12/12/03: Normal Noetherian integral domains are intersections
of DVRs. Geometric interpretation. [E: Section 11.2]
12/12/03: Normal Noetherian integral domains are intersections
of DVRs. Geometric interpretation. [E: Section 11.2]
12/10/03: Groebner bases (presentation by Hazem Hamdan). [http://www.math.umn.edu/~hamdan/grobnerpres.pdf;
E: Chapter 15]
12/08/03: Monomial ideals (presentation by Hyeung-joon
Kim). [E: Section 15.1 and J. Eagon and M. Hochster, R-sequences and
indeterminates. Quarterly J. of Math. 25 (1974), 61-71]
12/05/03: Noetherian valuation rings are DVRs. Examples of
general valuation rings. [E: Exercises 11.3-4; AM: Exercise 9.3, the
beginning of Section "Valuation rings" from Chapter 5]
12/03/03: A proof of the main theorem on DVRs (a Noetherian,
normal integral domain A with Spec A = {0, m}). General valuation
rings. [E: Section 11.2 through the proof of Theorem 11.2, Exercises
11.1-2; AM: Section "Discrete valuation rings" from Chapter 9,
Exercise 9.4, the beginning of Section "Valuation rings" from Chapter
5]
12/01/03: Properties of discrete valuation rings (DVRs). A
criterion for a DVR (a local Noetherian integral domain with a
principle maximal ideal). The main theorem on DVRs (a Noetherian,
normal integral domain A with Spec A = {0, m}; no proof yet). [E:
Section 11.2 before the proof of Theorem 11.2; AM: Chapter 9 before
the proof of Proposition 9.2]
11/26/03: Localization and primary ideals. The second
uniqueness theorem. Discrete valuation rings (DVRs): definitions. [E:
Section 3.3, especially Theorem 3.10.d and its proof, Section 11.1;
AM: Chapter 4 (after Theorem 4.5 through the end) and Chapter 9
through the beginning of Section "Discrete valuation rings"]
11/24/03: The cone example. Noether's theorem on the existence
of primary decomposition. Primary decomposition of I and Ass(A/I): the
first uniqueness theorem. [E: Section 3.3, especially Theorem 3.10 and
its proof; AM: Chapter 4 (through Theorem 4.5) and Section "Primary
decomposition in Noetherian rings" from Chapter 7]
11/21/03: Primary decomposition. Shortest primary
decomposition. Geometric interpretation in general. Primary
decompositions for I = (X^2, XY): geometric interpretation and
nonuniqueness. [E: Section 3.3, especially Corollary 3.8 and Theorem
3.10 (no proof yet), Sections 3.7 and 3.8; AM: Chapter 4 (through the
definition of primary decomposition, Example after Theorem 4.5 and
Remarks after Proposition 4.6)]
11/19/03: P-primary ideals. P-primary ideals and powers of
P. If the radical of Q is maximal, then Q is primary. Example of (X^2,
XY) in k[X,Y]. Primary ideals and Ass. [E: beginning of Section 3.3;
AM: beginning of Chapter 4 (through Proposition 4.2)]
11/17/03: Relation between of Ass M and Supp M. Decomposition
of Supp M as the union of irreducible closed sets corresponding to the
minimal primes containing Ann M; these minimal primes are also in Ass
M. Disassembling a module. Primary ideals. [E: Section 3.1, Section
3.2, especially Proposition 3.7, beginning of Section 3.3; AM:
beginning of Chapter 4 (through Example 1)]
11/14/03: The zerodivisors and Ass M. Ass M and union Ass L
and Ass M/L. Relation between of Ass M and Supp M. Geometric
interpretation. [E: Section 3.1; AM: Remark after Proposition 4.6;
Matsumura, Commutative Ring Theory: Theorems 6.1 and 6.5 and a remark
after it]
11/12/03: Geometric interpretation of Supp M. Associated
primes and the assassin Ass M. Properties of Ass M. [E: p. 67, Section
3.1, Proposition 3.4, Lemma 3.6; AM: Proposition 4.6]
11/10/03: Homework discussion: Problem 8. Towards primary
decomposition: Supp M and Ann M. [E: Lemma 13.2.c, p. 67; AM:
Exercises 5.16 and 3.19]
11/07/03: Homework discussion: Problems 15, 4, and 8. [E:
pp. 35-36, Lemma 13.2.c; AM: Exercise 1.28, Remark after Theorem 7.5,
Exercise 5.16]
11/05/03: Localization commutes with taking
quotients. Iterated localization. Examples. The idea of primary
decomposition. [E: Introduction to Chapter 3; AM: Exercise 3.4, the
first two paragraphs of Chapter 4]
11/03/03: Problem #7 (presented by Wenliang Zhang). Modules of
fractions. Exactness of S^{-1}. [E: Section 2.1, Proposition 2.5,
Corollary 2.6, Exercises 2.1, 2.8, 2.9; AM: Proposition 3.3 and
Corollary 3.4 in Chapter 3]
10/31/03: Restriction and extension of ideals. Ideals in A and
S^{-1} A. The description of Spec S^{-1} A as a subset of Spec
A. Localization at a prime ideal. Examples: Z_(p) and localization of
the ring of polynomials as germs of rational functions defined near
V(P), regular at a generic point of V(P). [E: Introduction to Chapter
2, Section 2.1, Exercise 2.3; AM: Section "Extending and contracting
ideals in rings of fractions" and Examples after Corollary 3.2 in
Chapter 3]
