Fundamental concepts in non-commutative
algebra, Math 80583
Announcements:
- October 20. As was decided on the last lecture, we will meet on Mondays for 3 hours at 9-12, and will cancel the one hour lecture on Thursdays.
Instructor:
· Michael Temkin
· E-mail: michael.temkin
mail.huji.ac.il
· Office: Ross 78A (tel. 02-6584575)
· Office Hours: by appointment
· Lectures: Mondays 9-11:45 at Mathematics 209.
Texts:
· I do not know a comprehensive book on the theory of valued fields, and one of the aims of the course is to write (a preparation to) such a book. Here are the notes I am preparing for this aim; they will have a major intersection with what is going on in class. I will try to be a little ahead of the course – we will see if I manage to hold this temp. Caution: not only I will add material as the class proceeds, any material in the notes can be freely modified, regrouped, rewritten, etc.
Grading:
· By a lecture given by students in the end of the course.
Lectures:
- October 15. Real valuations, completion of real valued fields, R and Q_p, Ostrowski’s theorem.
- October 22. Generalized Gauss valuations, Berkovich affine line and classification of points in it.
- October 29. Classification of Archimedean analytic fields (second Ostrowski’s theorem), normed vector spaces over analytic fields and equivalence of norms, Hensel’s lemma.
- November 5. Extension of valuations for complete valued fields, continuity of roots and completed algebraic closure, the field C_p.
- November 12. Krasner’s lemma, extensions of valuations for non-complete fields, conjugation of extended valuations, the tensor product formula, independence of valuations and the weak approximation.
- November 19. Spectral norm on finite extensions of real valued fields, henselian valued fields and henselization.
- November 26. Characterizations of henselian real valued fields, main properties of Berkovich spectrum (formulations). Started chapter 2: valued field. Ordered groups, valuations and valuation ring.
- December 3. Valuation rings and valuations, various characterizations of valuation rings, integral closure in terms of valuation rings, independence of valuations and finite intersections of valuation rings.
- December 10. Extensions of valuations: existence for arbitrary extensions, and transitivity of the Galois action for Galois extensions. Invariants of extensions of valued fields: fundamental inequality and Abhyankar’s inequality.
- December 17. Henselian valued fields and henselization.
- December 24. Limits and compositions of valued fields, the fundamental inequality.