Filter/Ultrafilter Methods in Commutative Algebra/ ring theory

Question

I have a rather broad question: Are there any useful results/ approaches known on study commutative unitary rings and their ideals with filter/ ultrafilter methods?

What I know:

1. In Boolean algebra ideals are dual objects to filters (that's of course a very special ring)

2. If $R$ is the direct product of a family of fields $F_{\alpha}$ indexed by a set $X$, ie $R= \prod_{\alpha \in X} F_{\alpha}$, then there exist a very strong relation: the proper ideals in $R$ are in bijective correpondence with the filters $F$ on $X$, see this discussion.

Is this the only interesting connection using filters as abstract toolbox to analyse structure of rings and their ideals or are there more?

More generally are there any general heuristics known when one may expect that the application of filter methods to some mathematical subarea could provide interesting results, and on the other hand when one should expect it's nearly useless (as seemingly here in the case for rings which are not fields). Can this 'dissonance' be explaned in heuristic terms?

My problem is that I have rather bad intuition for filters (...at latest when nonprincipal ultrafilters come into the game, my intuition tends instantly to say goodbye) and I'm wondering if it's possible to develop some kind of intuition when filter methods might provide some new insight at the studied topic, and when tendentilly not. For study of rings it seems that filter methods not provide something new. Is there a heuristic reason why at least in this case for rings & ideals applying filter methods tends (...except for the case of products of fields) not to unravel some new interesting insights about the structure?

Hope, that the question is too vaguely formulated, it's primarily about intuition.

Answer 1

There is a series of papers that tries to generalize the description of prime ideals in a product of fields that you mentioned in your question to products of more general rings. A reference for results of this type and for further literature might be this paper.

A short summary:

Let $\Lambda$ be a set and $(D_\lambda)_{\lambda \in \Lambda}$ a family of commutative rings.

1. The maximal ideals of the product $\prod_{\lambda} D_\lambda$ are all induced by ultrafilters on the disjoint union of the $\text{Max-Spec}(D_\lambda)$, the sets of maximal ideals of the $D_\lambda$.

2. If the $D_\lambda$ are domains then the minimal prime ideals of the product are in bijective correspondence with ultrafilters on the index set, just as for fields.

3. If all $D_\lambda$ are Prüfer domains (see the definition below), one can describe all prime ideals of the product in terms of the ultrafilters of point 1.

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Another phenomenon, where ultrafilters can be used in a very practical way, is the constructible topology. It is the unique compact Hausdorff topology on the spectrum of a commutative ring that refines the Zariski topology. Originally, it had been described by the open compact sets as a basis of clopen sets. However, this is not very practical for everyday use.

But the closure operation can be described very intuitively by ultrafilters (see the paper by Fontana and Loper and, for a very clever and short proof, the one by Finocchiaro) in the following way:

Let $A$ be a commutative ring and $Y$ be a subset of $\text{Spec}(A)$. For an ultrafilter $\mathcal U$ on $Y$, one defines $Y_{\mathcal U} = \{ a \in A \mid V(a) \cap Y \in \mathcal U \}$. This set forms a prime ideal of $A$ and is called the ultrafilter limit point of $Y$ with respect to $A$.

The closure of $Y$ in the constructible topology is the set of all ultrafilter limit points of $Y$.

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An integral domain $D$ is called a Prüfer domain if it satisfies one of the following equivalent conditions (see also the monograph on multiplicative ideal theory by Robert Gilmer):

• Every non-zero finitely generated ideal is invertible.
• The localization $D_M$ is a valuation domain for each maximal ideal $M$ of $D$.
• The set of ideals of $D$ with $+$ and $\cap$ is a distributive lattice.

Indeed, there are many more equivalent conditions. The Noetherian Prüfer domains are exactly the Dedekind domains.

Answer 2

I haven't read the following book, but it might be interesting to you.

Hans Schoutens
The Use of Ultraproducts in Commutative Algebra
Lecture Notes in Mathematics, volume 1999, 2010.

Here is the publisher's description of the book:

In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.

Let me copy the chapter titles here:

Introduction
Ultraproducts and Łoś’ Theorem
Flatness
Uniform Bounds
Tight Closure in Positive Characteristic
Tight Closure in Characteristic Zero. Affine Case
Tight Closure in Characteristic Zero. Local Case
Cataproducts
Protoproducts
Asymptotic Homological Conjectures in Mixed Characteristic