Cohomological Methods in Intersection Theory

TL;DR

This paper explores how motives can enhance cohomological methods in intersection theory, providing new proofs and generalizations of key formulas and dualities in algebraic geometry.

Contribution

It introduces motivic approaches to prove independence of ℓ results, offers a new motivic proof of the Grothendieck-Lefschetz formula, and extends duality and base change results for motives.

Findings

Motivic proof of the Grothendieck-Lefschetz formula

Enhanced duality for étale motives over schemes

Virtually integral property of Q-linear motivic sheaves

Abstract

These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to see how motives may be used to enhance cohomological methods, giving natural ways to prove independence of $\ell$ results for traces and zeta-functions, and constructions of characteristic classes (as $0$-cycles). This leads to the Grothendieck-Lefschetz formula, of which we give a new motivic proof. There are also a few additions to what have been told in the lectures: a proof of Grothendieck-Verdier duality of \'etale motives on schemes of finite type over a regular quasi-excellent scheme (which slightly improves the level of generality in the existing literature); a proof that $\mathbf{Q}$-linear motivic sheaves are virtually integral; a proof of the motivic generic base change formula.