Noether Normalization

The first post on this blog is titled “geometric intuition”, and discusses the geometry behind Noether’s normalization lemma. When I wrote it I didn’t yet understand all the pieces, as I was not very comfortable working with algebraic geometry. One year later, I’m still not comfortable, but a bit more than last year. So, I thought I would update last years post with my new knowledge, as well as generalize the intuition to schemes - which we introduced in the last post. ...

Introduction This spring I have been taking a graduate class in commutative algebra, and I have yet to do algebraic geometry in a proper way, and have only gotten a small taste while writing my bachelor thesis. So this entire semester, I have felt this hinting at a geometric picture from the algebra itself, but i didn’t have the insight to figure it out. That said, I now think I have the geometric picture for Noether normalization, which in term implies a geometric picture of Hilbert nullstellensatz and some other results. It took a long time to convert the algebra into geometry for me, and i still have much to learn regarding this. What i have started to figure out is the close relationship between ideals and varieties. I have for a while known that this is one of the main reasons to introduce commutative algebra into algebraic geometry, but i couldn’t see the picture myself. Anyway, lets start with some introductory stuff. ...