[FOM] Infinity in etale cohomology
colin.mclarty at case.edu
Wed Jul 25 21:12:33 EDT 2012
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.
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