Étale fundamental group

05 Feb 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Equivalent characterizations of étale algebras. Definition of unramified morphisms. A morphism is unramified at x if and only if x is isolated in the fiber, without nilpotents. Examples.

07 Feb 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Derivations. Kahler differentials. Presentation using the ideal of the diagonal. A morphism is unramified at x if and only if Kahler differentials vanish at x, if and only if the diagonal is an open embedding around x. Review of flatness (no proofs due to time constraints). Definition and basic properties. Over PIDs, flat = torsion free. Flatness is local. Flat, finitely generated over a local ring is free. Faithfully flat = flat + surjective. Flat + locally of finite presentation is open.

13 Feb 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Étale morphisms, properties, Galois coverings.

20 Feb 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Descent theory. Fpqc topology is subcanonical. Descent for various properties of morphisms. Quotients of finite étale covers by finite group actions.

21 Feb 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Galois categories and fibre functors. Examples: finite étale covers, the Galois category of a pro-finite group. Various basic properties of Galois categories. Fibre functors are strictly pro-representable.

27 Feb 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

The automorphism group of a fibre functor. Galois objects in Galois categories. Every Galois category is equivalent to the category of finite sets with a continuous action of the fibre functor. A pro-finite group is isomorphic to the automorphism group of the fibre functor of its associated Galois category.

28 Feb 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Any two fibre functors are isomorphic. Definition of étale fundamental group. Homomorphism of profinite groups vs functors on Galois categories. Finite algebras over complete noetherian local rings.

06 Mar 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Fundamental group of complete noetherian local rings. Cohomology and base change. Zariski's connectedness theorem.

07 Mar 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Stein factorization. Homotopy exact sequence for proper, separable morphisms.

20 Mar 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Fundamental group of the reduced subscheme, of a product. Base change along algebraically closed fieds. Homotopy exact sequence of a Galois cover.

21 Mar 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Homotopy exact sequence of a scheme over a field. Fitting ideals.

27 Mar 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Specialization of the fundamental group. Abhyankar's lemma

28 Mar 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Zariski-Nagata. Fundamental group is birational invariant. Riemann existence theorem. Fundamental group of curves.

03 Apr 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Yoneda's lemma. Abelian varieties. Isogenies.

04 Apr 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Picard schemes. Jacobian varieties. Abelian varieties over C. Tate modules. Lang-Serre.

10 Apr 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Introduction to anabelian geometry. Faltings (Tate conjecture), Neukirch-Uchida, isom, hom, section conjectures. Tamgawa, Mochizuki, Pop. Injectivity for the section conjecture.

11 Apr 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Mattuck's theorem. Real section conjecture. Hain, Koenigsmann. Tamagawa neighbourhoods.

15 Apr 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Brauer obstruction for rational points of the Picard scheme. Brauer-Severi varieties.

23 Apr 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Brauer obstruction vanishes on torsion when there is a Galois sections (Stix). Period and index. Brauer group of local fields. Roquette, Lichtenbaum.

24 Apr 2025 (2h 00m)

GIULIO Bresciani - Corso (attività didattica) - In presenza

Stix' theorems on period and index in light of the section conjecture over p-adic fields and over Q.