Math 557 Fall 2024 (Commutative Algebra)

Math 557 Fall 2024 (Commutative Algebra): Syllabus Summary
Instructor: Takumi Murayama

This is a summary of the course syllabus and does not contain every component of the official course syllabus on Brightspace. Please look at Brightspace for a complete syllabus.

Course Information

Math 55700: Commutative Algebra
Location and Times

Section 001: Math 215, MWF 3:30 PM – 4:20 PM

Office Hours (Tentative)

Monday 1:50PM – 3:20PM in Math 646
Friday 1:50 PM – 3:20PM in Math 646
Extra Zoom Office Hour: Tuesday 10:00AM – 11:00AM (link on Brightspace)

Course Webpage: Brightspace – Please check regularly for announcements.

Instructor Contact Information

Name of Instructor: Takumi Murayama
Office: Math 646
Purdue Email:

Please put "CA Math 557 Fall 2024" in the subject line. This will help me find your email more easily.
Emails are read between 10AM and 6PM and will generally be responded to within 24 hours. Emails sent after 6PM in any given day may not be read until the following day.

Course Description

Credit Hours: 3.00. This course is an introduction to commutative algebra. Commutative algebra is the study of commutative rings and modules and has interactions with various fields of mathematics, including algebraic geometry, number theory, and several complex variables. Planned topics include the following:

A review of commutative rings and modules.
Spec and the Zariski topology.
Localization.
Integral extensions and integral closure.
Noether normalization.
Hilbert’s Nullstellensatz and connections to affine algebraic geometry.
Chain conditions. Noetherian and Artinian rings and modules.
Tensor products and flatness.
Primary decomposition.
Normal rings, discrete valuations rings, and Dedekind domains.
The Artin–Rees lemma.
The Krull height theorem and dimension theory.
Completions.

Textbook: Course notes will be provided. The notes will largely draw from Melvin Hochster’s 2017 Math 614 lecture notes on commutative algebra, available here: https://dept.math.lsa.umich.edu/~hochster/614F17/614.html.

Optional Textbooks: All texts listed below have free access options for Purdue students.

A term of commutative algebra by Allen B. Altman and Steven L. Kleiman, available here: https://doi.org/10.13140/RG.2.2.31866.62400.
Introduction to commutative algebra by Michael F. Atiyah and Ian G. Macdonald, available via the Purdue library here: https://doi.org/10.1201/9780429493638
Undergraduate commutative algebra by Miles Reid, available via short term loan here: https://n2t.net/ark:/13960/s27c54tjnc0.

Learning Outcomes: We will cover the majority of Hochster’s 2017 Math 614 lecture notes. Topics to be covered are listed in the course description.

Note: Commutative algebra is best learned by solving as many exercises as possible. I expect homework solutions to be thorough and either typed or handwritten neatly. In addition, please read the Homework Guidelines on Brightspace.

Important Dates

Please see the Purdue University Fall 2024 Add/Drop Calendar for important administrative dates.

Learning Resources

Lecture notes will be provided on BrightSpace.

Copyright Policy

Effective learning environments provide opportunities for students to reflect, explore new ideas, post opinions openly, and have the freedom to change those opinions over time. Stu- dents and instructors are the authors of the works they create in the learning environment. As authors, they own the copyright in their works subject only to the university’s right to use those works for educational purposes. Students may not copy, reproduce, or post to any other outlet (e.g., YouTube, Facebook, or other open media sources or websites) any work in which they are not the sole or joint author or have not obtained the permission of the author(s).

Course Policies

Please make sure to keep up with the conventions and definitions from the lecture notes. While the core material comes from the textbook and I will try to incorporate the conventions in the textbook, there will inevitably be differences stemming from the instructor’s mathematical perspective. My suggestion would be to consider the lecture notes as the main text and the textbook as a very good supplementary reference that can provide additional examples and explanations.

The homework will be graded according to the conventions, notations, and definitions from the lecture notes.

Homework Information and Policies

There will typically be one homework assignment due each week.
The due date for homework will typically be Wednesday by 11:59PM (West Lafayette time). However, due dates may vary depending on the proximity to breaks. It is your responsibility to check the due dates and complete the assignments by the due dates.
Homework assignments must be submitted on Gradescope.
Each student will be allowed 2 late homework submissions, no questions asked. The student must give notice of late submission to the instructor by the homework due date. Additional extensions may be granted on a case-by-case basis.
Please note that there may be grading delays for late homeworks.
Collaboration is strongly encouraged for homework assignments. However, the work you turn in must be your own. Please list the names of any collaborators as well as any outside resources that you have consulted on the first page of your homework. Please make sure to read the section on Academic Honesty.

