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:
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)