Cohomological Methods in Intersection Theory

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