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.



5 Responses to “A finite presentation chapter?”

  1. As you probably know, I am currently teaching a commutative algebra course out of my (rapidly changing) commutative algebra notes. In my course I skipped the section on finitely presented modules, thinking it was a technicality and that in a first algebra course one should concentrate on Noetherian rings. Well, maybe the latter is true but I certainly wasn’t up to the task of systematically pretending that all rings were Noetherian. Even thinking about it makes me fear being haunted by the spirit of Brian Conrad. (Yes, I know he’s still alive.) So as a result I have been slipping in the “f.p.” hypothesis and then muttering apologies for it under my breath increasingly often. Especially, just over the last few days I have been touching up the portion of my notes on the relationship between finitely generated projective modules and finitely generated locally free modules, and none of this makes any sense without paying attention to issues of finite presentation. (Is this why this material is so hard to find in the standard commutative algebra texts? Of the five or so that I own, only Bourbaki really hits this head on.)

    So I would say…yes, of course you should have some material on finitely presented modules. Whether you need a whole chapter or not should become clear in the fullness of time.

    • Akhil Mathew Says:

      Well, “nobody but Bourbaki does this properly” is enough motivation for me to start writing this up.

      I think I may need to set up a “grab-bag” chapter full of random things that are too short for their own chapters, but should be in there nonetheless. For instance, there is a short chapter on “linear algebra over rings”, which right now covers some results of McCoy (taken from notes by Anton Geraschenko) that has nothing else in it.

      • Well, just to be sure, I didn’t say “nobody but Bourbaki does this properly”. It might be true, but I didn’t say it! (I would be willing to bet that some other standard text must treat it; I just haven’t found one yet.)

  2. Akhil Mathew Says:

    Right — I should remember not to use quotation marks when not literally quoting someone :).

  3. […] Comments Akhil Mathew on A finite presentation cha…Pete L. Clark on A finite presentation cha…Akhil Mathew on A finite […]

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