Math 621: Commutative Algebra and Algebraic Geometry
Spring 2023

Instr: Mike O'Sullivan


Schedule

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

[IVA] Cox, Little, O'Shea Ideals, Varieties, and Algorithms:
An Introduction to Computational Algebraic Geometry and Commutative Algebra
4th Ed., Springer-Verlag, 2015.
[OS] O'Sullivan Groups, Rings, and Fields,
Course notes, 2023. Secs 1-4 Secs 5-6 Secs 7-10 Secs 11-12 Secs 13

Day Topics Reading
Th. 1/19 The integers and polynomial ring over a field.
Similarities and differences.
Videos The polynomial ring in one variable: Main Properties IVA 1.5, OS. Ch. 1
Tu. 1/24 Varieties: Solutions of an ideal. IVA 1.1,2,5
Th. 1/26 Ideals IVA 1.3,4,IVA 2.1
Tu. 1/31 Parametrization and implicitization. IVA 1.3,4, IVA 2.1
Th. 2/2 Derivatives, Mason-Stother theorem. [OS] Secs 5,6
Tu. 2/7 Monomial Orderings.
Division Algorithm.		 [OS] Sec 7, [CLO] 2.2
[OS] Sec 8, [CLO] 2.3
Th. 2/9 Work on orderings, division. GB. [OS] Sec 7,8, [CLO] 2.3
Tu. 2/14 Hilbert Basis Theorem. Groebner Basis. [OS] Sec 9,10, [CLO] 2.4-6
Th. 2/16 Groebner Bases and Buchberger's algorithm. [OS] Sec 9,10, [CLO] 2.4-6
Tu. 2/21 Groebner Bases and computing in quotient rings.
Chinese Remainder theorem.		 [OS] Sec 9,10. See my video on the CRT
Th. 2/23 Groebner Bases and computing in quotient rings.
Chinese Remainder theorem. Examples		 [OS] Sec 9,10. See my video on the CRT
Tu. 2/28 Unique Factorization. [OS] Sec 11.
Th. 3/2 Irreducibility. [OS] Sec 12; [CLO] 3.1,2.
Tu. 3/7 Test 1. [OS] Test 1 prep
Th. 3/9 The Spectrum of a Ring. [OS] Sec. 13
Tu. 3/14 Elimination and Extension theorems. [CLO] 3.1-2
Th. 3/16 Elimination and Extension theorems.
Implicitization.		 [CLO] 3.1-3
Tu. 3/21 Hilbert's Nullstellensatz:
statement and proof elements.		 [CLO] 4.1-2.
Th. 3/23 Nullstellensatz proof
Irreducible varieties.		 [CLO] 4.1,2,5.
Tu. 4/4 Operations on ideals. [CLO] 4.3,4.
Th. 4/6 Problems on ideals: computing. [CLO] 4.3,4,5,6.
Tu. 4/11 Irreducible decomposition of varieties. [CLO] 4.6.
Th. 4/13 Problems on ideals and decomposition. [CLO] 4.3,4,5,6.
Tu. 4/18 Bezout's theorem as an introduction to algebraic geometry. Homogeneous ideals, cones and projective geometry [CLO] 8.1-3. and problems.
Th. 4/20 Homogenoeus ideals, cones, Bezout: problems. [CLO] 8.1-4.
Tu. 4/25 Local rings and multiplicity. .
Th. 4/27 No class: Schedule to talk to me about your project .
Tu. 5/2 Intersection multiplicity and Bezout: problems. .
Th. 5/4 Presentations:Emily, Antwon, Shadi .
Th. 5/11 Presentations:Evan, Kieran, Aurora, James, Derek .