MAT 493 Rings and Varieties
When abstract algebra and geometry are fused together, they form a branch of mathematics called algebraic geometry. This fusion comes by uniting certain geometric objects, called varieties, with their algebraic counterparts, called rings. In this course we will study this two-way interplay: How are the algebraic properties of a ring reflected as geometric properties of its corresponding variety? And how does the geometry of a variety influence the structure of its corresponding ring?
This course will blend theory with practice; examples, computations, and algorithms will be given as much emphasis as abstract principles and concepts.
Prime Ideals, Part 2: Prime ideals versus irreducibility; quotients which are integral domains and prime ideals; maximal => prime => radical.
Student presentations. Radical Ideals, Part 2: The ideal of a variety is radical; The Strong Nullstellensatz; The Ideal-Variety Correspondence; definitions of nilpotent and reduced; coordinate rings are reduced.
Homework 9 due (problem 3 from 04.10, problems 3 and 5 from 04.17). Read Section 5, Chapter 4 of CLO.
Finish Maximal Ideals, Part 2.
Same reading assignment as 04.12.
Maximal Ideals, Part 1: Definition of evaluation homomorphisms; definition of maximal ideals; kernels of evaluation homomorphisms are maximal ideals.
Same reading assignment as 04.12.
Student presentations. Finish Coordinate Rings, Part 3.
Homework 8 due (problem 3 from 03.27, problem 2 from 04.05, and problem 5 from 04.08). Read Section 1, Chapter 4 of CLO.
Coordinate Rings, Part 1: Definition of polynomial maps; example; equivalence relations and ideals; definition of coordinate rings; coordinate rings are commutative rings; coordinate rings are in bijective correspondence with certain equivalence classes.
Read Section 2, Chapter 5 of CLO.
Homework 7 due (problems 6 and 9 from 03.13, and problem 5 from 03.25).
Finish Dickson's Lemma.
Same reading assignment as 03.06.
Student presentations. Continue Dickson's Lemma.
Homework 5 due (problem 5 from 02.27, problems 1 and 4 from 03.04). Same reading assignment as 03.06.
Student presentations. Finish Examples of Monomial Orders.
Homework 4 due (problems 1 and 2 from 02.20). Same reading assignment as 02.27.
Student presentations. Begin An Intro to Monomial Orders: k[x] is a PID, k[x_1,...,x_n] is not when n>=2; the Ideal Membership and Description Problems; definition of monomial order.
Homework 3 due (problems 2 and 4 from 02.13, problem 4 from 02.15). Same reading assignment as 02.20, being sure to look at Definition 7 and Lemma 8 on page 59 and 60.
Continue Ideals, part 2.
Read Section 5, Chapter 1 of CLO.
Student presentations. Finish (Affine) Linear Systems.
Homework 1 due (problem 1 from 02.01, problems 2 and 4 from 02.04).
Finish An Introduction to Irreducibility and Dimension. (Affine) Linear Systems (Mathematica notebook): Definition of affine linear system; examples.
Same reading as last time.
Finish Polynomials. Begin Affine Varieties: Pictures; the definition of affine varieties; the union and intersection rules; examples.
Read Section 2, Chapter 1 in CLO.
Introduction to class. Begin Polynomials: A word on ground fields; definition of polynomial rings and related matters (e.g., monomials, degrees, etc.); formal polynomials vs. polynomial functions.