Étale cohomology and the weil Conjectures

This course will focus on étale cohomology and the Weil conjectures. I plan to record all lectures and make them available on YouTube.

We will use the following resources:

Time permitting, we may cover some other advanced topics.

Here is the Zoom link for our meetings, which will take place on Tuesdays and Thursdays from 935am-1050am Eastern; the first course meeting will be Thursday, August 20. The password is the order of the symmetric group on 4 letters. If you have any trouble joining, let me know.

Office hours will take place Mondays at 10am, at this link (with the same password, namely the order of the symmetric group on four letters).

There is also a class Discord server for discussions; let me know if you need an invite link.

  • Lecture 1 (video, notes): Introduction, the Weil conjectures, rationality of the zeta function for curves, Serre’s Kähler analogue

  • Lecture 2 (video, notes): Proof of Serre’s Kähler analogue, review of étale morphisms, intro to sites

  • Lecture 3 (video, notes): Sites, sheaves on sites, examples of sites, the étale and fppf topologies

  • Lecture 4 (video, notes): Morphisms of sites, fppf descent part I

  • Lecture 5 (video, notes): Fppf descent part II, beginning of the study of the category of sheaves

  • Lecture 6 (video, notes): Pushforwards, stalks, sheafification, the category of sheaves is abelian

  • Lecture 7 (video 1, video 2, notes): Enough injectives, derived functor cohomology, Čech cohomology

  • Lecture 8 (video, notes): Čech cohomology continued, Čech-to-derived spectral sequence, étale cohomology of quasicoherent sheaves, Artin-Schreier exact sequence, cohomology of the constant sheaf \(\mathbb{F}_p\).

  • Lecture 9 (video, notes): Čech cohomology of fields, torsors and descent, Grothendieck’s generalization of Hilbert’s theorem 90

  • Lecture 10 (video, notes): Hilbert's theorem 90, étale cohomology of \(\mathbb{G}_m\) in low degrees, the Kummer sequence, cohomology of \(\mu_\ell\) and \(\mathbb{Z}/\ell\mathbb{Z}\) in low degrees, statement of étale cohomology for curves, Leray spectral sequence

  • Lecture 11 (video, notes): Leray spectral sequence continued, computing derived pushforwards, strict henselizations and stalks of derived pushforwards, Weil-Divisor exact sequence, cohomology of the sheaf of divisors, reduction to Galois cohomology, intro to Brauer groups

  • Lecture 12 (video, notes): Brauer groups, Azumaya algebras, Severi-Brauer varieties, twisted sheaves, basic properties.

  • Lecture 13 (video, notes): Tsen’s theorem, Tate’s theorem, étale cohomology of curves concluded

  • Lecture 14 (video, notes): Extension by zero, compactly supported cohomology, compactly supported cohomology of curves, intro to proper base change

  • Lecture 15 (video, notes): More proper base change, compactly supported cohomology, cohomology with supports, Gysin sequences and purity

  • Lecture 16 (video, notes): Gysin sequences and a sketch of the proof of purity, cohomology of projective space, elementary fibrations, intro to Artin comparison

  • Lecture 17 (video, notes): Artin comparison, étale \(\pi_1\)

  • Lecture 18 (video, notes): More on étale \(\pi_1\)

  • Lecture 19 (video, notes): Finiteness, \(\mathbb{Z}_\ell\)-sheaves, etc.

  • Lecture 20 (video, notes): Smooth base change, smooth and proper base change, cospecialization, lifting varieties to characteristic zero, Kunneth

  • Lecture 21 (video, notes): Cup products, Kunneth, projection formula, cycle class maps, the Tate conjecture

  • Lecture 22 (video, notes): Chern classes, Poincaré duality, trace maps

  • Lecture 23 (video, notes): Poincaré duality, Verdier duality, Lefschetz fixed point formula

  • Lecture 24 (video, notes): Lefschetz fixed point formula, rationality of Weil zeta functions, integrality, intro to RH

  • Lecture 25 (video, notes): Digression on Frobenii, preliminary reductions for RH, intro to the Grothendieck-Lefschetz trace formula

  • Lecture 26 (video, notes): Perfect complexes, statement of the Grothendieck-Lefschetz trace formula, intro to the main lemma

  • Lecture 27 (video, notes): The MAIN LEMMA, Lefschetz pencils, sketch of proof of RH, beginning of proof of Main Lemma

  • Lecture 28 (video, notes): Proof of the MAIN LEMMA, cohomology of Lefschetz pencils

  • Lecture 29 (video, notes): Example of Lefschetz pencil, cohomology of Lefschetz pencils continued, vanishing cycles, Kazhdan-Margulis

  • Lecture 30 (video, notes): Last class — End of the proof, Weil II, applications to semisimplicity of geometric monodromy and to Chebotarev