[Paper Review] Fréchet Modules and Descent

Oren Ben-Bassat, Kobi Kremnizer|arXiv (Cornell University)|Feb 26, 2020

Commutative Algebra and Its Applications4 citations

TL;DR

This paper develops a homological framework for Ind-Banach modules over Banach rings, unifying functional analysis and algebraic geometry via nuclearity, metrizability, and derived descent. It establishes that nuclear and metrizable modules commute with countable limits under projective tensor products, enabling a descent theory for quasi-coherent modules in Banach algebraic geometry and proving equivalence of Stein and dagger algebras in archimedean and non-archimedean settings.

ABSTRACT

We study several aspects of the study of Ind-Banach modules over Banach rings thereby synthesizing some aspects of homological algebra and functional analysis. This includes a study of nuclear modules and of modules which are flat with respect to the projective tensor product. We also study metrizable and Fréchet Ind-Banach modules. We give explicit descriptions of projective limits of Banach rings as ind-objects. We study exactness properties of projective tensor product with respect to kernels and countable products. As applications, we describe a theory of quasi-coherent modules in Banach algebraic geometry. We prove descent theorems for quasi-coherent modules in various analytic and arithmetic contexts.

Motivation & Objective

• To extend homological algebra techniques from complex Banach spaces to general Banach rings, enabling a synthesis of functional analysis and algebraic geometry.
• To characterize modules for which the projective tensor product commutes with countable limits, identifying metrizability and nuclearity as key conditions.
• To develop a descent theory for quasi-coherent modules in Banach algebraic geometry, particularly in analytic and arithmetic contexts.
• To establish equivalences between archimedean and non-archimedean formulations of Stein and dagger algebras via limit constructions.
• To generalize results from Meyer and Prosmans-Schneiders to arbitrary Banach rings, including non-archimedean cases.

Proposed method

• Uses the category of Ind-Banach modules over a Banach ring R, generalizing classical Banach space theory to arbitrary base rings.
• Applies quasi-abelian category theory and relative homological algebra to define derived functors and tensor products in this setting.
• Introduces nuclearity via Definition 4.9 and proves permanence properties such as the two-out-of-three rule for strict exact sequences.
• Defines metrizability (Definition 5.5) and proves that metrizable modules satisfy a canonical limit-compatibility with projective tensor products.
• Constructs projective limits of Banach rings as ind-objects and proves that such limits commute with the projective tensor product under nuclearity and metrizability.
• Applies descent theorems (Theorem 7.9) to show equivalence of categories of modules on limits of Stein/dagger algebras in both archimedean and non-archimedean settings.

Experimental results

Research questions

• RQ1 Under what conditions does the projective tensor product with a module commute with countable products or limits?
• RQ2 How can nuclearity and metrizability be used to characterize modules with good homological behavior in the context of Banach rings?
• RQ3 Can descent theorems for quasi-coherent modules be established in analytic and arithmetic settings using this framework?
• RQ4 To what extent do archimedean and non-archimedean formulations of Stein and dagger algebras agree in the derived category?
• RQ5 How do the derived tensor product and flatness interact in the context of Ind-Banach modules over general Banach rings?

Key findings

• Nuclear modules are flat with respect to the projective tensor product, as shown in Lemma 4.19.
• A module commutes with countable products under the projective tensor product if and only if it is metrizable, as proven in Lemmas 5.18 and 5.19.
• The projective limit of a system of Banach rings can be described as an ind-object in the category of Ind-Banach modules.
• The object $\underset{r<1}{\lim}\ \underset{l\in\mathbb{N}}{\operatorname{colim}}^\leq 1\mathbb{Z}\{\left(\frac{x}{r}\right)^{\frac{1}{l}}\}$ is flat over $\mathbb{Z}$, as shown in Lemma 8.5.
• The derived tensor product $\left(\underset{l\in\mathbb{N}}{\operatorname{colim}}^\leq 1\mathbb{Z}_{p}\{\left(\frac{x}{r}\right)^{\frac{1}{l}}\}\right)\widehat{\otimes}^\mathbb{L}_{\mathbb{Z}}R$ is isomorphic to $\underset{r<1}{\lim}\ \underset{l\in\mathbb{N}}{\operatorname{colim}}^\leq 1R\{\left(\frac{x}{r}\right)^{\frac{1}{l}}\}$, as established in Lemma 8.6.
• The categories of modules on the archimedean and non-archimedean limits of Stein/dagger algebras are equivalent, as shown via descent and isomorphism of limits in the appendix.