Today I learned a new basic fact on morphisms of schemes, namely the result mentioned in the title of this post. I started wondering about this question as I was thinking about separation conditions for algebraic stacks. Namely: it appears that the standard definition for a separated algebraic stack is one whose diagonal is proper, and I was wondering if we could get away with just requiring the diagonal to be universally closed and separated. It turns out we can due to the result of the title. After trying to think about it for a bit I decided to look for it on the web, and I quickly found a mathoverflow question asking exactly whether universally closed implies quasi-compact for morphisms of schemes, as well as the proof provided by Bjorn Poonen!
Note that all posts on mathoverflow are under CC-BY-SA, which is (unfortunately) not compatible with GFDL which is the license that the stacks project is under. Moreover, they ask to link back to their site, see here; and actually I think they are really stretching the meaning of the license since I think no linking should be required (IANAL). Anyway, I asked Bjorn if he agreed to relicense his material, and he said “Yes, that’s fine”. This means I need not link to their site if I do not want to (I did anyway).
It seems that CC-BY-SA is winning over GFDL in some respects, so I may switch the stacks project over to it in the future (there are still not too many authors so it shouldn’t be difficult to do). If I do this then I imagine I am allowed to take any latex code submitted to Mathoverflow by mathematicians and add it to the stacks project as long as I make sure to attribute it to the author of the comment. But for the moment, contacting the author of the comment and asking for permission directly makes more sense. Of course this is a bit difficult to do since it often isn’t clear who the author is especially for some very prolific contributors on Mathoverflow such as BCnrd…
Anyway, on a completely different note: I finally figured out how to set up the Makefile so that I can run the latex compiles in parallel. You will know why this is a problem if you’ve ever tried to write such a Makefile. If not it probably makes sense to stop reading this now. It is really quite simple (and I’m sure it is an often used trick). Instead of running
latex stem.tex you execute a bash script which
- creates a temporary directory using
- copies all aux files and temp.toc to the temporary directory
- creates symbolic links in the temporary directory to stem.tex and stem.bbl (and maybe some style files, etc)
- changes directory into the temporary directory
- moves stem.dvi, stem.aux, stem.toc back to the main directory
- removes temporary directory
Anyway, using this I was able to cut overall compilation time in half. On the server at work the times were
make dvis -j1takes 1m42s
make dvis -j2takes 50s
make dvis -j3takes 33s
make dvis -j4takes 25s
Not too shabby.