I am not much of a mathematician, and I think it is unlikely I will ever be. I know some basic facts about analysis and algebra but certainly nothing beyond that. This makes me a little sad. Recently I’ve been trying to read Hartshorne’s book on algebraic geometry. Unfortunately given the extremely suspect state of my intuition combined with some shoddy recall, this has been a struggle to say the least. I have a tendency to bounce around from subject to subject a bit much though, so this time I will try to make a concerted effort to keep with it, even if that means I have to go back and re-learn basic math. To that end I’m going to write a few posts about algebra and how I attempt to understand the structure. There is absolutely nothing novel about this and it is more for me than for anyone else, but if I am able to pursue algebraic geometry in more depth I also hope they will serve as a small reference for any readers wishing to follow along. Also in this vein it should be noted that any general comments I make should be taken with heaps of salt as they are mostly just guesses about what I think may be important.
I won’t start entirely from the beginning even though that wouldn’t be entirely useless to me. Here are the Wikipedia pages for groups, rings, ideals, and fields. In the introduction of page on fields you can see the chain of algebraic structures. These names sometimes come up, but since I personally don’t do a good job of keeping them straight I’ll try to mention definitions when they do. We also will skip over the definitions of normal subgroups, quotient groups, and homomorphisms. Algebraic geometry (at least in my limited understanding) has commutative algebra as a prerequisite so the article on commutative rings is also relevant, but I’ll actually write a bit about that myself. To clarify one possible issue, I will always assume that rings always have a one.
A very fair question is what applications algebraic geometry has or what it even is about. To this I have no good answer. The Wikipedia page lists some areas it is useful in, but I must be clear that I have zero understanding of how it is applied in those fields. Hartshorne’s book itself makes a very brief statement but then returns to the question after the first chapter. I will attempt to do the same. At the very least I have been informed that it finds some application in biology, and despite my abhorrence of the subject, it’s hard not to find that a little interesting. With our minimal assumptions of knowledge I will write up a little bit about basic algebra and the most basic of definitions in topology before attempting to dive into Hartshorne. If you’re interested in a real resource, in school I worked out of Algebra by Michael Artin and Abstract Algebra by Dummit and Foote, both of which I thought were perfectly reasonable.
Even when I get to algebraic geometry my commentary will of course be less insightful than Hartshorne or any other author (or most math students). I hope there will be some value in watching someone struggle to learn the topic rather than reading the perspective of someone who knows it inside and out. As always, any comments are more than appreciated.