Math 423/502 - Algebra II

Problem set #1

Problem set #2

Problem set #3

Problem set #4

Problem set #5

Problem set #6

Problem set #7

Tue, Jan 5. Definition of affine space and closed algebraic sets in the affine space. Examples of closed algebraic sets.

Thu, Jan 7. Correspondence between algebra and geometry. Statement of Hilbert Nullstellensatz.

Tue, Jan 12. Proof of the Hilbert Basis theorem. Definition of Zariski topology on the affine space. Irreducible closed sets and corresponding prime ideals.

Thu, Jan 14. Decomposition into irreducible components. Zariski topology is Noetherian. Statement of the full Hilbert Nullstellensatz and reduction to to lemmas.

Tue, Jan 19. Proof of Hilbert Nullstellensatz. Commutative algebra of finite ring extensions. Statement of Noether normalization.

Tue, Jan 26. Proof of Noether normalization. Separable field extensions in the case of positive characteristic. Primitive element theorem.

Thu, Jan 28. Polynomial functions and polynoimal maps. Correspondence between affine varieties and k-algebras.

Tue, Feb 2. Finite maps and Noether normalization. Products of varieties. Rational functions. Domains of definition.

Thu, Feb 4. Rational maps. Birational equivalence. Quasi-affine varieties.

Tue, Feb 9. Projective spaces. Homogeneous coordinates. Projective varieties.

Thu, Feb 11. Graded rings and homogeneous ideals. Projective hilbert nullstellensatz. Rational functions on projective varieties.

Tue, Mar 2. Rational and regular maps of projective varieties.

Thu, Mar 4. Examples of rational and regular maps of projective varieties. Rational normal curve, Veronese map, Segre map, blowup of a point.

Tue, Mar 9. Tangent spaces and smoothness of affine varieties.