SYLLABUS

Course Overview

The lecture notes and problem sets from the 2008 version of the course are available on OCW. We will deviate from that version a bit, with slightly more focus on the topics that are most crucial for algebraic geometry (18.725) and number theory (18.785), but the core will remain the same.

Prerequisites and corequisites

A year of algebra covering groups, rings, and fields at the level of 18.701/18.702. In particular, you should be very comfortable with rings, ideals, and modules.

There are no corequisites for this course, but this course is a corequisite for 18.725 and 18.785, which motivates the choice of topics and order of presentation. Having said that, if you are taking 18.725 and/or 18.785 you may well encounter references to commutative algebra results that we have not yet covered; I realize this is less than ideal, but it is unavoidable (arguably 18.705 should be a prerequisite rather than a corequisite).

Undergraduates considering this course should read the Undergraduates section below.

References

Below is a list of references that I think provide good treatments of the subject. I will assign specific readings week by week, but I encourage you to peruse them more broadly than that; I think it is really helpful to see the same topic from different perspectives. We will initially follow the treatment in Jeffries lecture notes, using Altman-Kleiman as an ample source of problems (both for psets and exams).

The following textbooks should all be electronically accessible from MIT:

    A Term of Commutative Algebra, A. Altman and S. Kleiman, 2013 (errata; 2019 edition).
    Commutative Algebra with a View Toward Algebraic Geometry, D. Eisenbud (errata).
    Commutative Ring Theory, H. Matsumura.
    Undergraduate Commutative Algebra, M. Reid.
    Local Algebra, J.-P. Serre.
    Computations in Algebraic Geometry with Macaulay2, D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels.

The following lecture notes are freely available online and cover most of the material in this course:

    Math 614: Commutative Algebra (Michigan, Fall 2018), J. Jeffries (course page).
    Math 614: Commutative Algebra (Michigan, Fall 2020), M. Hochster (other notes).
    Commutative Algebra I, Commutative Algebra II, Homological Algebra, E. Grifo.
    Commutative Algebra (TU Kaiserslautern, 2013/14), A. Gathmann.
    A Primer of Commutative Algebra, J.S. Milne.
    18.705 Commutative Algebra (MIT OpenCourseWare, Fall 2008), S. Kleiman.

The following online resources are also useful:

    The Stacks Project, Chapter 10: Commutative Algebra, A.J. de Jong et al.
    The Rising Sea: Foundations of Algebraic Geometry, R. Vakil.
    Commutative Algebra (video lectures), R. Borcherds.

Problem Sets

There will be weekly problem sets. They will only count for 10% of your grade, but they are the primary vehicle through which you will learn the material in this course; if you master them the other 90% will be easy! I recommend the following approach: (1) attempt to solve them entirely by yourself, (2) discuss the problems you are stuck on with a friend, (3) ask your favorite LLM to give you a hint, (4) look at the posted solution and make sure you understand it. Problem sets will be self-graded. You will get full credit for every solution you turn in.

Grading

Your grade in this course will be determined by your performance on in-class quizzes (60%), a final exam (30%), and problem sets (10%).

Undergraduates

Motivated undergraduates with adequate preparation are welcome to register for this course, but should do so with the understanding that it is a graduate level course and the pace may be faster than you are accustomed to. You should be prepared to do any extra reading necessary to acquaint yourself with background material that is unfamiliar to you.

Disability Accommodations

Please contact Disability and Access Services as early in the term as possible, if you have not already done so. If you already have an accommodation letter, please be sure to submit a copy to Mathematics Academic Services. Even if you do not plan to use any accommodations, if there is anything I can do to facilitate your learning, please let me know.