Algebraic Geometry

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. ...

The first two posts ([1],[2]) I ever did on this blog - now over a year ago - were posts about algebraic geometry. In particular we explored the geometric implications of some of the algebraic results I was learning in my commutative algebra class. Last summer I also wrote a post about sheaves, and left it off by claiming to soon write about schemes. If you scroll through the blog we have covered a bunch of different topics, but the blog post on schemes, seems to have fallen through the cracks. Today we will rectify this situation. I have my algebraic geometry exam this week, so this is both an explainer-post, and a “making sure I understand the course material”-post. These types of posts have in fact become common on this blog, but hopefully that is ok. ...

This summer I’m participating in Ravi Vakils pseudocourse on algebraic geometry, AGITTOC. Hence this summer serves as a wonderful opportunity to learn and write about cool mathematics. For long I have wanted to dive deeper into this abstract topic after just dipping my toes in during my bachelor thesis, and now it is time. Ravi though us in the first lecture that we shouldn’t study abstract objects without a cause, i.e. we need to ask ourselves why we want to learn about the objects, or the mathematics that lies ahead. I want to study algebraic geometry because I really like algebraic topology, and a lot of the concepts and notions of algebraic topology are abstracted in algebraic geometry, and several concepts gets a new viewpoint or gets some new tools to use to study them. I thought I would start off by discussing one of the fundamental objects of study in algebraic geometry, namely sheaves. These objects abstract, concretize and formalize several other mathematical notions, some of which we already know. One of these in particular is sheaf cohomology which can be viewed to generalize both singular, deRham and Cech cohomology. We are going to look into this cohomology theory in a later post. ...

Yesterday I wrote a geometric explanation of Noether’s normalization lemma, which you can find here. I’m going to use the geometric machinery developed in that post, so it can be useful to read that first. One useful result that is often stated as a corollary to Noether’s normalization lemma is Zariski’s lemma. It is a corollary of the algebraic form of the normalization lemma, so i thought there ought to be a geometric version of it as well, which I think I have found. Zariski’s lemma holds true even for non algebraically closed fields, but I think the geometric picture becomes much clearer for algebraically closed fields. ...

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. ...