# Math 8510: Commutative Algebra I

## Course logistics:

Meeting times: MWF, 3-3:50pm in Room 12 of MSB.

email: firstname.lastname@youknowwhat.edu

Office Hours: By appointment or Wednesdays from 1-2pm and Fridays from 4-5pm.

This will be a first course in commutative algebra. Broadly speaking, commutative algebra is the study of commutative rings (such as number fields, p-adic numbers, polynomial and power series rings over a field) and modules over such rings. In this course, we will begin with general properties of commutative rings and modules over them and slowly specialize by imposing finiteness conditions on the modules and rings. Historically, these finiteness conditions arose as characteristics shared by commonly occurring commutative rings in number theory, invariant theory and algebraic geometry. If you are interested in the history of commutative algebra, here is a link to Chapter 1 of David Eisenbud's book, Commutative Algebra with a view toward Algebraic Geometry, where he discusses the roots of the subject.

We will roughly cover the contents in the book by Atiyah-MacDonald, although we will not closely follow any book. I like the book by Atiyah-MacDonald because it is concise. The strength of the book is in its extensive exercises, many of which are not just a routine application of results from the earlier parts. I will assume that all students have taken (an equivalent version) of both semesters of the graduate algebra sequence taught at MU, and so, are comfortable with rings, ideals, modules and the theory of field extensions. Familiarity with the language of categories and functors will be assumed. I will also assume you know basic point set topology.

You will have to work hard in this course. There will be regular assignments.

Here are some free resources that you can use in addition to notes that will be provided.

Notes by Mel Hochster for the analog of Math 8510 taught at University of Michigan.

A book by Allen Altman and Steven Kleiman that is meant to be an expansion of Atiyah-MacDonald's classic text.

The Stacks project chapter on commutative algebra. This is written as a reference but is easy to navigate. It often does things in a level of generality that is hard to find elsewhere.

The first five sections of Chapter I of Ravi Vakil's notes should give you more than enough background from category theory. I do not plan to spend a lot of time developing this language. Hence, this may be good preparatory reading for the course if you are unfamiliar with the language of categories and functors. Additionally, Ravi's book develops commutative algebra concurrently with algebraic geometry, which has the benefit of providing context for many of the notions we will discuss in this course. It may be good for you to peruse this book when we cover a topic in class.

## Grades

Grades will be based on weekly quizzes (20%), assignments (50%) and a final paper (30%).

## Topics covered

Week 1: Commutative rings and algebras, finiteness conditions on algebras, operations on ideals, prime ideals, maximal ideals, Zorn's Lemma, minimal primes, multiplicative sets, Oka families, nilpotence, radical ideals, nilradical.

Week 2: Zariski topology, quasi-compactness, irreducibility, distinguished opens, spectrum Nullstellensatz, clopen subsets of spectra, idempotents, connectedness, Jacobson radical, prime avoidance, generating sets of modules, free modules, direct sum, exact sequences, short exact sequences, snake lemma, finitely generated modules, finitely presented modules.

Week 3: Finiteness conditions on modules and their behavior in short exact sequences, annihilators, Nakayama's Lemma and applications, Cayley-Hamilton theorem.

Week 4: Representable functors, Yoneda's Lemma, tensor products, adjunction, tensor-Hom adjunction, right exactness of tensor products, base change/extension of scalars, extension-restriction of scalars adjunction.

Week 5: Tensor products of algebras as a pushout, flat modules, filtered colimits of flat modules, bimodules, flatness and base change, ideal theoretic criterion of flatness, flatness in short exact sequences, universally injective maps, universally exact sequences, flatness and universal exactness, equational criterion of flatness, linear independence of elements in a module over a local ring, finitely generated flat modules over a local ring.

Week 6: Faithful functors, faithfully flat modules, ideal theoretic criterion of faithful flatness, faithful flatness over local homomorphisms of local rings, base change of faithful flatness, descending properties of modules along a faithfully flat ring map, localization of rings and modules at a multiplicative set in terms of a representable functor, explicit construction of localization, exactness of localization, localization of a ring is a flat algebra, ideals of a localized ring, prime ideals of localization.

