Algebra przemienna, wyklad z cwiczeniami, semestr jesienny 2014/2015

Commutative algebra, lectures and exercises, Fall 2014/2015

Jaroslaw Wisniewski

Tuesdays, 12:15 - 15:45, room 3320, Banacha 2

picture
orderings, 6 choose 3
source: Wikipedia

Oficjalna strona przedmiotu

Books and notes

  • Atiyah, Macdonald, Introduction to commutative algebra, Addison-Wesley
  • Allufi, Algebra, Chapter 0, AMS
  • Cox, Little, O'Shea, Ideals, varieties, and algorithms. Undergrad Texts in Math. Springer
  • Milne, Fields and Galois Theory
  • Reid, Undergraduate Commutative Algebra, LMS Students texts 29
  • Bialynicki-Birula, Algebra, PWN
  • Bialynicki-Birula, Zarys algebry, PWN
  • Brynski, Jurkiewicz, Zbior zadan z algebry
  • Eisenbud, Commutative algebra with a view, Springer GTM150
  • Lang, Algebra, PWN

Topics:

  1. Basic info about categories and functors. Review on rings and modules.
  2. Modules: homs and tensors. Localization of rings and modules. [Atiyah, Macdonald, ch. 2, 3; Reid, ch. 2, 6]
  3. Noetherian rings and modules, basic properties. Hilbert Basis thm. [Atiyah, Macdonald, ch. 6,; Reid, ch. 3].
  4. Extensions of fields and rings: algebraic, integral, finite. Noether Normalization thm. [Atiyah, Macdonald, ch.5, 6; Reid, ch. 4].
  5. Hilbert Nullstellensatz. Affine varieties. Irreducibility. [Reid, Ch.5].
  6. Rings of polynomials, orders, division algorithm, Groebner bases, symmetric polynomials. [Cox, Little, O'Shea, Ch. 2]
  7. Linear group actions on polynomial rings and rings of invariants, Reynolds operators, theorem of Hilbert. [Cox, Little, O'Shea, Ch. 7].
  8. Extensions of fields. Algebraic closure. Normal extensions. [Milne; Aluffi, Ch. VII].
  9. Separable extensions. Introduction to Galois theory. [Milne; Aluffi, Ch. VII].
  10. Applications of Galois theory. [Aluffi, Ch VII].
  11. Normalization of f.g. k-algebras. Disrete valuation rings. [Reid, ch. 8, ]
  12. Decomposition of modules and ideals in noetherian rings. [Reid, ch. 7, Atiyah, Macdonald ch. 7]

Problem sheets:

  1. Set 1: review on rings and ideals, due Oct. 7th.
  2. Set 2: basics on categories and functors, due Oct. 14th.
  3. Set 3: presheaves, sheaves, limits, due Oct. 21st.
  4. Set 4: modules, homs and tensors, due Oct. 28st.
  5. Set 5: Notherian, Artinian, rings, due Nov. 4th.
  6. Set 6: Monomial ideals in rings of polynomials, due Nov. 11th.
  7. Set 7: Integral extensions and normalization, due Nov. 18th.
  8. Set 8: Ideals in polynomial rings, symmetric polynomials, due Nov. 25th.
  9. Set 9: Group actions and rings of invariants, due Dec. 9.
  10. Set 10: Introduction to fields, due Dec. 16.
  11. Set 11: Extensions of fields, due Jan 13.

Written exam: Monday, February 23, at 3170.

General assessment criteria. The participants will prepare homework (a dozen of problems each week) to be discussed at the exercise sessions. The problems for the Tuesday's session will be posted on Friday the latest. In addition there will be two mid-term exams. Evaluation based on the students' work during the semester plus the final exam (written and oral). Read details in the small print in Polish below.

SAGE presentations:
Register if you want to use Sage at MIMUW

  1. Wstep, powtorzenie z teorii grup
  2. Porzadki, bazy Groebnera; przyklad zastosowania: znajdowanie ekstremum funkcji (autor: Maria Donten-Bury)
  3. Niezmienniki grup skonczonych oraz invariants of finite groups.

Programy i pakiety: