⇤Math diary
Aug 30, 2023: 24
I’ve been pretending to do AG for years now, but this is the first time I’m actually taking an AG class. And its a lot. I have very little algebra background, so I’m needing to pick up all the requisite commutative algebra on the fly. I needed to do this eventually, but man its a bit of a trudge. I’m really thinking through things, trying to make them my own.
For instance, I think I finally have my own little way of thinking about the Zariski topology, and a slightly new visualization for the spectrum of a ring. Since I’m short on time today, let me relent myself with some motivation for Zariski.
What is algebraic geometry? the peons among you may say “its the study of polynomials and the geometry of their zero sets!” Gross. Disgusting. Who cares about polynomials anyways, i wanna study shapes. (Tbh this attitude kept me away from AG for a long time). Those who might be described as “based” would say, algebraic geometry is a geometric study of rings and their structure. In particular, their ideals. Call the ring $R$, and choose some element $a$. What is the ideal generated by $a$? Well, its the collection of all possible elements $r_n a^n + \dots + r_1 a^1$. In other words, Ideals are exactly the set of elemetns of $R$ achievable by polynomials! In fact, everything in a ring must involve polynomials, as the only algebraic option is multiplication and addition, which must both be finte. So, the two perspectives arent so different!
We define $\text{Spec}(R)$, the set of prime ideals of $R$. This will be our space paramertizing all the interesting sub-rings of $R$. What topology do we give it? From the ideal perspective, one is extremly natural. Some prime ideals are included in others, so we can form closed sets on the space of prime ideals as the chains of all prime ideals containing the given ideal. In other words, the zariski topology on $\text{Spec}(R)$ is induced by the inclusion topology of ideals on $R$.
But what about from the polynomial perspective? well, we can look back at topologies in general. Usually these are presented thru open sets. dually you can think of functions, and their ability to seperte prestesps. Think about this like a physical observable. A point $A$ is not contained in the closure of a set $B$ if you can find a continous function which vanishes on $B$ and equal 1 on $A$. In other words, the two can be seperated using continous functions. Now apply this to Zariski topology: Instead of allowing all possible functions, we only take polynomials. Snce polynomials are defined by their local information, they are bad at seperating points. The Zariski topology from the set of polytnomials is very loose, with huge open sets, because polynomials are such global critters. Any polynomial zero set in a ball determines the whole polynomial, there is a not-local topology. And so it goes.
(I was mid falling assleep writing this, if you couldnt tell)