MA8202 Kommutativ Algebra, hoesten 2006

News:

Program is there!

On Friday, 10.11, no lecture,
but -- the TASK session!

On Friday (03.11)
we have to move to room 822 !

Time for Lectures :

  • wen - 8:15-10:00, r.734
  • fri - 10:15-12:00, r.734
    (but r.822 on 22.09, 3.11).
  • Last lecture -- 24.11.

  • Examen -- 01.12.

  • Lecturer: prof. Alexei Rudakov,
    * the office room number 854,
    * e-mail: rudakov@math.ntnu.no,
  • Lectures will be in English !
    --
  • Examen will be oral on Dec.1, 2006.

    • TEXTBOOK:
  • [CRT] : "Commutative ring theory" by Hideyki Matsumura, Cambridge Univ.Press.
    TASKS. More tasks:_ps __pdf
    "Final" PROGRAM (Nov.3).
    We go through the main topics in the first five chapters of [CRT]:
  • Commutative rings, ideals, units. Maximal and prime ideals. [Th.1.1,1.2]
    Local rings. Radicals. Ring of formal power series.
    Modern Ch.remainder Th. [Th.1.3,1.4]
    Units in the ring of power series. [p.4]
  • Rings of fractions. Localization of rings and modules. [pp.20-21,25]
    Ideals in A_S. [Th.4.1]
  • Modules, homomorphisms. Finitely generated modules.
    "Determinant trick" theorem and Nakayama lemma. [Th.2.1,2.2]
    Modules over a local ring. [Th.2.3]
    Epimirphism theorem of Vasconselos. [Th.2.4]
  • Length of a module. Properties of length. [p.12]
    Short exact sequences and long exact sequences.
    "Exactness via Hom" propositions.
    The length identity for a long exact sequence.
    Localization M-->M_S is exact functor.
  • Chain conditions. Noetherian rings and modules. [Th.3.1]
    The ring of polynomials over a Noetherian ring. [Th.3.3]
    The ring of power series over a Noetherian ring. [Th.3.3]
    If M is Noetherian, then A/ann M is Noetherian. [Th.3.5]
    Artinian ring has finite number of maximal ideals and finite length. [Th.3.2]
  • Spectrum of a ring. Zarisski topology. [pp.24-25]
    Support of a module. Support for modules in an exact sequence.
  • Associated primes. Basic properties. [Th.6.1,6.2,6.3]
    Fundamental theorems for a finite module over an Noetherian ring. [Th.6.4,6.5]
  • Tensor products: bilinear maps, definition of the tensor product, "right-exactness". [Appendix A, pp.266-269]
    Localisation equals tensor product [Th.4.4].
  • Open sets defined by a module [Th.4.10(i)].
  • Irreducible closed sets. Noetherian spaces [p.29, Exerc.4.9-4.11]. "V(I) irr iff rad(I) prime."
  • Combinatorial and Krull dimensions [pp.30-31]. Examples.
  • Algebraic and transcendent elements. A theorem about an algebraic extension: [Th.5.1].
  • Algebraically independent sets. Transcedence bases.
    Transcedence degree theorem: all transcedence bases have the same number of elements.
    "Not a field" theorem: [Th.5.2].
  • The Hilbert Nullstellensatz theorems: [Th.5.3, 5.4, 5.5]. "dim = tr.deg" [Th.5.6]
  • Algebraic integers. Example: Gauss integers and sums of squares.
    Integral elements: [Th.9.1], integral extensions, integral closure [p.64].
    Some results about integral closure: [p.65: Exam.1,3, Th.9.2].
  • Fundamental lemmas about integral extensions and "no inclusions"-theorem [p.66, L.1,2, Th.9.3(i),(ii)].
  • The case of the finite Galois extension: "primes lying over" [Th.9.3(iii), only for finite Galois extension ].
  • "Going up"-theorem: [Th.9.4(i)]. "dim A = dim B for an integral extension"[p.69, Exerc.9.2].
  • Valuations (discrete only), basic properties.
    The ring and ideal of a valuation, the factorisation formula. (Compare with [p.75, p.77, Th.10.7]).
  • The approximation theorem and "independence of valuations" (and related lemmas).
  • Fundamental theorem about DVR: [Th.11.2, we suppose R is a domain.]
  • Normal rings [p.64], height of a prime ideal [p.30], and
    the fundamental theorem for Noetherian normal domains: [Th.11.5, Th.12.4(i)].
  • Fractional ideals, invertible ideals: [p.80, Th.11.3(1,3),Th.11.4].
  • Dedekind rings, the fundamental factorization theorem: [p.82, Th.11.6, we give different proof for (3)].
  • The Krull-Akizuki theorem and Corollary -- "factorization" for algebraic numbers: [pp.84-85, Th.11.7, Cor.].
  • Graded rings and Hilbert function [sec.13, pp.92-96].
  • Semilocal rings and Samuel function [sec.13, pp.97-98], the Artin-Rees argument.
  • The fundamental "dim=deg" theorem [Th.13.4, pp.98-100] and the Krull dimension theorem (Hauptidealsatz) [Th.13.5, p.100].
  • Returning to graded rings: homogeneous prime ideals and "dim A = dim gr(A)" theorems [Th.13.7(i), 13.8, 13.9, pp.101-102].

    The page is under construction !! Last change 03.11.2006