Archive for the 'Uncategorized' Category

A finite presentation chapter?

March 15, 2011

There are a whole slew of results in commutative algebra and algebraic geometry that are essentially elaborations on a standard set of tricks for finitely presented objects. For instance, one has the following fact: if \{ A_\alpha\} is an inductive system of rings, then any finitely presented module over the colimit descends to one of the A_\alpha. Moreover, the category of f.p. modules over the colimit is the “colimit category” of the categories of f.p. modules over the A_\alpha. Similarly, any f.p. algebra over the colimit descends to one of the A_\alpha. This, together with fpqc descent, is behind Grothendieck’s extremely awesome proof of Chevalley’s theorem that a quasi-finite morphism is quasi-affine; this trick, in EGA IV-3, is what lets him reduce to the case where the target scheme is the Spec of some local ring. ┬áSo I think it would be fun to have a whole bunch of these sorts of results.

On the other hand, I’m not sure whether it would be pedantic to devote an entire chapter to them. There are probably more important things in commutative algebra proper, and the above results are really cleaner if we can use the language of schemes a bit (then we can talk about quasi-coherent sheaves on projective limits, and even derive ZMT!), though it is an open question exactly how much we should delve into algebraic geometry.

Thoughts?

Yes, we still exist

March 5, 2011

I realize nothing has been posted on this blog in approximately a month now, but this project is still being updated. The academic year makes it difficult to get much work done on outside projects. The CRing project is still different from what it was a month ago, though no sweeping changes have occurred.

Since we have a lot of submitted material now that has yet to be edited into the project, I plan to work on that a bit during spring break. Other people should feel free to be involved in this!