Math 621: Commutative Algebra and Algebraic Geometry
Spring 2025

Instr: Mike O'Sullivan


Schedule

I will try to keep to this schedule but will update it as needed.

[CR] E. Clader, D. Ross, Beginning in Algebraic Geometry , 2025.
[OS1] O'Sullivan On Beyond Z : Groups, Rings and Fields, 2025.
[OS2] O'Sullivan Notes on Algebraic Geometry, 2025.
Day Topics Reading
Tu. 1/21 Ideals and Varieties, an introduction to the course. [CR] Ch. 1
Th. 1/23 The integers and K[x] .
Key properties.		 [OS1 1.1, 1.3] [CR 0.1, 0,2, 0.5]
Tu. 1/28 Group work: Division, factorization etc [OS1 1.1, 1.3] [CR 0.1, 0,2, 0.5]
Th. 1/30 Fields, rings, algebras, ideals.
Homomorphisms, quotient rings; maximal, prime, radical ideals. 		[CR 0.3-4, 3.4] [OS Ch. 4, 5].
Tu. 2/4 Group work.
Function fields, K -algebra homomorphisms		 [CR 0.3-4, 3.4] [OS Ch. 4, 5].
Th. 2/6 Noetherian rings. Monomial orderings, Groebner bases. [CR 2.2]; [OS2].
Tu. 2/11 Group work.
Monomial orderings, Groebner bases.		 [CR 2.2]; [OS2].
Th. 2/13 Unique Factorization.
Irreducible Decomposition of Varieties.		 [CR: 0.6, 2.1,3,4].
Tu. 2/18 Group work
Unique Factorization.
Irreducible Decomposition of Varieties.		 [CR: 0.6, 2.1,3,4].
Tu. 2/20 Coordinate rings of varieties.
[CR: 3.1-4].
Tu. 2/25 Polynomial maps of varieties
and ring homomorphisms. 		[CR: Ch 3,4].
Th. 2/27 Group work
Polynomial amps and ring homomorphisms. 		[CR: Ch 3,4].
Tu. 3/4 The Nullstellensatz and
Noether Normalization. 		[CR: Ch 5].
Th. 3/6 Group work
Nullstellensatz and Noether basis. 		[CR: Ch 5].
Tu. 3/11 Tangent spaces to varieties. [CR: Ch 6].
Th. 3/13 Group work
Tangent spaces to varieties. 		[CR: Ch 6].