[Paper Review] On commutative nonarchimedean Banach fields

Kiran S. Kedlaya|arXiv (Cornell University)|Feb 29, 2016

Advanced Topics in Algebra15 references3 citations

TL;DR

This paper establishes that a commutative nonarchimedean Banach ring which is algebraically a field (a Banach field) can be topologized by a multiplicative norm—making it a nonarchimedean field—under certain conditions, particularly for uniform Banach rings and perfectoid rings. The key result resolves a foundational question in nonarchimedean geometry by showing that any perfectoid ring which is a field is necessarily a perfectoid field.

ABSTRACT

We study the problem of whether a commutative nonarchimedean Banach ring which is algebraically a field can be topologized by a multiplicative norm. This can fail in general, but it holds for uniform Banach rings under some mild extra conditions. Notably, any perfectoid ring whose underlying ring is a field is a perfectoid field.

Motivation & Objective

• To determine whether a commutative nonarchimedean Banach ring that is algebraically a field (a Banach field) can be equipped with a multiplicative norm, making it a nonarchimedean field.
• To investigate the conditions under which such a Banach field must be a nonarchimedean field, especially in cases where the initial topology is not induced by a multiplicative norm.
• To resolve a foundational issue in perfectoid theory: whether a perfectoid ring whose underlying ring is a field must be a perfectoid field.
• To clarify the relationship between the algebraic structure of a ring and its topological properties in nonarchimedean functional analysis.
• To extend the theory of uniform Banach rings and perfectoid rings by showing that certain classes of Banach fields are necessarily nonarchimedean fields.

Proposed method

• Uses the theory of uniform Banach rings and the existence of a bounded multiplicative seminorm to show that under mild conditions (e.g., nondiscrete valuation), a Banach field is a nonarchimedean field.
• Applies the perfectoid tilting correspondence to reduce the problem in characteristic 0 to the case of characteristic p, where perfectness and uniformity imply the existence of a multiplicative norm.
• Employs the operator norm construction to normalize the norm so that |1| = 1, ensuring topological equivalence and simplifying analysis.
• Uses the inverse limit structure of perfectoid rings via the map x ↦ x^p to analyze the behavior of elements and their inverses in the tilting process.
• Applies the Ax–Sen–Tate theorem to constrain possible subfields in perfectoid rings, showing that only Q_p or finite extensions can serve as base fields.
• Constructs explicit examples of perfectoid rings with quotients isomorphic to completions of Q_p(μ_{p^∞}) and Q_p(p^{1/p^∞}) to demonstrate that such rings are not Banach algebras over perfectoid fields.

Experimental results

Research questions

• RQ1 Can every commutative nonarchimedean Banach field be topologized by a multiplicative norm, i.e., is it necessarily a nonarchimedean field?
• RQ2 Under what conditions does a uniform Banach ring with a field underlying ring admit a multiplicative norm?
• RQ3 Is a perfectoid ring whose underlying ring is a field necessarily a perfectoid field?
• RQ4 Can a perfectoid ring be a Banach algebra over a perfectoid field if its quotients are completions of different p-adic extensions?
• RQ5 What is the role of the tilting equivalence in transferring properties from characteristic p to characteristic 0 in perfectoid rings?

Key findings

• A uniform Banach ring whose underlying ring is a field and whose valuation is nondiscrete is necessarily a nonarchimedean field, as shown in Theorem 3.7.
• Any perfectoid ring which is a field is a perfectoid field, resolving a foundational question from Scholze’s original definition (Theorem 4.2).
• The completion of a field with respect to a submultiplicative norm need not be a field, showing that the multiplicative norm condition is essential (Example 2.15).
• The tilting correspondence allows reduction from characteristic 0 to characteristic p, where perfectness and uniformity imply the existence of a multiplicative norm.
• Explicit constructions show that some perfectoid rings are not Banach algebras over any perfectoid field, even though they admit quotients isomorphic to completions of Q_p(μ_{p^∞}) and Q_p(p^{1/p^∞}).
• The maximal ideal quotient of a perfectoid ring is a Banach field, and if the quotient is of characteristic p, it is a perfectoid field; however, this does not hold in characteristic 0 without additional assumptions.