Group Cohomology and Galois Cohomology

This Ph.D. course aims at discussing the basic theory and tools of cohomology applied to algebra and number theory. The (co)homological study of groups and modules historically started in algebraic topology by work of Hurewicz on aspherical spaces. With the advent of pure algebraic definitions, it became a central topc in the study of representation theory and algebraic number theory, e.g. in the modern formulation of class field theory. The course will give the basic definition and properties of cohomology of groups, both explicitly and via homological algebra. It will then deepen specific aspects related to algebra and number theory depending on the interests of the audience.

Organization. The course will start with a couple of introductory lectures and then will continue with seminars prepared and presented by the participants. These seminars will be shared among the participants taking into account backgrounds and expertise).

#	Content	Speaker	Date and Time
1	Introduction 1	Amos Turchet	Apr 9, h 11:00, Aula C308
2	Introduction 2	Amos Turchet	Apr 16, h 11:00, Aula C308
3	Cohomology via Cochains	Riccardo Bernardini	Apr 23, h 11:00, Aula C308
4	Restriction, Corestriction, Inflation	Danilo Avaro	Apr 30, h 11:00, Aula C308
4		TBA	May 7, h 11:00, Aula C308
4		TBA	May 14, h 11:00, Aula C308

• Brown, Kenneth S.. Cohomology of Groups. Switzerland, Springer New York, 2012;
• Gille, Philippe, and Szamuely, Tamás. Central Simple Algebras and Galois Cohomology. Singapore, Cambridge University Press, 2017;
• Harari, David. Galois Cohomology and Class Field Theory. Switzerland, EDP Sciences, 2020;
• Serre, Jean-Pierre. Galois Cohomology. Germany, Springer Berlin Heidelberg, 2002;;
• Serre, Jean-Pierre. Local Fields. Germany, Springer New York, 2013;
• Sharifi, Romyar. Group and Galois cohomology (notes), LINK;
• Sharifi, Romyar. Homological Algebra (notes), LINK;
• Stein, William. A short course on Galois Cohomology (notes), LINK;
• Washington, Lawrence C. Galois cohomology. In "Modular forms and Fermat’s last theorem". Springer New York, 1997. 101-120.
• Weiss, Edwin. Cohomology of Groups. United States, Academic Press, 1969.

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Lecturer: Amos Turchet