Topics in Galois Theory, Spring Term 2015
Schedule: Wednesday, 17:00 - 18:20, room 311 at the Faculty of Mathematics of HSE (Vavilova 7), first class: 04.02.2015
Instructor: Florian Heiderich
Topics planned to be discussed:
- Recall of the classical theory of finite and infinite Galois extensions of fields
- Grothendieck's approach to the Galois theory of field extensions
- The algebraic fundamental group and Grothendieck's Galois theory
- Galois theory of commutative rings
- Introduction to Hopf algebras and affine group schemes
- Hopf Galois theory of extensions of commutative and non-commutative rings
- Tensor categories
- Representation theory for Hopf algebras and Tannaka duality
Prerequisites: Basic knowledge in algebra, commutative algebra, algebraic geometry, topology and category theory.
Literature:
Grothendieck Galois theory:- Francis Borceux and George Janelidze. Galois theories, volume 72 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2001.
- Tamas Szamuely. Galois groups and fundamental groups, volume 117 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2009.
- Alexander Grothendieck. Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1, Lecture notes in mathematics 224, Springer-Verlag, Berlin). A TeXed version is available from arXiv
- Moss E. Sweedler. Hopf algebras. Mathematics Lecture Note Series. W. A. Benjamin, Inc., New York, 1969
- Susan Montgomery. Hopf algebras and their actions on rings, volume 82 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 1993
- Pierre Deligne. Catégories tannakiennes. In The Grothendieck Festschrift, Vol. II, volume 87 of Progr. Math., pages 111–195. Birkhäuser Boston, Boston, MA, 1990
- Pierre Deligne and James S. Milne. Tannakian categories. In Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, Vol. 900, pages 101–228. Springer, 1982. A TeXed version is available from the homepage of James Milne.
- Saunders MacLane. Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1971.
- Tom Leinster. Basic category theory, Cambridge University Press, Cambridge, 2014.