# Math 602 Syllabus

This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.

#### Prerequisites

Math 601 or the equivalent.

#### Syllabus

In any given instance the course will cover most (but possibly not all topics) below.

1. Basic algebraic notions: ideals (maximal, prime, radical, primary, irreducible), commutative rings (Noetherian, reduced), finitely generated algebras, algebra homomorphisms, Hilbert basis theorem, primary decomposition.
2. Basic topology: spectrum of a commutative ring (as a topological space), Noetherian topological spaces, irreducible topological spaces, Krull dimension.
3. Classical geometric notions: affine n-space over an algebraically closed field, affine algebraic sets, polynomial maps between affine algebraic sets.
4. Groebner bases: monomial orders, definition of Groebner  basis, monomial ideals, uniqueness of reduced Groebner bases, applications using relevant computer software.
5. Modules: Hom and tensor product, exactness properties, standard examples of flatness, extension of scalars, tensor products of algebras.
6. Localization: multiplicative sets, rings of fractions, universal mapping property, behavior of ideals under localization, geometric interpretation of localization at a prime ideal, Nakayama's Lemma, total ring of fractions, modules of fractions, local criterion for a module to be zero.
7. Finiteness and integrality: finite and integral extensions of rings, going up, preservation of Krull dimension in finite extensions, Noether normalization, Hilbert Nullstellensatz, dimension of affine space.
8. Numerical invariants: Hilbert function, Hilbert series, Hilbert polynomial, dimension via Hilbert polynomial and via transcendence degree.
9. Filtrations: Artin-Rees Lemma, I-adic completion, Krull's intersection theorem, dimension via chains of prime ideals, systems of parameters, Hilbert polynomials. Associated graded rings, regular local rings, formal power series rings including unique factorization.
10. Global aspects of dimension: going down, local dimension equals global dimension for finitely generated algebras over a field, height of an ideal, Krull's principal ideal theorem.
11. Resolutions and syzygies: free and injective resolutions, Betti numbers, syzygies, Koszul complexes, Hilbert syzygy theorem, relation to Hilbert series.

#### References

• Introduction to Commutative Algebra, Atiyah & MacDonald.
• Commutative Algebra: with a view toward Algebraic Geometry, Eisenbud.