What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

Question

On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis holds. He doesn't give any clue as to the proof of this fact or why the continuum hypothesis got involved. On page 92 he mentions some examples of Osofsky and says the continuum hypothesis gets involved there because of non-constructible ideals over uncountable rings. I think this explains at least the "why" of the appearance of the continuum hypothesis (though I would welcome more details on this!), but it leaves me with some other questions:

How is the continuum hypothesis used in this proof?

Why wouldn't the proof work without the continuum hypothesis?

I will understand if the above have to do with some work of Osofsky that is not widely known. If I can't get answers for those questions, perhaps I can still get help on the below. I got involved with this because I wanted to understand an example of a ring that is von Neumann regular but not semisimple (and an infinite product of fields is such an example). I had hoped all such examples would have weak dimension zero (to be VNR) and right global dimension 1. In particular, I wanted to know that the global dimension of $A = \prod_{i=1}^\infty \mathbb{F}_2$ was $1$. According to this MO answer and its comments, $Spec A$ is the Stone-Cech compactification of $\mathbb{N}$. Now I'm concerned that things from set theory which I try to avoid thinking about will come into play in this example as well as in the above ring $R$.

What is the global dimension of $\prod_{i=1}^\infty \mathbb{F}_2$? Do we need to assume the continuum hypothesis at any point? What about an uncountable product of $\mathbb{F}_2$?

Answer 1

In [Osofsky, B. L. Homological dimension and cardinality. Trans. Amer. Math. Soc. 151 1970 641--649. MR0265411 (42 #321)] she proved that the global dimension of a countable product of fields is $k+1$ iff $2^{\aleph_0}=\aleph_{k}$. In particular, if the continuum hypothesis holds, so that $2^{\aleph_0}=\aleph_1$, the global dimension of such a product is exactly $2$.

Because the AMS is nice, you can see the paper here.

Answer 2

In answer to Tilemachos Vassias, it is not at all unnatural to have the Continuum Hypothesis related to questions on dimension. For example, Sierpinski showed that the Continuum Hypothesis is equivalent to the statement that the plane can be partitioned into two pieces, one of which is countable on every vertical section and the other countable on every horizontal section --- this establishes a connection with dimension 2. A striking result that continues in this direction is due to Jacek Cichoń and Michał Morayne, "On differentiability of Peano type functions. III." Proc. Amer. Math. Soc. 92 (1984), no. 3, 432–438. There they show that the inequality $2^{\aleph_0}\leq \aleph_n$ is equivalent to the assertion that there exists an onto function $f:{\bf R}^{n}\to{\bf R}^{n+m}$ such that at each point of ${\bf R}^n$ at least $n$ coordinates of $f$ are differentiable. However, I believe that Barbara Osofsky was the first to realize that this phenomenon occurs outside of pure set theory.

Answer 3

Since Tilemachos Vassias asked (in a comment to Mariano Suárez-Alvarez's answer) about "the appearance of set theory in seemingly random and completely unexpected places in mathematics," I'd like to give a more general answer than the one given by Juris Steprans, who concentrated on dimension. I've come to expect set theory to appear in any area of mathematics that gets beyond the consideration of countable or "essentially countable" (e.g. separable, in topology) structures. This expectation is not so much based on the foundational role of set theory, mentioned in Fernando Muro's comment, but on its role as the study of combinatorial structures on infinite sets. It is not surprising (to me) that when one analyzes problems or structures in depth, combinatorial issues arise. Indeed, it happens surprisingly often that analysis of a problem reduces it entirely to a combinatorial question, and then one expects to use set-theoretic tools. In some cases, that leads to independence results; in other cases, the tools produce solutions in ZFC. One difference between these two sorts of outcomes is that we know how to prove independence results only when uncountability is involved in an essential way (unless you count applications of Gödel's incompleteness theorems, but they're not really set theory), whereas the use of infinite combinatorics to prove results in ZFC can also arise in essentially countable situations.