[FOM] Grothendieck foundations: Zariski and coherent cohomology

[Colin McLarty]

> When you say sheaves below, I presume you mean on the Zariski site?

Yes, in effect.  Actually for this work I write using the Hartshorne
framework of sheaves on topological spaces, not sites, so on the
Zariski topology.


> 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?

I probably should not have said Zorn's lemma.  That is a remnant from
how I found the proof eventually.  Really it is a well ordering of the
universe.

I have to be able to select a (well, arbitrarily many) homomorphism
out of a class of them (i.e. arbitrarily many classes), where the
class(es) does not form a set.

So this lets me do Baer's construction of injective embeddings for all
modules on any Noetherian ring, not only for countable modules or
anything like that.  Of course the only sets that ZF[0] can prove
exist are countable, but this way if you add stronger assumptions
about say arbitrarily large modules the proof is still good for all of
them.

best, Colin


>
> 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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