Characteristic-p Methods
Instructor: Manoj Kummini.
Email: mkummini AT cmi DOT ac DOT in.
http://www.cmi.ac.in/~mkummini/index.html
Lectures: Tue, Wed and Thu, 0910-1025; Lecture Hall 4
Office hours: TBD.
We will study singularities of rings and of varieties in prime
characteristic using the Frobenius map. We hope to cover the following
topics: F-purity and F-splitting; tight closure; F-rationality and its
relation to pseudo-rational singularities; test ideals; reduction to prime
characteristic.
Prerequisites: knowledge of primary decomposition and associated primes,
integral extensions, completion, dimension theory of noetherian local
rings.
Important information:
- Plan of lectures:
http://www.cmi.ac.in/~mkummini/courses/2015_16/term2/schedule_char_p.txt
- Notes:
http://www.cmi.ac.in/~mkummini/courses/2015_16/term2/char_p.pdf
- I will use Moodle to make announcements and post your marks.
You can directly access the course page at
http://moodle.cmi.ac.in/course/view.php?id=149
. Otherwise, look for `Characteristic-p Methods' under
`Mathematics / Jan-Apr 2016'.
- References: There are a few papers/expository
articles/textbooks from which I will draw the material.
- M. Hochster and J. Roberts: Rings of invariants of reductive
groups acting on regular rings are Cohen-Macaulay, Adv. Math., 1974
- C. Huneke: Tight closure and its applications, CBMS Lecture Notes in
Mathematics, vol. 88, American Math. Soc., Providence, 1996.
- C. Huneke: Tight closure and geometry, in `Six Lectures in Commutative
Algebra', Birkhauser.
- K. Schwede and K. Tucker: A survey of test ideals,
http://www.arxiv.org/abs/1104.2000
- K. E. Smith: Vanishing, singularities and effective bounds via prime
characteristic local algebra.
http://www.math.lsa.umich.edu/~kesmith/paperlist.html
- K. E. Smith: Tight closure and vanishing theorems, ICTP School on
Vanishing Theorems and Effective Methods in Commutative Algebra.
http://publications.ictp.it/lns/vol6/vol6toc.html
- K. E. Smith and W. Zhang: Frobenius splitting in commutative algebra,
http://www.arxiv.org/abs/1409.1169
- Grading:
Two take-home exams (30% each) |
60% |
Presentation/written final exam |
20% |
Oral exam |
20% |
CMI |
Manoj's Home-page |
Last modified:
Tue Jan 5 12:10:25 IST 2016