Suggestions for further topics in Commutative Algebra

Question

I am currently taking a semester long course in Commutative Algebra. We have covered a lot of dimension theory, and today finished proving Zariski's Main Theorem, which was the professor's original goal. However, he designed the course in such a way that a lot of basic topics have been omitted or very briefly touched upon. We have covered stuff about localization, proved a lot of results about finite ring maps and their fibres, and have done some of the standard stuff on integral extensions. In dimension theory the main theorem he proved was Krull's Hauptidealsatz. We concentrated on dimension 0 rings a lot. But, his entire focus seems to have been to build just enough theory to be able to prove ZMT.

Today he asked us if we wanted to do some more commutative algebra (we have , or whether we would like to start with scheme theory. I personally want to do some more commutative algebra. There are a few topics I want to know more about like:

1) Valuations

2) Flatness (I find this a hard topic to understand)

3) Some basic homological algebra

What are some important topics that a student taking a course on Commutative Algebra should know, and has been left out from whatever I have mentioned has been taught so far. To be honest though, I am very happy with how this course has turned out so far, because I learned a lot interesting results, without requiring too much background. I am sure I would not have learned some of these results in a more standard CA course, that built the whole theory from ground up.

I ask this question because often in some of the undergrad courses I have taken so far, important topics were completely omitted (for e.g. in my algebra course we did not do tensor products). But, somehow by the time you enter grad school you are already supposed to be familiar with these topics. Although the CA course that I am taking now is a graduate course, I would, however, not want to be ignorant of important topics within this field. I understand that within the span of a semester it is not possible to cover every important topic in CA. But, your input would be very helpful.

(I am not sure if this is an appropriate forum for a question like this. But, I hope it is not too vague)

Answer 1

This answer may be viewed as a bit odd and out of fashion by some, but one direction in which you could go--which intersects with commutative algebra, algebraic number theory, and discrete mathematics--is factorization theory.

Back when you were memorizing your multiplication tables in elementary school, the Fundamental Theorem of Arithmetic made that relatively easy in that there was only one way to break down a positive integer (greater than 1) into primes (which can be thought of as "building blocks", in a sense).

The generalization of this property that is discussed in the standard undergraduate abstract algebra sequence is, of course, the concept of the unique factorization domain (UFD), also called a factorial domain. The canonical example of an integral domain that does not enjoy unique factorization is the following example in $\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5}\,\vert\,a,b\in \mathbb{Z}\}$:

$6=2\cdot3 = (1-\sqrt{-5})(1+\sqrt{-5})$.

One can use the standard norm function for rings of algebraic integers to show that 2,3, and $1\pm \sqrt{-5}$ are all irreducible in $\mathbb{Z}[\sqrt{-5}]$, and that the two factorizations of 6 above are distinct.

What's interesting, though, is that it turns out that while nonunit elements of $\mathbb{Z}[\sqrt{-5}]$ may not factor uniquely into irreducibles, any two factorizations of a fixed nonzero nonunit element have the same number of irreducible factors. Such a domain is called a half factorial domain (or HFD). In a short, two page paper in the Proceedings of the AMS, L. Carlitz characterized all HFDs amongst rings of algebraic integers via the (ideal) class group. Such rings have unique factorization precisely when the class group is trivial and are HFDs precisely when the class group is isomorphic to $\mathbb{Z}_2$. So, in the context of rings of algebraic integers, the size of the class group gives a measurement of "how far" we are from unique factorization.

Since Carlitz's seminal 1960 paper, there has been a lot of work done in factorization theory (and yes, this was my area of research while I was in academia, so I'm not exactly unbiased). Currently, the only book on the topic is Geroldinger and Halter-Koch's Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory. Make no mistake, this book is an extremely dense read, but it's very thorough and has an excellent bibliography.

Other sources (some of which are a bit more accessible to the newcomer) include:

• The stuff on Scott Chapman's website. Scott has run a wildly successful REU program, and many of the projects have been in factorization theory.
• Some research papers and course materials on Jim Coykendall's website.

Answer 2

Well, here are some of my thoughts. Valuations is an important topic in its own right however you will find most elementary applications of this theory in algebraic number theory and perhaps theory of curves. So, the material contained in Atiyah Macdonald (and perhaps Matsumura) is standard (not very difficult). The more general setting of valuations is discussed in Bourbaki Commutative algebra however the presentation may seem too pedantic.

Homological Algebra on the other hand is a very very useful tool. It takes sometime to get an intuition and get a feel for the basic techniques (which may seem ad hoc). It is a central tool for algebraic geometry and commutative algebra itself.

Moreover things like flatness, regularity etc can be formulated in terms of derived functors of homological algebra. As an added bonus the same formalism (of homological algebra) applies to many other areas e.g group theory (group homology-cohomology),lie algebras, essentially any abelian category.

Some references of homological algebra (in some generality) will be:

1. Weibel's Book (it covers a lot of ground so might seem little un-motivated at places).

2. Cartan Eilenberg (wonderful book with a little outdated terminology)

3. Tohuku by Grothendieck.

4. Parts of Homological Algebra by Gelfand Manin.

Of course it is usually a good practice to come back to commutative algebra (after a little excursion into homological world) and re-examine the basic homological invariants again.