Math 252: Commutative Algebra and Algebraic Geometry (Spring 2002)

Instructor

Description

A list of notions from algebraic geometry which will be covered is included below. We will restrict attention to algebraic varieties over algebraically closed fields. Mostly we will work with the category of affine varieties, although projective varieties will be touched upon.

Geometric topics

The Zariski topology on affine space, affine varieties, Noetherian topological spaces, Krull dimension, irreduciblity, polynomial maps between affine varieties, closed embeddings, dominant maps, product varieties, principal open subsets, affine algebraic groups, tori, affine toric varieties, the tangent space, finite maps, closed maps, quotients by finite group actions, constructible sets, images of polynomial maps, bounds on dimensions of intersections, the singular locus, open maps, flat maps, dimensions of fibers of polynomial maps, projective varieties, Hilbert functions, Bezout's theorem for intersection with a projective hypersurface, tangent cones, foundations of the theory of divisors on a smooth complex analytic space or variety (divisors will only be introduced in the algebraic geometry course; we just do the commutative algebra needed to get the theory going).

In order to treat these geometric topics and others we cover the following

Algebraic Topics

Prime, maximal, and radical ideals, algebras and the category of finitely generated, reduced algebras over an algebraically closed field, statements of the basis theorem and the Nullstellensatz, contraction and extension of ideals, Chinese Remainder Theorem, review of Groebner bases and applications. Basic operations on modules, a detailed treatment of tensor product of modules over a commutative ring, tensor products of algebras. A fairly thorough treatment of rings of fractions. The Zariski cotangent space. Cayley-Hamilton theorem for endomorphisms of modules and variants, a fairly thorough treatment of integral extensions, Krull dimension, Noether normalization, proof of the Nullstellensatz, rings of invariants, transcendence degree of field extensions, a very breif treatment of integral closure, going down, height of prime ideals, catenary rings, Nakayama's lemma, Hauptidealsatz, dimension theory for Noetherian local rings, a brief introduction to flat morphisms, a structure theorem for finitely generated modules over a Noetherian ring, generic flatness, regular local rings, the Jacobian criterion. Graded rings, homogeneous ideals, homogenization and dehomogenization, Hilbert polynomials, multiplicity of modules at prime ideals. Associated graded ring, I-adic completion, Artin-Rees Lemma, Krull's intersection theorem, Hensel's lemma, power series rings, Weierstrass preparation theorem, unique factorization in geometric regular local rings.

The following basic topics in commutative algebra are either not treated at all in Math 252 or are treated very briefly, because they receive more thorough treatment in the algebraic number theory course: Primary decomposition, integral closure, Dedekind domains, discrete valuation rings, valuation theory, ramification theory, discriminant and different, orders in number fields, Picard groups, the spectrum of a ring.

The following topics are strictly algebraic geometry topics and are treated in the algebraic geometry course which has Math 252 as a prerequisite: Coherent sheaves, quasi-projective varieties and their morphisms, rational maps, Weil and Cartier divisors, intersection theory.

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Homework

Prerequisites

Last modified: 17 October 2001