Week 7: Localization at a prime, fibres of a ring map, local properties, reduced rings, flatness is local, going down for flat ring maps, integral elements, integral ring maps, integral closure, finite ring maps, base change of integral ring maps, compositions of integral ring maps, integral extensions, integral maps and maximal ideals.

Week 8: Integral ring maps are universally closed on Spec, finite ring maps have finite fibres, going up property of integral ring maps, incomparibility property for integral ring maps, Krull dimension of a ring, height of a prime ideal, height of an ideal, co-height of an ideal, integral ring maps and Krull dimension, integral ring maps and height vs co-height of primes, integral elements and localization, integral closedness and localization, being integral is a local property, being integrally closed is a local property, definition of a normal ring.

Week 9: Characterization of a normal domain, irreducible components of a normal ring are disjoint, factoring monic polynomials after a ring extension, minimal polynomials and normality, characterization of integral elements when you adjoin a variable, normality is preserved under adjoining finitely many variables, the total ring of fractions, decomposition of the total ring of fractions of a ring with finitely many minimal primes, normal rings are integrally closed in their total ring of fractions, characterization of normal rings with finitely many minimal primes, some background from Galois theory of algebraic field extensions that are not necessarily finite.

Week 10: Maximally dominating ring maps, going down for maximally dominating integral ring maps, action of Galois group on the integral closure of a normal domain in an algebraic extension of its fraction field, orbits of the Galois action and fibres of the induced map on Spec, noether normalization over a domain, noether normalization over a field, making multivariate polynomials essentially monic in one of the variables.

Week 11: Hilbert's weak and strong nullstellensatz, corresponding statements for finitely generated algebras over a field, dimension theory of finitely generated algebras over a field, Jacobson rings, catenary rings, chain conditions on modules over a ring, noetherian and artinian modules, equivalent characterizations of noetherian modules, behavior of noetherian and artinian modules in short exact sequences, characterization of noetherian modules over a noetherian ring, essentially of finite type algebras, Hilbert's basis theorem, essentially of finite type algebras over a PID are noetherian.

## Lecture notes

Notes (these will be updated over time). If you find typos or mistakes, please email me.

## Assignments

Please note that you are only allowed to work on these assignments with other people who are officially registered for this course. You are not allowed to ask graduate students who have already taken Commutative Algebra or Algebraic Geometry for help. Use of AI, Chegg, MathStackExchange and MathOverflow is prohibited. While I will assume you are judiciously doing all the problems, you only need to submit solutions to the starred (*) exercises.

Assignment 1, Due August 28th 30th at the start of class.

Assignment 2, Due September 9th at the start of class.

Assignment 3, Due September 23rd at the start of class.

Assignment 4, Due October 7th at the start of class.

Assignment 5, Due October 23rd at the start of class.

Assignment 6, Due November 6th at the start of class.

## Quizzes

Your weekly quizzes can be accessed here.

## Final paper guidelines

The final paper comprises 30% of your grade for this course. It is intended to be an original exposition of some topic in commutative algebra (of your choice) that has not been covered in class, in the assignments or in the notes above. Although I am not requiring you to Tex your homeworks (I think it's a good idea to do so), the final paper must be written using LaTex. It must be self-contained (although you may freely use concepts developed in class with approriate citations), have font that is 11 or 12 pt, have one inch margins, and it should be between 8 and 12 pages including the bibliography. Please do not assume that I know any commutative algebra outside of what I have taught you this semester. Detailed citations are required and should be pulled from Mathscinet (https://mathscinet.ams.org/mathscinet/publications-search), which you have access to through the university. I am happy to chat with you about your choice of topic. The final paper is due via email by 11:59pm on December 11. The paper must be written in your words and the use of AI is prohibited.

## Miscellaneous resources

If you like to think in terms of pictures, you may find the following enlightening: Visualizing algebraic concepts by Ravi Vakil.

If you find diagram chases daunting, you may find the following system informative: How to write diagram chases by Anton Geraschenko.