Math 115 Number Theory:
Galois Cohomology and Descent

Syllabus

Main course texts:

List of other useful texts:

Weekly Syllabus and Homework

Updated May 21, 2024.

Week Date Topics Reading Work
1 Tue 26 Mar Reminders on field theory and Galois theory. Algebraic and separable closures. Direct limits. Infinite Galois theory. M 7, GS 4.1, S I.1.1-1.4, Sh I, IV.1
Thu 28 Mar Inverse limits. Profinite groups. Profinite completion. M 7, GS 4.1, S I.1.1-1.4, Sh I, IV.1
2 Tue 02 Apr Continuous actions. Discrete G-sets. Galois actions. Etale algebras. Embeddings. Grothendieck's Galois theory. M 8, P 1.3.1-1.3.4 Group Work 1
Thu 04 Apr Embeddings. Grothendieck's Galois theory. Galois descent for vector spaces. M 8, P 1.3.1-1.3.4
3 Tue 09 Apr Quaternion algebras. Central simple algebras. Brauer group. Galois descent for central simple algebras. GS 1.1-1.2, 1.5, 2.1, 2.4-2.6, P 1.5.1-1.5.4
Thu 11 Apr The category of discrete modules over a group. Crossed homomorphisms. Nonabelian H1. GS 2.3, 4.2, S I.5.1-5.2, III.1.1
4 Tue 16 Apr Twisted forms of tensors and nonabelian H1. GS 2.3, 4.2, S I.5.1-5.2, III.1.1
Thu 18 Apr Applications of twisted forms: generalized Hilbert 90, central simple algebras, quadratic forms, etale algebras, and G-Galois algebras. Galois cohomology of the symmetric group and finite groups. GS 2.3, 2.7, P 1.3.5, S 5.5
5 Tue 23 Apr Longish exact sequence. Roots of unity. Kummer theory. Discriminant. Elliptic curves. GS 2.3, 2.7, P 1.3.5, S 5.5 Group Work 2
Thu 25 Apr Classical group cohomology. Review of right derived functors and Ext. GS 3.1-3.2, S I.1.1, Sh II.1
6 Tue 30 Apr Right derived functors of contravariant functors. Standard resolution. Inhomogeneous cochains. GS 3.1-3.2, S I.1.1, Sh II.1-II.3 Group Work 3
Thu 02 May Profinite group cohomology. Restriction. Inflation. Extension of the longish exact sequence. GS 3.1-3.2, S I.1.1
7 Tue 07 May Cohomological Brauer group. Finiteness of period. Period-index problem. GS 4.4-4.5
Thu 09 May Discrete valuations. Completions. Ramification theory of discrete valuation rings. GS A.6, S2 II.3
8 Tue 14 May Ramification theory of discrete valuation rings. GS A.6, S2 II.3
Thu 16 May Ramification theory of discrete valuation rings. Unramified extensions and extensions of the residue field. GS A.6, S2 II.3
9 Tue 21 May Brauer group of complete discretely valued fields. Hasse invariant GS 6.3, S 4.3, S2 XII
Thu 23 May Rational points. Adeles. Obstructions. Descent obstruction. Brauer-Manin obstruction.
10 Tue 28 May Final presentations.
Thu 30 May Final presentations.


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