Vanishing theorems for étale sheaves. 


In algebraic geometry, Serre proved that on an affine variety, coherent sheaves have vanishing higher cohomology. It is a generalization of the classical Mittag-Leffler conditions. 
For topological sheaves, there are two classical vanishing theorems, the one of Artin on affine varieties, and the one stemming from the Artin-Schreier theory on proper varieties in positive characteristic, both in cohomological degrees beyond the dimension. 

I’ll explain a vanishing theorem of a new kind due to Peter Scholze,  showing vanishing on projective  varieties beyond the dimension, for cohomology with torsion coefficients. He proved  it with his perfectoid methods, which yields the result in characteristic zero. Our proof, more classical in flavour, yields his vanishing theorem in any characteristic.