Department Mathematik


Summer Semester 2017

Seminar: Hermitian K-theory and applications

Link to the course catalogue.
Organizers Andreas Rosenschon and Anand Sawant
Time and place Wednesdays 14-16 in Room B 252.
Short description We will study basics of Witt and Grothendieck-Witt groups of a commutative ring and their relationship with projective modules over the ring. As an application of these methods, we will study some of the recent results on free-ness of stably free modules over certain rings.
  • P. Balmer: Witt groups. Handbook of K-theory. Vol. 1, 2, 539-576, Springer, Berlin, 2005. Link
  • M. Schlichting: Hermitian K-theory of exact categories. J. K-Theory 5 (2010), no. 1, 105-165.
  • J. Fasel, R. A. Rao and R. G. Swan: On stably free modules over affine algebras. Publ. Math. Inst. Hautes Études Sci. 116 (2012), 223-243.
  • The seminar will be conducted in English. The first meeting will be on May 3, 2017.
  • The seminar is intended for Master students (prerequisites: commutative algebra and algebraic geometry).
  • There will be no lecture on June 28.
Plan of lectures
  • May 03: Lecture 1: Overview of the topics that will be covered in the seminar; organization. (Anand Sawant)
  • May 10: Lectures 2,3: Introduction to quadratic forms and Witt groups. (Claudia Stadlmayr, Killian Rückschloß)
  • May 17: Lecture 4: Introduction to algebraic K-theory. (Tariq Syed)
  • May 24: Lectures 5,6: Witt groups of an exact category with duality. (Ismail Achmed-Zade, Killian Rückschloß)
  • May 31: Lecture 7: Hermitian Q-construction and the Grothendieck-Witt space of an exact category with duality. (Monica Flamann)
  • June 07: Lecture 8: Unimodular rows and the Vaserstein symbol - I. (Tariq Syed)
  • June 14: Lecture 9: Unimodular rows and the Vaserstein symbol - II. (Tariq Syed)
  • June 21: Lecture 10: Completion of some unimodular rows. (Andrei Lavrenov)
  • Proof of the Fasel-Rao-Swan theorem. (3 lectures, Anand Sawant)
    • July 05: Lecture 11: Structure of the proof and reductions.
    • July 12: Lecture 12: Grothendieck-Witt groups of affine schemes, Karoubi periodicity theorem and the Gersten-Grothendieck-Witt spectral sequence.
    • July 19: Lecture 13: Divisibility of certain Grothendieck-Witt groups and completion of proof.