Étale Cohomology
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Roughly speaking, étale cohomology is an algebraic analogue of singular cohomology that extends sheaf cohomology. As we study topological spaces over Zariski topology (therefore associated to the small Zariski site) by considering sheaves and sheaf cohomology, having a stronger topology admits more open subsets, which makes studying small étale site over étale topology using schemes and étale cohomology the “correct” analogue. It turns out that étale cohomology also has strong connections with many other cohomologies, e.g., Galois cohomology and Čech cohomology. We will justify the said analogue by studying flasque sheaves and the higher direct image functor. Finally, we will show that étale cohomology has properties similar to those in the Eilenberg-Steenrod axioms. [Notes]