Meeting times: MWF, 2-2:50pm in Room 12 of MSB.
email: firstname.lastname@youknowwhat.edu
Office Hours: By appointment or Wednesdays from 3-4pm and Fridays from 3-4pm.
This is the third in a series of courses in commutative algebra. The goal is to use the theory we have developed in the previous academic year to undertake a study of local properties of ring homomorphisms. More specifically we will develop the theory of (formally) unramified, smooth and etale homomorphisms. In order to prove non-trivial structural results about these homomorphisms, we will first prove an algebraic and very general version of Zariski's Main Theorem (ZMT). ZMT in turn rests on the theory of quasi-finite homomorphisms, which is a generalization of the notion of finite ring maps. So we will also develop the theory of quasi-finite ring maps as well. Like completions there is another important procedure that one can perform on a local ring known as its henselization. We will develop the theory of henselization and henselian local rings. Time permitting we will talk about Artin approximation and how questions about arbitrary noetherian rings often reduce to the case of finite type algebras over a field or a DVR.
Course notes:
They will be posted here.
References:
There are no required texts for this course. However, here are some resources:
Notes by Melvin Hochster on similar topics.
The Stacks Project chapters 10, 15, 16.
The first chapter of the book Néron Models.