Filter/Ultrafilter Methods in Commutative Algebra/ ring theory

commutative-algebrafiltersmodel-theoryring-theory

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.

Best Answer

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

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