[FOM] Grothendieck foundations: Zariski and coherent cohomology
David Roberts
david.roberts at adelaide.edu.au
Wed Nov 30 16:51:18 EST 2011
Hi Colin,
When you say sheaves below, I presume you mean on the Zariski site?
And perhaps intermediate to etale cohomology you might try Nisnevich
sheaves, which are important for several reasons (A^1-homotopy theory
for example). There is a nice characterisation of a Nisnevich cover
equivalent to the usual one given by Lurie, which I reproduce here
http://mathoverflow.net/questions/78431/nisnevich-topology-on-non-locally-noetherian-schemes
and which I find more intuitive and simple. You can ignore the non-Noetherian
stuff, it was a bit of a red herring in the end.
And regarding the use of global choice over ZF[0]: is it equivalent
(over ZF[0]) to a form of Zorn's lemma weaker than usually stated?
Or can you isolate what fact you use in the proof of enough injectives
to be a choice principle on its own?
Regards,
David
On 30 November 2011 00:37, Colin McLarty <colin.mclarty at case.edu> wrote:
> I can confirm one conjecture from my previous post, but the proof is
> more involved than I expected so it leaves me a bit less sure of the
> other conjecture.
>
> Harvey's suggestion of using ZF[0] (with suitable choice principle) is
> working really well. In the present case I began with a quick and
> dirty idea for n=1 or 2, and when it shook out it was n=0. So this is
> at the strength of 2nd order arithmetic.
>
> Specifically, the case is derived functor cohomology for all sheaves
> of modules on any Noetherian scheme. This includes coherent
> cohomology of Noetherian schemes, which is the tool of Hartshorne's
> book _Algebraic Geometry_ and the central tool of all cohomological
> number theory. It would be nice to also get etale cohomology on this
> foundation, and I still suspect that can be done. But I do not know.
>
> On the other hand, this foundation does not prove existence of some
> important examples: real complex and p-adic numbers. It proves the
> theorems hold for them if they exist. That is another issue. This
> foundation does prove existence of all the "arithmetic schemes"
> (schemes of finite type over the integers).
>
> I sketch the key issues in the proof, as I now have it. It brings
> coherent cohomology to the ambit of reverse math, where I have not
> followed.
>
> 1) It currently takes global choice (over ZF[0]) to prove every
> module over any Noetherian ring has an injective embedding. We use a
> kind of Zorn's lemma argument for submodules of a given module,
> without supposing there is a set of all those submodules. I tried
> hard to get by with less choice, but I have no evidence now that it
> *cannot* be done with less.
>
> 2) The natural context for this foundation is the Noetherian case.
> That is the most important case in practice, especially for coherent
> cohomology. The point for us is that ZF[0] even without choice proves
> every set has a set of all its finite subsets, so every ring has a set
> of all its finitely generated ideals -- which is all ideals, for a
> Noetherian ring. An obvious inner model of countable sets shows that
> this foundation does not prove all countable rings have sets of all
> their ideals.
>
> 3) The chief result now is that every sheaf of modules on a Noetherian
> scheme has an injective embedding, not only quasi-coherent modules.
> This goes by proving the structure sheaf of rings on any Noetherian
> module has a set of all sheaf ideals, not only quasi-coherent ideals.
> And that works by proving that every sheaf of ideals on the sheaf of
> rings R is determined by finite data relative to R, so there is a set
> of all those data. The simplest proof I can find uses Krull's
> Principle Ideal Theorem, roughly saying any single equation on an
> algebraic space defines a subspace of codimension 1 or 0. It serves
> to sharply limit how far a sheaf of ideals can deviate from being
> quasi-coherent, in the Noetherian case.
>
> colin
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