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.
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.
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.
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.