Math 627B: Modern Algebra II
Spring 2005

Instr: Mike O'Sullivan


I will try to keep to this schedule but will update it as needed.
Day Topics Sections in IVA
Mon. 1/24 Introduction: some problems in algebraic geometry. .
Wed. 1/26 Polynomial rings and affine varieties. I.1,2
Mon. 1/31 Parametrization and Implicitization.
Wed. 2/2 Ideals in polynomial rings. I.4.
Mon. 2/7 Univariate polynomial rings versus multivariate polynomial rings. I.5.
Wed. 2/9 Monomial Orderings II.1,2
Mon. 2/14 Division in multivariate polynomial rings. II.3
Wed. 2/16 Dickson's Lemma. II.4
Mon. 2/21 Hilbert basis theorem, Grobner basis
and applications.
Wed. 2/23 S-polynomials and Grobner basis. II.6,8
Mon. 2/28 Buchberger's algorithm. The syzygy module. II.6,8,9
Wed. 3/2 Bases for the syzygy module. II.9
II.1, 2
Mon. 3/7 Elimination theorem. Extension theorem
Closure theorem
Wed. 3/9 Parametrization III.3
Mon. 3/14 Sylvester matrix, resultants. II.5
Wed. 3/16 Proof of the Extension theorem. III.6
Mon. 3/21 Hilbert's Nullstellensatz.
The algebra-geometry dictionary.
Zariski topology.
Wed. 3/23 Sums, products and intersections of ideals.
Ideals and homomorphisms of rings.
Computation of the intersection of two ideals.
Mon. 4/4 Ring homomorphisms and quotients.
The coordinate ring of a variety.
Wed. 4/6 Finite varieties.
Radical, prime, maximal ideals.
Mon. 4/11 Irreducible varieties and minimal decomposition. V.5,6
Wed. 4/13 Short project presentations. .
Mon. 4/18 Colon ideal and set difference of two varieties.
Project presentations.
Wed. 4/20 p-adic numbers and formal power series. .
Mon. 4/25 Cone varieties,homogeneous ideals, the cone over a variety. VIII.1-4.
Wed. 4/27 Projective geometry. VIII.1-4.
Mon. 5/2 Polynomial maps on a variety. The coordinate ring. V.1-4.
Wed. 5/4 Rational maps and parametrization. .
Mon. 5/9 Shayne, Raymond, V.1-4.
Wed. 5/11 Val, Marcel, Jeremy. .
Mon. 5/16 Colleen, Dan, David, Alina. .