Blog posts

This is part 1 of a series leading up to and exploring model categories. For the other parts see the series overview. My main mathematical interest for the last couple years has been algebraic topology. I feel it suits my needs for intuition, and graphical picturing of what happens. A concept I have been learning more rigorously recently is fibrations, and how to use them in computing homotopy groups and homology groups of different spaces. There is something fun and exciting about computing the homology and homotopy groups of new spaces, as it usually requires different techniques and insight every time, and fibrations have certainly presented some new tools for my calculation toolbox. Since fibrations gives us nice tools, it would be nice to understand them better, and that is my plan for this post. As a remark, all spaces used and mentioned will be topological spaces, and all maps will be continuous. ...

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. ...