for questions about etale cohomology of schemes, including foundational material and applications.
Etale cohomology is a cohomology theory of schemes that uses sheaves on the étale site, instead of the underlying topological space. It was introduced by Grothendieck as part of his program to prove the Weil conjectures for varieties over finite fields, with significant contributions from Artin, Verdier, and Deligne.
Applications of étale cohomology include the solution to the Weil conjectures by Grothendieck and Deligne, cohomological obstructions in arithmetic, and the determination of representations of finite groups of Lie type by Deligne and Lusztig.
Etale cohomology with torsion coefficients applied to smooth complex projective varieties compares well with singular cohomology with the same coefficients. In order to get a well-behaved theory for varieties over finite fields with non-torsion cohomology groups, one applies a projective limit of cohomology groups, and inverts torsion.