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Ultraproducts of fields, II June 28, 2008

Posted by Alexandre Borovik in Uncategorized.
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I continue my post on ultraproducts. So, we want to understand in what sense an ultraproduct of finite fields F_i of unbounded order is a limit at infinity of finite fields. The answer now should be obvious: since ultraproducts are residue fields for maximal ideals in the cartesian product

R = \prod F_i,

the topology in question should be the canonical topology (Zariski topology) of the spectrum of the ring \mbox{Spec}(R). It instantly follows from the description of ideals and maximal ideals in R that this is the Stone topology on the set of ultrafilters on \mathbb{N}, or, what is th same, the Cech-Stone compactification \beta\mathbb{N} of the set \mathbb{N} with descrete topology. Therefore the answer is: an ultraproduct is the limit in the Cech-Stone compactification of a discrete countable set.

I have to admit that at this point I reached limits of my knowledge of set-theoretic topology and had to dip into Wikipedia. It happened that \beta\mathbb{N}, and even more so its non-principal part {\mathbb{N}}^* = \beta{\mathbb{N}} \setminus {\mathbb{N}} is characterised by some unique properties: If the continuum hypothesis holds then {\mathbb{N}}^* is the unique Parovicenko space, up to isomorphism.

According to Wikipedia, a Parovicenko space is a topological space X satisfying the following conditions:

  • X is compact Hausdorff
  • X has no isolated points
  • X has weight c, the cardinality of the continuum (this is the smallest cardinality of a base for the topology).
  • Every two disjoint open Fσ subsets of X have disjoint closures
  • Every nonempty Gδ of X has non-empty interior.

As you can see, {\mathbb{N}}^* is uniquely characterised by very natural properties.

It is yet another manifestation of of one of the most pardoxical properties of mathematical infinity: canonicity of workable constructions in the infinite domain.

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Comments»

1. dcorfield - July 4, 2008

I wonder how much of what is natural about \mathbb{N}^* can be put in category theoretic terms. The \beta you mention is a functor, Stone-Cech compactification, from Top (the category of topological spaces) to CHaus (the category of compact Hausdorff spaces). It is left adjoint to the underlying functor going the other way.

So, Hom(\beta X, K) \cong Hom(X, U K). In the case of \beta \mathbb{N} a map to a compact Hausdorff space is determined by a map from \mathbb{N} to that space.

But does the formation of \mathbb{N}^* by removing the principal part, correspond to a category theoretic construction?


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