Etale Cohomology: Grothendieck’s Contribution

Étale Cohomology: Grothendieck’s Contribution K.Srinivasa Raghava, [email protected] August 21, 2024 1 Étale Cohomology: Grothendieck’s Contribution Étale cohomology is one of the most significant tools introduced by Alexander Grothendieck in algebraic geometry. It addresses limitations in classical cohomology theories, particularly for varieties over fields of positive characteristic, like finite fields. This theory enabled deep results, such as the proof of the Weil conjectures. Here, I am trying to explain the outlines of étale cohomology in a simple way, making it accessible for all general readers to understand, even without deep prior knowledge of algebraic geometry or cohomology theories. 1.1 Motivation Classical cohomology theories, such as singular and de Rham cohomology, are well-suited for complex varieties but do not work well for varieties over finite fields. Étale cohomology overcomes this by constructing a cohomology theory using the étale topology. 1.2 Étale Topology In the étale topology, open sets correspond to étale morphisms, which are unramified and smooth. This allows us to generalize classical cohomology techniques to more general schemes. 1.3 Étale Sheaves and Cohomology i Étale cohomology groups, denoted Hét (X, F), are the derived functors of the global sections functor applied to a sheaf F on the étale site of a scheme X. i Hét (X, F) = Ri Γét (X, F) 1 1.4 Example: Étale Cohomology of a Point Consider a point x = Spec(Fq ). The étale cohomology groups of x with coefficients in Z/nZ are: ( Z/nZ if i = 0 i Hét (x, Z/nZ) = 0 if i > 0 1.5 Weil Conjectures and Grothendieck’s Approach Grothendieck applied étale cohomology to solve the Weil conjectures, relating the number of points on a variety over a finite field to its topology. The zeta function of a variety X over Fq is given by: ! ∞ X |X(Fqn )| n t Z(X, t) = exp n n=1 1.6 Lefschetz Trace Formula One of the key results derived from étale cohomology is the Lefschetz trace formula, which relates the number of points on a variety over a finite field Fq to the traces of the Frobenius morphism acting on the étale cohomology groups: |X(Fq )| = 2 dim XX i (−1)i Tr(Frobq | Hét (X, Ql )) i=0 1.7 Conclusion Étale cohomology provided a way to connect algebraic geometry with number theory and topology, bridging the gap between fields in positive characteristic and classical cohomological methods. Grothendieck’s contribution was essential for proving the Weil conjectures and continues to influence modern developments in mathematics. 2