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I have renamed and rewritten the article (formerly: affine Serre’s theorem), hoping to make it easier to digest, and adding in some references. I may come back to it with a statement that quasicoherent sheaves over a general scheme are obtained by a descent construction, involving categories of modules attached to affine open subschemes.
I like “quasicoherent sheaves over affine schemes” more, but was mindful of the nLab tendency to avoid plurals in titles.
The entry is about a particular adjoint equivalence (for graded rings there is also a famous projective version of the theorem, also due Serre).
If the title is that it is about quasicoherent sheaves (of modules) over affine schemes in general then it should start with a definition of that category of sheaves and so on, not just the Serre’s theorem (what is the scope if the title is affine Serre’s theorem), so it changes the scope of the entry radically (or it otherwise misleads the reader if basics are not there). There are plenty of other important theorems and constructions on quasicoherent sheaves over affine schemes (derived categories, base change etc.), so the change of title can have serious implications.
Added:
In the formalism of functor of points, the equivalence turns into a definition: affine schemes are defined as the opposite category of the category of commutative rings (with the functor $Spec$ now being tautologically defined as the identity functor), and the category of quasicoherent modules over $Spec R$ is now defined as the category of $R$-modules. This assignment defines a stack of categories over the site of affine schemes with the Zariski topology.
The functor of points approach carries over to quasicoherent modules over non-affine schemes: given such a scheme $X$, a quasicoherent module over $X$ is a morphism of stacks from $X$ to the stack of quasicoherent modules defined above. In concrete terms, this boils down to picking an open cover of $X$ and defining a quasicoherent module using cocycle data.
Thanks, Dmitri! I was just thinking about that.
I’ll note though, while we’re on the topic, that the entry on functorial geometry = functor of points is not in very satisfactory shape in my opinion. It lacks a definition, or details about the transition between that and the older point of view that involves locally ringed spaces.
Zoran, we do have an entry for quasicoherent sheaves and the category of such, so I’m not quite sure there’s really any problem. But: please feel free either to expand on the entry in the way you find appropriate, or rename the page to something better. As I said earlier in another thread, I don’t know how the result is named in the literature or even if it has a name, and in particular I haven’t seen any source outside the nLab that calls it “the affine Serre theorem”, which was the original title and also what you called it just now. If you know that to be a name in the common parlance, it would be great if you could name a source. Thanks!
Is “quasicoherent module” not terminological abuse for “quasicoherent sheaf of $O(X)$-modules. And that’s what the entry is about.
I thought I had seen that terminology (abusive or not; these are conventions, and categorical contexts for the term “module” are quite flexible, as you know; it should be understood that here it’s internal to a category of sheaves). And already the title is getting annoyingly long! I’ll change “modules” to “sheaves”, but this is the last title change I’m going to make. (I won’t object if someone wants to change it again, but I’m done here.)
that title looks good to me.
(I don’t think comment #5 is reasonable: For one, not every entry that mentions quasicoherent sheaves needs to recall their definition – on the contrary.)
I have briefly crossed-link this entry here with several related entries, including:
haven’t seen any source outside the nLab that calls it “the affine Serre theorem”
The thing is that there are 3 “versions” of Serre’s theorem, namely 2 projective and one affine. The projective version is so well known by the name of “Serre’s theorem” that that one has hundreds of references (and I should certainly write an entry on this), sometimes but more rarely called by full name Serre’s theorem on Proj (or for Proj). It says essentially the same thing but for graded rings and graded modules (either in coherent or in quasicoherent version, where the former has some finiteness assumptions). So when one means affine, one in context sometimes just says “Serre’s theorem”, but when one needs to clarify what is usually in discussions (I had hundreds of those with experts as I was working for 15 years in noncommutative generalizations of study of geometry in terms of categories of quasicoherent sheaves and it is standard way to distinguish from projective Serre’s theorem in discussions), one adds “affine” what is usually not needed or used in print. Thus in print you will find Serre’s theorem only for all 3 theorems, while far more rarely the affine version will be named in any way. An alternative modifier is Serre’s theorem on/for quasicoherent sheaves on affine schemes.
I do not think that the entry “quasicoherent sheaf” is a good substitute for “quasicoherent sheaves over affine scheme” or over spectrum as the sheaf theoretic construction here is very specific and elementary (and can be done in two different ways), while the notion of quasicoherent sheaf in general may be done tautologically, has various levels of generality and the entry there is very abstract and not usable to discuss the Serre’s theorem as most of those who taught and written on the subject know. One approach for spectra is via stalks and another via sheafification of presheaf picture, first say in Hartshorne’s book and second in Neeman’s.
Redirects also Serre’s theorem on quasicoherent sheaves over affine schemes. Added a link to newly created Serre’s theorem on Proj.
Zoran, as I said, I don’t plan on fiddling with the title any more, but if you’d like to yourself, no objection from me. All I had wanted to do originally is try to bring the original entry into better shape (= easier and more pleasant for me personally to read). Thanks for the new entries.
I will not change the title either, I have just added an extra redirect and two related entries.
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