Etale and motivic cohomology and ultraproducts of schemes

TL;DR

This paper investigates how etale cohomology, algebraic cycles, and motives behave under ultraproducts of schemes, aiming to transfer properties between different characteristics and explore implications for algebraic geometry.

Contribution

It introduces methods to transfer cohomological and cycle-theoretic statements across characteristics using ultraproducts, extending previous work on scheme enlargements.

Findings

Proves transfer principles for etale cohomology under ultraproducts.

Shows independence of l for Betti numbers in etale cohomology.

Provides bounds on algebraic cycle complexity.

Abstract

This paper is a continuation of the authors article "Enlargements of schemes" (Log. Anal.1 (2007), no. 1, 1-60) We mainly study the behaviour of etale cohomology, algebraic cycles and motives under ultraproducts respectively enlargements. The main motivation for that is to find methods to transfer statements about etale cohomology and algebraic cycles from characteristic zero to positive characteristic and vice versa. We give one application to the independence of $l$ of Betti numbers in etale cohomology and applications to the complexity of algebraic cycles.