Characteristic-p Methods

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.

• Plan of lectures: http://www.cmi.ac.in/~mkummini/courses/2015_16/term2/schedule_char_p.txt
• 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%