10/29/03: Rings of fractions S^{-1} A and their
properties. [E: Section 2.1 through top of p. 61, Exercises 2.2, 2.7;
AM: Chapter 3 through Corollary 3.2]
10/27/03: Varieties versus Spec A: summary. [E: Section
1.6]
10/24/03: Decomposition of a variety into finitely many
irreducibles. The Zariski topology on Spec A. An "easy strong
Nullstellensatz": I(V(J)) = rad J for Spec A. Spec A for a Noetherian
ring. [E: Subsection 2 on p .88, Section 3.8, Corollary 2.12, Exercise
1.24; AM: Proposition 1.14, Exercises 1.15-16, 1.18-20, 6.7-9]
10/22/03: The Nullstellensatz and Spec A. The Zariski topology
on k^n and a variety. The Zariski topology is Noetherian. [E:
p. 32-34, Subsection 2 on p. 88; AM: Exercises 6.5 and 6.8]
10/20/03: A proof of the Nullstellensatz. Irreducible
varieties and prime ideals. [E: p. 32, Subsection 2 on p. 88; AM:
Exercises 7.14 and 1.19]
10/17/03: Affine varieties. Maximal ideals of a finitely
generated algebra over an algebraically closed field and points of the
corresponding variety. The case of an algebraically nonclosed
field. The Nullstellensatz (no proof yet). Comparison to the "easy"
Nullstellensatz: rad J = the intersection of primes containing J. [E:
Sections 1.6, 4.5 (Corollary 1.9 and Theorem 1.6), and 13.2; AM:
Exercises 1.27 and 7.14]
10/15/03: Field extensions. The weak Nullstellensatz. Maximal
ideals of k[X_1,...,X_n]. Maximal ideals of k[X_1,...,X_n] and points
of k^n when k is algebraically closed. [E: Section 13.2 (Corollary
13.12.i only); AM: Proposition 5.23, Corollary 5.24, Exercises
5.17-19, Proposition 7.9, and Corollary 7.10]
10/13/03: Proof of the Noether normalization lemma, continued
(proof of Main Claim). Geometric interpretation: example of the
hyperbola XY = 1. [E: Section 13.1 (Lemma 13.2 and Theorem 13.3); AM:
Exercise 5.16]
10/10/03: Review of Problem Set 1: Problem 4.27 from [E]
(presented by Wenliang Zhang). Noether normalization. [E: Section 13.1
(mainly Theorem 13.3); AM: Exercise 5.16]
10/08/03: Review of Problem Set 1. Proposition 4.11 from [E]:
if the coefficients of the product of two monic polynomials are
integral, then the coefficients of each are integral. [E: Proposition
4.11; AM: Exercise 5.8]
10/06/03: Integral closure, integrally closed rings, normal
domains, normalization, the ring of integers of a number
field. Examples. Normalization and desingularization: preview. [E:
pp. 117-123, Section 4.2 (Propositions 4.10 and 4.11, Corollary 4.12),
Section 4.3; AM: Definition of an integrally closed domain (which we
called a normal domain) from Section "Integrally closed domains" in
Chapter 5]
10/03/03: Integral elements, integral extensions, and finite
algebras. Examples. Tower Laws. [E: pp. 117-119, Section 4.1
(Corollaries 4.5 and 4.6); AM: Section "Integral dependence" from
Chapter 5]
10/01/03: The Hilbert Basis Theorem. Algebraic and integral
dependence. Algebras over rings, extension rings. [E: Section 1.4,
Introduction to Chapter 4, p. 13; AM: the first section of Chapter 7,
the first page of Chapter 5, Section "Algebras" from Chapter 2]
09/29/03: The ring of germs of continuous functions,
continued. Noetherian modules. Exact sequences. Modules over
Noetherian rings. [E: pp. 16-17, 28, Proposition 1.4, Exercises 1.1,
1.3; AM: Section "Exact sequences" from Chapter 2, Chapter 6 through
Proposition 6.6]
09/26/03: The ascending chain condition (ACC). Noetherian
rings. Examples of non-Noetherian rings. The ring of germs of
continuous functions. [E: p. 27; AM: Propositions 6.1, 7.1, and
7.2]
09/24/03: The Cayley-Hamilton Theorem. A generalization and
corollaries. Nakayama's Lemma. [E: Section 4.1 (Theorem 4.3,
Corollaries 4.4, 4.7, 4.8, and Warning); AM: Section "Finitely
generated modules" from Chapter 2]
09/22/03: Modules. Generators of modules. Free
modules. Examples. [E: Section 0.3; AM: Chapter 2 through the
beginning of Section "Finitely generated modules"]
09/15/03: Local rings. Examples. Power series rings. [E:
Introduction to Chapter 2; AM: Chapter 1, the end of Section "Prime
ideals and maximal ideals", Exercise 5]
09/10/03: Zorn's Lemma. The existence of maximal and prime
ideals. Nilpotents and nilradical. The radical of an ideal. [E:
Section 2.3, p. 33; AM: Chapter 1, Sections "Prime ideals and maximal
ideals", "Nilradical and the Jacobson radical", and the "radical" part
of "Operations on ideals"]
09/08/03: The description of Spec k[X,Y] and Spec
Z[X]. Geometric interpretation. [E: Section 1.6; AM: Exercise 16 in
Chapter 1]
09/05/03:
Spec A. Examples, Spec k[X,Y] and Spec Z[X]. [E: pp. 54-55; AM:
Exercise 16 in Chapter 1]
09/03/03:
Introduction. Ideals, prime and maximal ideals. [E:
1.1, 1.2, 0.1; AM: Chapter 1 through the beginning of
section "Prime ideals and maximal ideals"]