• cohomological dimension
• étale cohomology

In this work we will answer, to some degree, the question: What happens to étale cohomology above the dimension of a scheme? The first three statements are classic results in étale cohomology: The first two theorems exhaust the argument for schemes of finite type over a field k for torsion sheaves. The third one is a milestone in étale cohomology and it is known as Artin vanishing theorem and deals with the vanishing of cohomology for affine schemes. The last one was proven fairly recently by Peter Scholze and it is about the vanishing of cohomology in projective towers of Frobenius covers on the projective space.