Math 627B: Modern Algebra II
Spring 2009

Instr: Mike O'Sullivan


I will try to keep to this schedule but will update it as needed.

IVA is

Cox, Little, O'Shea Ideals, Varieties, and Algorithms:
An Introduction to Computational Algebraic Geometry and Commutative Algebra

2nd Ed., Springer-Verlag, 1997.

H is

Hungerford Abstract Algebra: An Introduction, Hungerford. 2nd Ed., Harcourt, 1997.

A is

Ash, Abstract Algebra: The basic graduate year , online.

Day Topics Sections in IVA
Mon. 1/26 The polynomial ring in one variable:
Algbebraic structure
IVA 1.5, H. Ch. 4
Wed. 1/28 The polynomial ring in one variable:
Algorithms: Division, Euclidean, Radical
Geometry: Roots of a univariate polynomial.
IVA 1.5, H. Ch. 4
Mon. 2/2 Ideals in polynomial rings.
Varieties: Solutions of a ideals.
IVA 1.1-2, 4
Wed. 2/4 Parametrization and Implicitization.
I.2,3. II.1
Wed. 2/11 Monomial Orderings
Division in multivariate polynomial rings.
II.2, 3
Fri. 2/13 A menagerie of monomial orderings.
Division Algorithm in Magma.
II.2, 3
Mon. 2/16 Dickson's Lemma. II.4
Wed. 2/18 Hilbert basis theorem, Grobner basis. II.5-8
Mon. 2/23 S-polynomials and Grobner basis.
Buchberger's algorithm.
Wed. 2/25 Rings, ideals, prime and maximal ideals
Homomorphisms, quotient rings.
H Ch. 6
A Ch. 2
Mon. 3/2 Unique factorization domains. A Ch. 2
Wed. 3/4 Localization, rings of fractions. A Ch. 2
Mon. 3/9 R a UFD implies R[x] a UFD. A Ch. 2
Wed. 3/11 Modules over a ring. A Ch. 4.1,2,3,7 (just bits of each)
Mon. 3/16 Presentations of a module and exact sequences. A Ch. 4.1,2,3,7 (just bits of each)
Wed. 3/18 The syzygy module for monomial ideals.
Relation to reduction by G
Mon. 3/23 Sylvester matrix, resultants. III.5,6
Wed. 3/25 Proof of the Extension theorem. III.5,6
Mon. 4/6 Implicitization of parametrically
defined varieties.
Wed. 4/8 Radical ideals and Hilbert's Nullstellensatz.
Zariski closure.
Mon. 4/13 The algebra-geometry dictionary.
Sums, products and intersections of ideals.
Wed. 4/15 Computation of the intersection of two ideals.
Proof of the closure theorem. Ideal quotients, and the geometric interpretation.
Mon. 4/20 Irreducible varieties and minimal decomposition. IV.5,6
Wed. 4/22 Polynomial maps on varieties.
The coordinate ring of a variety.
Mon. 4/27 Computing in the coordinate ring of a variety.
Finite varieties.
Wed. 4/29 Cone varieties, homogeneous ideals, the cone over a variety. VIII.1-4x.
Mon. 5/4 Projective geometry. VIII.1-4.
Wed. 5/6 Projective geometry and parametrized varieties. VIII.1-4.
Mon. 5/11 No class: discuss projects with me. .
Wed. 5/13 No class: discuss projects with me. .