Exam Information and Policies

There will be one timed take-home midterm exam and one 30–40 minutes oral final exam. More details on the exams will be provided closer to the exam dates.

Exam Dates

Midterm Exam: The exam will be available on Gradescope at 11:59PM (West Lafayette time) on Friday, October 4 and will close one week later at 11:59PM on Friday, October 11. Students may take the exam at any time during this one week period. More details will be available on Brightspace closer to the exam.
Final Exam: Monday, December 9 or Wednesday, December 11. Individual scheduling will be available a week prior to the exam.

Grades

Your total score will be determined as follows:

50% Homework
20% Midterm Exam
30% Final Exam

Students who get at least 97% of the total points in this course are guaranteed an A+, 93% guarantees an A, 90% an A-, 87% a B+, 83% a B, 80% a B-, 77% a C+, 73% a C, 70% a C-, 67% a D+, 63% a D, and 60% a D-; for each of these grades, it’s possible that at the end of the semester a somewhat lower percentage will be enough to get that grade.

Accessibility/Accommodations

Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247, as soon as possible.

If the Disability Resource Center (DRC) has determined reasonable accommodations that you would like to utilize in this class, you must send your Course Accommodation Letter to the instructor. Instructions on sharing your Course Accommodation Letter can be found by visiting: https://www.purdue.edu/drc/students/course-accommodation-letter.php. Additionally, you are strongly encouraged to contact the instructor as soon as possible to discuss implementation of your accommodations.

Attendance Policy

Attendance is not required. However, it is the student’s responsibility to keep up with the course in the event that the student misses any lectures. I will not re-lecture any material.

Illness Policy

Students should not come to class if they feel very ill.

Lecture Schedule and Homework Due Dates for Math 557:

