Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I have expanded a bit at Serre-Swan theorem: gave it an actual Idea-section, mentioned more variants (over general ringed spaces, in higher geometry) and added more references.
Thanks for collecting the matrial, I was in particular not aware of the paper of Morye. On the other hand, once I have a better internet connection than at the moment, I will try to soften a bit the treatment of Serre’s part of the pair of the theorems.
First of all, while morally it can be interpreted toward the duality, this is just a nomenclature: both sides of Serre’s theorem are about functions/algebraic/quantity side. The basic affine Serre’s theorem is the equivalence of the category of modules over global sections $\Gamma\mathcal{O}$ to the category of $\mathcal{O}$-modules (all over arbitrary affine scheme, Serre has done it for affine varieties). This is easy and it is about affiness (cf. fundamental theorem on morphism of schemes!). Both sides of equivalence involve the sheaf $\mathcal{O}$. The thing which enters the Serre-Swan duplex is to single out the finitely generated projectives. Now there is a theorem that finite rank locally free module over a commutative unital ring is the same as finitely generated projective. Finite rank means finite rank at every prime ideal.
Now, higher categorical version is an analogue but not quite the generalization. Namely, most importantly, equivalence of categories at the level of a abelian category of (quasi)coherent sheaves is a stronger statement than the equivalence of stable $(\infty,1)$-categories of complexes of sheaves (what is on smooth quasiprojective varities the same as derived equivalence). Second, one needs to take care with finiteness conditions, typically finiteness or boundness conditions are put on cohomologies. Third, on the level of categories of complexes there are also theorems which talk precisely about the categories of complexes at the level of ordinary (not infinity) equivalence, and go beyond the affine case. The prime example is GAGA theorem between analytic and algebraic category, which needs complexes and is 1-categorical.
Hi Zoran,
notice that the higher categorical version that you mention is not the one currently indicated in the entry. The entry presently just mentions something very simple, which isn’t even a full blown higher geometry statement yet: it just says that if you have a non-projective module and enough projectives, then you can resolve it by a complex of projective modules.
Concerning the statement of Serre’s theorem: I didn’t touch this. But it would indeed be nice if you or somebody found the time to expand on it and state it in more detail.
At Serre-Swan theorem there used to be a shy pointer to a version of the theorem in differential geometry. I made that more explicit in the entry, and then in fact I gave it a little minimum entry of its own
to go along with a list of theorems that say that differential geometry is “more algebraic” than one might superficially seem to have a right to expect (notably the embedding of smooth manifolds into formal duals of R-algebras).
Added a link under Related Concepts to Quillen-Suslin theorem, which has just been created.
I have made the two parts of the statement at smooth Serre-Swan theorem more explicit.
I have been just citing this from Nestruev’s book. That book never requires the base manifold to be compact, it seems. I need to think about this again. Do we need to require a compact base manifold for the second part of the statement to be true?
If I recall correctly that book uses the Withney embedding theorem to show that any vector bundle can be complemented to a trivial one plus the fact that the global sections functor is additive. Looks like it doesn‘t require compactness.
Thanks!! Much appreciated.
I have added that as a remark to the entry, here. Please feel free to expand.
1 to 8 of 8