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 $\mathbb{F}_2$?
Best Answer
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.