Math 125 Current Problems in Number Theory:
Galois Cohomology and Descent
Syllabus
Main course texts:
List of other useful texts:
Weekly Syllabus and Homework
Updated March 1, 2022.
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Week
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Date
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Topics
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Reading
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Work
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1
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Tue 04 Jan
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Reminders on field theory and Galois theory. Algebraic and separable
closures. Direct limits. Infinite Galois theory. Krull topology.
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M 7, GS 4.1, S I.1.1-1.4, Sh I, IV.1
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Thu 06 Jan
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Topological groups. Profinite groups. Inverse limits. Topology of
profinite groups.
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M 7, GS 4.1, S I.1.1-1.4, Sh I, IV.1
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2
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Tue 11 Jan
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Profinite topology vs. profinite groups. Continuous actions.
Discrete G-sets. Galois actions. Etale algebras. Grothendieck's Galois theory.
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M 8, P 1.3.1-1.3.4, Sh I.1
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Group Work 1
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Thu 13 Jan
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Etale algebras. Embeddings. Grothendieck's Galois theory II. Galois
descent for vector spaces.
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M 8, P 1.3.1-1.3.4
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3
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Tue 18 Jan
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Quaternion algebras. Central simple algebras.
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GS 1.1-1.2, 1.5, L III.1, P 1.5.1-1.5.2
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Thu 20 Jan
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Reduced norm.
Artin-Wedderburn theory (Grant presenting).
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GS 2.1, 2.6, P 1.5.3
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4
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Tue 25 Jan
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Cancelled.
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Thu 27 Jan
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Brauer group. Period, index, degree.
Biquaternion algebras, Albert form, examples
of Amitsur-Rowen-Tignol. Merkurjev's theorem.
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GS 1.5, 2.4-2.5, L 1.5.4
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5
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Tue 01 Feb
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The category of discrete modules over a group. Crossed homomorphisms.
Nonabelian H1. Twisted forms of tensors.
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GS 2.3, 4.2, S I.5.1-5.2, III.1.1
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Group Work 2
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Thu 03 Feb
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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. Roots of unity. Kummer theory.
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GS 2.3, 2.7, P 1.3.5, S 5.5
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6
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Tue 08 Feb
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Classical group cohomology. Review of derived functors and
Ext. Standard resolution.
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GS 3.1-3.2, S I.1.1, Sh II.1
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Group Work 3
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Thu 10 Feb
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Inhomogeneous cochains. Profinite group cohomology. Group extensions (Richard presenting).
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GS 3.1-3.2, S I.1.1, Sh II.1-II.3
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7
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Tue 15 Feb
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Restriction.
Inflation. Extension of the longish exact sequence.
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GS 3.1-3.2, S I.1.1
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Group Work 4
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Thu 17 Feb
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Cohomological Brauer group.
Finiteness of period. Period-index problem.
Cohomology of cyclic groups (Zach presenting).
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GS 4.4-4.5
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8
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Tue 22 Feb
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Ci fields. Cohomological dimension.
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P 1.2, Sh IV.3, GS A.6, S2 II.3
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Thu 24 Feb
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Make the most of the HMF days!
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9
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Tue 01 Mar
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Tsen-Lang theory.
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P 1.2, Sh IV.3
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Thu 03 Mar
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Discrete valuations. Completions.
Ramification theory of discrete
valuation rings. Affine group schemes (Juanita presenting).
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GS A.6, S2 II.3
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Thu 04 Mar
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Ramification theory of discrete
valuation rings. Unramified extensions and extensions of the
residue field.
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GS A.6, S2 II.3
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10
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Tue 08 Mar
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Brauer group of discretely valued fields. Induced representations and
Shapiro's Lemma (Matt presenting).
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GS A.6, S2 II.3
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Group Work 5
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Tue 10 Mar
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Hasse-Brauer-Noether exact sequence. Cup products (Varun
presenting). Artin-Schreier theory (Haochen presenting). Fourier
analysis on locally compact groups (Steve presenting).
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