Week	Date	Section(s)	Homework Due Dates (at 11:59PM)
1	8/19	Introduction, Ideals, Universal Properties
	8/21	Hochster Lectures 9/6, 9/8, 9/11 Ideals (radical, prime, maximal), Spec(R)
	8/23	Hochster Lectures 9/6, 9/8, 9/11 Zorn’s Lemma, More on Spec(R)
2	8/26	Hochster Lectures 9/15, 9/18 Category Theory, Modules
	8/28	Hochster Lectures 9/18, 9/20 Modules, Algebras, Localization Part 1	Homework 1
	8/30	Hochster Lectures 9/20 Localization Part 2, Restriction of Scalars, Local Properties, Reduced rings
3	9/2	No Class (Labor Day)
	9/4	Hochster Lectures 9/22, 9/25 Reduced Rings, etc. More on Spec(R)	Homework 2
	9/6	Hochster Lecture 9/25 Dimension, Integral Extensions Part 1
4	9/9	Hochster Lectures 9/25, 9/27 Integral Extensions Part 2: Lying Over (Incomparability)
	9/11	Hochster Lecture 9/29 Integral Extensions Part 3: Lying Over, Going up, Chinese Remainder Theorem, Fibers, Primes of k-algebras	Homework 3
	9/13	Hochster Lecture 10/2 Integral Extensions Part 4: Height, Normal Domains, Going Down Part 1
5	9/16	Hochster Lecture 10/4 Integral Extensions Part 5: Going Down Part 2
	9/18	Hochster Lecture 10/6 Algebraic Sets, k-algebras, Algebraic Independence, Noether Normalization, Hilbert Nullstellensatz	Homework 4
	9/20	Hochster Lectures 10/6, 10/9 Noether Normalization, Hilbert Nullstellensatz
6	9/23	Hochster Lectures 10/9 Dimension Part 1: UFDs, transcendence degree/basis, k-algebras. DVRs
	9/25	Hochster Lectures 10/11, 10/13 Dimension Part 2: Chain Conditions, Noetherian Rings/Modules Part 1	Homework 5
	9/27	Hochster Lectures 10/13, 10/18 Noetherian Rings/Modules Part 2: Hilbert Basis Theorem, Noetherian Induction, Hilbert Nullstellensatz Part 2	Last Lecture on Midterm Exam Material
7	9/30	Hochster Lecture 10/20 Noetherian Topological Spaces, Affine Algebraic Geometry Part 1: Algebraic Sets Regular maps, etc.
	10/2	Hochster Lecture 10/23 Affine Algebraic Geometry Part 2: Anti-equivalence of algebraic sets and reduced f.g. k-algebras	Homework 6
	10/4	No Class - Class Release for Midterm Exam	Midterm Exam Opens
8	10/7	No Class (Fall Break)
	10/9	Hochster Lectures 10/23, 10/25 Semigroup Rings, Power Series Rings
	10/11	Hochster Lectures 10/25, 10/27 Tensor Products Part 1: Basic Properties, Flatness	Midterm Exam Closes
9	10/14	Hochster Lectures 10/30, 11/1 Tensor Products Part 2: Tensor products with multiple factors, Extension of Scalars (Base Change), Localization, Flatness
	10/16	Hochster Lectures 10/30, 11/1 Tensor Products Part 3: Presentations, Properties of Hom, Tensor-Hom Adjunction, etc.
	10/18	Hochster Lectures 11/6, 11/3 Tensor Products Part 4: Hom Commutes with Flat Base Change, Tensor Products of Algebras, Ideal Criterion for Flatness
10	10/21	Hochster Lectures 11/3, 11/6 Projective Modules, Local Properties Part 1, Support
	10/23	Hochster Lectures 11/8, 11/10 Local Properties Part 2, Nakayama’s Lemma
	10/25	Hochster Lectures 11/10, 11/13 Primary Decomposition Part 1: Primary Ideals, Colon Ideals, Existence of Irredundant Primary Decompositions in Noetherian Rings	Homework 7
11	10/28	Hochster Lectures 11/13, 11/15 Primary Decomposition Part 2: Uniqueness Statements, Associated Primes, Length, Prime Cyclic Filtrations
	10/30	Hochster Lectures 11/15, 11/17 Primary Decomposition Part 3: Behavior Under Localization and Quotients, Associated Primes	Homework 8
	11/1	Hochster Lecture 11/17 Primary Decomposition Part 4: Applications, Primary Decomposition of Modules
12	11/4	Hochster Lectures 11/20, 11/22 Primary Decomposition Part 5: Primary Decomposition in the General Case, Primary Ideals in Polynomial Ring Extensions
	11/6	Hochster Lectures 11/20, 11/22 Primary Decomposition Part 6: Finite Length, Structure Theorem for Artinian Rings	Homework 9
	11/8	Hochster Lectures 11/22, 11/27 Primary Decomposition Part 7: Krull’s Principal Ideal Theorem, Prime Avoidance Krull’s Height Theorem
13	11/11	Hochster Lecture 11/27, 12/4 Primary Decomposition Part 8/Dimension Part 3: Krull’s Height Theorem, Systems of Parameters, Krull Dimension of Noetherian Modules
	11/13	Hochster Lecture 11/29, 12/4 Theorem on pg. 131 Normal Noetherian Rings Part 1: Dedekind Domains, DVRs, Krull’s (Serre’s) Criterion for Normality, Divisor Class Group	Homework 10
	11/15	Hochster Lectures 12/4, 12/6 Normal Noetherian Rings Part 2: Divisor Class Group, Normality under Ring/Field Extensions, Trace, Separability
14	11/18	Hochster Lectures 12/6 Normal Noetherian Rings Part 3: Trace for R-linear Endomorphisms of Free R-modules, Separability
	11/20	Hochster Lectures 12/8, 12/11 Normal Noetherian Rings Part 4: Properties of Dedekind Domains, Direct Limits	Homework 11
	11/22	Hochster Lecture 12/11 Completion Part 1: Inverse Limits, I-adic Topology, Cauchy Sequences, I-adic Completions
15	11/25	Hochster Lecture 12/11 Completion Part 2: Artin-Rees Lemma, Properties of Completions	Last Lecture of Final Exam Material
	11/27	No Class Thanksgiving Break	Homework 12 (automatic extension to 12/1)
	11/29	No Class Thanksgiving Break
16	12/2	Hochster Lectures 11/29, 12/1 Affine Algebraic Geometry Part 3: Coproducts in the Category of K-algebras, Dimension of Algebraic Sets,
	12/4	Hochster Lecture 12/1 Affine Algebraic Geometry Part 4: Existence of Partitions of Unity What next? Towards Schemes.
	12/6	Final Exam Review
17	12/9	Final Oral Exam
17	12/11	Final Oral Exam