Math 627B: Modern Algebra II
Spring 2006

Instr: Mike O'Sullivan


I will try to keep to this schedule but will update it as needed.
Day Topics Sections in IVA
Wed. 1/18 Introduction: to algebraic geometry:
classification, singularities, parametrization.
Mon. 1/23 Algebra: algebraically closed fields, polynomial rings,
  field of fractions.
Geometry: varieties, their intersection and union.
Wed. 1/25 Parametrization and Implicitization.
Mon. 1/30 Ideals in polynomial rings. I.4.
Wed. 2/1 Univariate polynomial rings versus multivariate polynomial rings.
The Euclidean algorithm.
Mon. 2/6 Monomial Orderings II.1,2
Fri. 2/10 Division in multivariate polynomial rings.
Division Algorithm in Magma.
Wed. 2/15 Dickson's Lemma. II.4
Fri. 2/17 Hilbert basis theorem, Grobner basis. II.5-8
Wed. 2/22 S-polynomials and Grobner basis.
Buchberger's algorithm.
Fri. 2/24 Rings and modules, presentations and syzygys. II.6,8,9
Mon. 2/27 The syzygy module for monomial ideals.
Relation to reduction by G
II.1, 2
Wed. 3/1 Improving Buchberger's algorithm.
Elimination and extension.
Mon. 3/6 Elimination, extension and closure theorems.
Sylvester matrix
III.2, 5
Wed. 3/8 Sylvester matrix, resultants.
Proof of the Extension theorem.
Mon. 3/20 Implicitization. III.3
Wed. 3/22 Radical ideals and Hilbert's Nullstellensatz.
Zariski closure.
Mon. 3/27 The algebra-geometry dictionary.
Sums, products and intersections of ideals.
Wed. 3/29 Computation of the intersection of two ideals.
Proof of the closure theorem.
Mon. 4/3 Ideal quotients, their computation
and the geometric interpretation.
Wed. 4/5 Cancelled. .
Mon. 4/10 Irreducible varieties and minimal decomposition. IV.5,6
Wed. 4/12 Ring homomorphisms and quotients.
The coordinate ring of a variety.
Mon. 4/17 The spectrum of a ring,
the Zariski topology
Wed. 4/19 Finite varieties. Polynomial and rational maps of varieties. V.5
Mon. 4/24 Bezout's theorem as an introduction to algebraic geometry. .
Wed. 4/26 Cone varieties,homogeneous ideals, the cone over a variety. VIII.1-4.
Wed. 5/1 Projective geometry. VIII.1-4.

IVA is

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

2nd Ed., Springer-Verlag, 1997.