Sorry that probably seems like a completely random question. I'm trying to look at either BA or DL from the point of view of order theory and that kind of stuff and figure out things like what are the prime ideals and so forth. In a Boolean algebra an element generates a prime ideal if and only if its complement is an atom. So, it appears that there are prime ideals of the form ↓a⟨K(n)⟩.

Thanks John. So we might say something like $\langle K(n)\rangle$ are the atoms of the Bousfield lattice?

Yau's "The Shape of Inner Space" is really fascinating, it's a little tricky at times with some advanced geometry, but I think they do a pretty good job making it digestible, and it basically explains the mathematics behind string theory. I've been reading it recently. Most people are interested in knowing about what is going on in string theory.

One question. Hochster's construction is believable, but rather unpleasantly intricate. Is there any simpler way to do this? Is there a nice way of describing the necessary ring? I don't see any in Hochster's paper..

Yeah absolutely. I'm working a little minor project with them right now actually. Hopefully I can actually figure something out!

Good point. This gives some intuitive evidence for the fact that Hochester's construction can give a ring that is an algebra over any field (since every space is in some sense a module over the one-point space).

Hi Cory. So you posted this in 2009 and I don't know if you're still around. However, I wanted to ask if you know what the connection is between so-called coherence spaces and the coherent spaces of topology, which are precisely those spaces which can be realized as the spectrum of a ring and so are very interesting indeed. Also, having closed sets be closed under arbitrary intersection is not a problem. In fact, it is REQUIRED. It might be too strong if open sets were closed under arbitrary intersection.

Wow thanks very much Greg. Do you recall meeting in Barcelona? Your answer clears up a lot of my confusion.

Man that's great! Thanks to both of you. I have requested Johnstone's book from the library (it appears that someone has already checked it out!). And wow, Pete that was incredibly quick and concise! Many thanks.