This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.
Math 601 or the equivalent.
In any given instance the course will cover most (but possibly not all topics) below.
- Basic algebraic notions: ideals (maximal, prime, radical, primary, irreducible), commutative rings (Noetherian, reduced), finitely generated algebras, algebra homomorphisms, Hilbert basis theorem, primary decomposition.
- Basic topology: spectrum of a commutative ring (as a topological space), Noetherian topological spaces, irreducible topological spaces, Krull dimension.
- Classical geometric notions: affine n-space over an algebraically closed field, affine algebraic sets, polynomial maps between affine algebraic sets.
- Groebner bases: monomial orders, definition of Groebner basis, monomial ideals, uniqueness of reduced Groebner bases, applications using relevant computer software.
- Modules: Hom and tensor product, exactness properties, standard examples of flatness, extension of scalars, tensor products of algebras.
- 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.
- 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.
- Numerical invariants: Hilbert function, Hilbert series, Hilbert polynomial, dimension via Hilbert polynomial and via transcendence degree.
- 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.
- 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.
- Resolutions and syzygies: free and injective resolutions, Betti numbers, syzygies, Koszul complexes, Hilbert syzygy theorem, relation to Hilbert series.
- Introduction to Commutative Algebra, Atiyah & MacDonald.
- Commutative Algebra: with a view toward Algebraic Geometry, Eisenbud.