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. 

Syllabus.

Project description.

05.01 Wed

Prime Ideals, Part 2: Prime ideals versus irreducibility; quotients which are integral domains and prime ideals; maximal => prime => radical.

04.29 Mon

Finish Radical Ideals, Part 2. Prime Ideals, Part 1: Definition of prime ideal; prime ideals = prime numbers in Z; hierarchy of maximal, prime, radical ideals.

Same reading assignment as 04.26.

04.26 Fri

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.

04.24 Wed

Radical Ideals, Part 1: A correspondence between points and maximal ideals over an algebraically closed field; definition of radical ideals; example; the radical of an ideal is an ideal.

Read Section 2, Chapter 4 of CLO.

04.22 Mon

Finish Maximal Ideals, Part 2.

Same reading assignment as 04.12.

04.17 Wed

Maximal Ideals, Part 2: Motivation for studying maximal ideals; the Weak Nullstellensatz.

Same reading assignment as 04.12.

04.15 Mon

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.

04.12 Fri

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.

04.10 Wed

Coordinate Rings, Part 3: Definition of kernel; preimages and images of ideals; The Lattice Isomorphism Theorem; natural projection onto quotient; ideals of quotients.

Same reading assignment as 04.05.

04.08 Mon

Coordinate Rings, Part 2: Definition of quotient rings; quotient rings are commutative rings; definition of ring homomorphisms and isomorphisms; the coordinate ring of a variety is isomorphic to a quotient ring.

Same reading assignment as 04.05.

04.05 Fri

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.

04.01 Mon

Some Applications of Grobner Bases: Solving polynomial equations; implicitization; examples.

Read Section 1, Appendix A of CLO and Section 1, Chapter 5 of CLO.

03.29 Fri

Student presentations.

Homework 7 due (problems 6 and 9 from 03.13, and problem 5 from 03.25).

03.27 Wed

Buchberger's Algorithm: Buchberger's Algorithm; examples; Macaulay2 demo; definition, existence, and uniqueness of reduced Grobner bases.

Read Section 8, Chapter 2 of CLO.

03.25 Mon

Properties of Grobner Bases: the division algorithm with Grobner bases produces unique remainders; implications for the Ideal Membership Problem; definition of S-polynomials; Buchberger's Criterion.

Read Section 7, Chapter 2 of CLO.

03.15 Fri

Student presentations. Grobner Bases: Definition of Grobner bases; an example.

Homework 6 due (problems 2, 3, and 4 from 03.06). Read Section 6, Chapter 2 of CLO.

03.13 Wed

The Hilbert Basis Theorem: A quote from Hilbert; ideals generated by leading terms; the Hilbert Basis Theorem; definition of ascending chains; polynomial rings are noetherian.

Same reading assignment as 03.06.

03.11 Mon

Finish Dickson's Lemma.

Same reading assignment as 03.06.

03.08 Fri

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.​

03.06 Wed

Dickson's Lemma: Definition of monomial ideals; some lemmas; some pictures; Dickson's Lemma; definition of minimal monomial basis; existence and uniqueness of minimal monomial bases.

Read Section 5, Chapter 2 of CLO.

03.04 Mon

The Multivariable Division Algorithm: Statement of the multivariable division algorithm; examples.

​Read Section 4, Chapter 2 of CLO.

03.01 Fri

Student presentations. Finish Examples of Monomial Orders.

​Homework 4 due (problems 1 and 2 from 02.20). Same reading assignment as 02.27.

02.27 Wed

Finish An Intro to Monomial Orders. Begin Examples of Monomial Orders: Definitions of lexicographic, graded lexicographic, and graded reverse lexicographic orders; examples.

​Read Section 3, Chapter 2 of CLO.

02.22 Fri

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.

02.20 Wed

Finish Ideals, part 2. The Division Algorithm: Definition of leading term; The Division Algorithm; example.

​Read Section 2, Chapter 2 of CLO.

02.18 Mon

Continue Ideals, part 2.

​Read Section 5, Chapter 1 of CLO.

02.15 Fri

Student presentations. Ideals, part 2: Definition of ideal of a variety; examples; summary of the ideal-variety correspondence.

​Homework 2 due (problem 1 from 02.06, problem 1 from 02.11, and problem 6 from 02.13).

02.13 Wed

Ideals: Definition of ideal; definition of generating sets; example of equality of ideals; definition of the variety of an ideal.

​Same reading assignment as last time.

02.11 Mon

Parameterizations: Examples; definitions of rational and polynomial parameterizations.

​Start reading Section 4, Chapter 1 in CLO.

02.08 Fri

Student presentations. Finish (Affine) Linear Systems.

​Homework 1 due (problem 1 from 02.01, problems 2 and 4 from 02.04).

02.06 Wed

Finish An Introduction to Irreducibility and Dimension. (Affine) Linear Systems (Mathematica notebook): Definition of affine linear system; examples.

​Same reading as last time.

02.04 Mon

An Introduction to Irreducibility and Dimension: Definition of irreducibility; examples; definition of dimension; examples; The Codimension Principle.

​Read Section 3, Chapter 1 in CLO, up to the CAGD stuff on page 20.

02.01 Fri

Finish Affine Varieties.

​Reread sections 1 and 2 of Chapter 1 in CLO. Then investigate this article on codimension.

01.30 Wed

Finish Polynomials. Begin Affine Varieties: Pictures; the definition of affine varieties; the union and intersection rules; examples.

​Read Section 2, Chapter 1 in CLO.

01.28 Mon

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.