Angus Macintyre, Edinburgh
Weil cohomologies, compactness, and uniformities
Rutgers Logic Seminar
Abstract
The notion of Weil Cohomology Theory in algebraic geometry abstracts the
principal properties needed to establish rationality of Weil zeta
functions,and,if augmented by Grothendieck's Standard Conjectures,leads
to an essentially formal proof of the Weil Conjectures.To construct even
one such theory was a major achievement.A rather careful modeltheoretic
analysis will show that the notions is what modeltheorists call first
order(it certainly does not look that way at first glance),and this
opens the possibility of averaging cohomology theories to construct new
ones.This in turns reveals that the Standard Conjectures,liberally
construed,encode uniformities concerning notions of intersection
theory,not explicitly spelled out in the literature.This in turn
connects to older issues of bounds in polynomial ideals,sometimes
conveniently proved via model theory.A systematic study of the
modeltheoretic aspects is connected with issues of effective(or uniform)
computation fo zeta functions of varieties.In addition it leads to new
issues in difference algebra,concerning the action of a generic
difference operator on cohomology.
I will give a leisurely introduction to these ideas.
References
- A. Macintyre, Weil cohomology and model theory. Connections between model theory
and algebraic and analytic geometry, 179--199, Quad. Mat., 6,
Aracne, Rome, 2000.
- H. Schoutens', Uniform bounds in algebraic geometry and commutative
algebra. Connections between model theory and algebraic and analytic
geometry, 43--93, Quad. Mat., 6, Aracne, Rome, 2000.
- Ivan Tomasic,
A new Weil cohomology theory
- Kleiman's two articles on the Standard Conjectures:
-
Algebraic Cycles and the Weil Conjectures,in Dix Exposes sur la
Cohomologie des Schemas,North Holland(1968),359-386,
- The Standard Conjectures,in Motives,vol 1(ed Janssen et
al),Proceedings of Symposia in Pure Mathematics 55 (1994),AMS,3-20.
- For the Weil Conjectures in the form needed for Model Theory,the
Springer book by Freitag and Kiehl is useful.
- Étale cohomology and the Weil conjecture. Translated from the German by Betty S. Waterhouse and William C. Waterhouse. With an
historical introduction by J. A. Dieudonné. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 13. Springer-Verlag, Berlin, 1988. xviii+317
pp. ISBN: 3-540-12175-7