The Duality Between Algebra and Geometry

Niven Achenjang bio photo By Niven Achenjang Comment

If you’re somehow seeing this right now, look away. It’s not finished, and I’m not sure when/if it will be

I talked a little bit about the topic of this post’s title in a recent post, but I want to stress that this duality between algebra and geometry goes beyond 1 this $\spec$ business. In particular, I’m going to discuss a more “topological” setting where we see nice interplay between algebra and geometry 2: relating a (compact, Hausdorff) topological space to its ring of (real-valued) continuous functions.

Prelim on Separation Axioms

This might just be because there’s no point-set topology class at my school 3, but I get the sense that too many people don’t know about the theory of separation axioms for topological spaces. Sadly, I do not think I have enough space in this post to develop this theory, but I can state some needed highlights.

$X``=” C^0(X)$

Swan’s Theorem

  1. I don’t actually know that much about this area, so maybe what I’m gonna talk about in this post is somehow the same as this $\spec$ stuff. That would surprise me though. 

  2. It really bothers me that topology and geometry are very similar in general character, but the only word I know for capturing both of them at once is “geometry”. Like, sometimes I say “geometry” and mean “topology/geometry” but other times I say it and mean just “geometry”. How’s anyone supposed to understand what I’m saying? 

  3. Which is really a shame 

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