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.  So 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?