[FOM] Infinity in etale cohomology
David Roberts
david.roberts at adelaide.edu.au
Thu Jul 26 17:49:48 EDT 2012
Dear Colin,
Have you considered proving the Nisnevich cohomology exists in your system?
I believe the Nisnevich site is give by a coverage (in the sense defined in
Johnstone's Elephant) which would be amenable to your methods.
Regards,
David
On Jul 27, 2012 4:00 AM, "Colin McLarty" <colin.mclarty at case.edu> wrote:
> I have found an unexpectedly simple and robust occurrence of infinity
> in etale cohomology. (To forestall possible confusion I am not
> talking about large cardinals here, just about infinity.)
>
> Kummer, Dedekind, and Kronecker realized that in order to understand
> divisibility in arithmetic you need to consider not only divisibility
> of numbers by numbers but also divisibility of numbers or ideals by
> ideals. Ideals are infinite sets. But this is not a serious
> occurrence of infinity since ideals are finitely specified. In this
> context each ideal is determined by a finite list of numbers (as
> Kronecker and Dedekind both knew).
>
> A natural evolution led number theorists to scheme theory and sheaves.
> The result I recently announced here and posted on the ArXiv as
> arXiv:1207.0276v1 shows the basic methods of cohomology given (for
> example) in Hartshorne's Algebraic Geometry can be interpreted in
> second order arithmetic. All the relevant constructions can be
> determined by finite amounts of arithmetic data.
>
> I now know the analogous claim for etale sheaves is false, even on
> Noetherian schemes used throughout current number theory. The proof
> is at the end of my paper on the ArXiv as arXiv:1207.0276v2. That is
> version 2 of the earlier paper. The title is lightly changed to
> "Zariski cohomology in second order arithmetic".
>
> Second order arithmetic will not prove every Noetherian scheme has a
> set of all ideals of its etale structure sheaf, or that each ideal has
> a set of all linear maps to a given module. The counterexamples
> involve schemes that number theorists use constantly and they are
> elementary constructions -- but the specific ideals I can prove are
> not finitely generated are not individually important in practice.
>
> This does not sink the project of interpreting etale cohomology in
> second order arithmetic. It means some restrictions (not yet
> formulated) will be needed or else some much more light-weight proof
> of sufficiency of injectives will need to be found.
>
> colin
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