- Math 8211, Commutative and
Homological Algebra
- Lectures:
- TuTh 2:30 p.m. - 3:45 p.m. in Ford Hall 130
- Prerequisites:
- Math 8202
- Instructor:
- Christine Berkesch Zamaere
- Office: Vincent Hall 250
- Email: cberkesc -at- math.umn.edu (For faster response time, please include "Math8211" in the subject line.)
- Office hours:
- Tuesday 1:30 p.m. - 2:20 p.m.,
- Thursday 1:30 p.m. - 2:20 p.m.,
- or by appointment (in Vincent Hall 250).
- Textbook:
- Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud, 1995. There is a copy on reserve in the Mathematics library in Vincent Hall.
- Description:
- This is an introductory course in commutative and homological algebra. We will dicuss topics including localization of rings, primary decomposition, completions, and dimension theory.
- Assessment:
- The course grade will be based on the homework and a project.
- Homework:
- You are encouraged to collaborate on the homework. When doing so, you may (and should!) turn in a single assignment for a small group of up to 4 people. In placing your name on an assignment with others, you are agreeing that each person listed has made a substantial contribution to the solutions provided.
- Projects:
- You are required to work in a groups of 2-3 for this project. The project has two parts, a paper and a presentation. Additional information can be found here.
- Disabilities:
- Students with disabilities, who will be taking this course and may need disability-related accommodations, are encouraged to make an appointment with me as soon as possible. Also, please contact U of M's Disability Services to register for support.
-
September 2 First Math 8211 meeting September 18 Project topic choice due October 7 Project outline due October 23 Project progress report due November 18 Project paper rough draft due November 25 Project paper peer review due December 4, 9 In-class project presentations December 9 Project paper final draft due, Final 8211 meeting
- Below is a list of topics and corresponding reading for the
semester. No topic next to a date means that we will
continue with the previous topic. Note that lectures more than a
week in advance are subject to change.
-
Date Topic Textbook references 09-02 Introduction and connections, Modules 1.1, 1.2, 1.3, 1.6, 0.3 09-04 Fractions 2.1 09-09 Noetherian and Artinian rings 2.4 09-11 Hilbert's basis theorem 1.4 09-16 Hom and Tensor 2.2 09-18 Associated primes, Prime avoidance 3.1, 3.2 09-23 Primary Decomposition 3.3 09-25 More primary decomposition 3.6, 3.8 09-30 Nakayama's Lemma and the Cayley--Hamilton Theorem, Normalization 4.1, 4.2 10-02 Primes in an integral extension 4.4 10-07 The Nullstellensatz 4.5 10-09 Graded rings, Hilbert functions 1.5, 1.9 10-14 Associated graded rings 5.1 10-16 The blowup algebra, The Krull Intersection Theorem, Free resolutions 5.2, 5.3, 1.10 10-21 Macaulay2, Flat families demo.m2, 6.1 10-23 Tor, Flatness 6.2, 6.3 10-28 Direct and inverse limits A6 10-30 11-04 Completions, Cohen Structure Theorem 7.1, 7.2, 7.4 11-06 Maps from power series rings 7.6 11-11 Resolutions A3.2, A3.3, A3.4 11-13 Homotopies and long exact sequences A3.5, A3.6, A3.7, A3.8 11-18 11-20 Derived functors A3.9, A3.10, A3.11 11-25 Dimension theory 8.1, 9.0 11-27 No meeting: Thanksgiving break 12-02 Dimension zero, Presentations 9.1 12-04 Presentations 12-09 Presentations
- Please include the statement of each
problem before its solution and use a separate page for each problem.
-
Assignment Due date Problem Set 1 Thursday, September 18 Problem Set 2 Thursday, October 2 Problem Set 3 Thursday, October 16 Problem Set 4 Thursday, October 30 Problem Set 5 Thursday, November 13 Problem Set 6 Thursday, December 4
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