Moreover, in this case we have $|g|=\max \{|q|,|r|\}$, where we endow $A\langle T \r

Theorems & Definitions (39)

  • Lemma 1.1: Relative Weierstrass division
  • proof
  • Lemma 1.2: Relative Weierstrass preparation
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Theorem 1.5: Relative Noether normalization
  • proof
  • ...and 29 more