Étale cohomology and the Weil Conjectures

This course focused on étale cohomology and the Weil conjectures. Lectures were recorded and made available on YouTube.

We will use the following resources:

• Milne’s Lectures on étale cohomology

• Milne’s Étale cohomology

• Freitag–Kiehl’s book

• Katz’s lectures on the Weil conjectures

Time permitting, we may cover some other advanced topics.

The class met online on Tuesdays and Thursdays from 9:35–10:50am Eastern, beginning August 20, 2020. Office hours and class discussion also took place online.

• 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