So I’ve built up localization starting there, and I’ve built up the theory of prime ideals scattered throughout, but ending here. I also just assume the basics of topology in my posts, so we are in the perfect position to talk about a very fascinating construction and incredibly useful tool that combines all these things.
Warning: I have just started learning about this stuff, so it could be riddled with confusion or error. Luckily, I’m just posting the basics which some readers probably know like the back of their hand and will hopefully point out problems.
Of course what I’m referring to is Spec. As usual let’s assume that R is a commutative ring with 1 (I don’t think we need the 1). Then . So we have the collection of all (proper) prime ideals of the ring. Other than prime ideals being my favorite type of ideal, this seems to be useless right now.
Let’s put a topology on our set now (the “points” of our space are prime ideals). Let be any ideal. Define . Then we define the closed sets of the topology to be the family of all such sets, i.e. are the closed sets. This is known as the Zariski topology.
To check that these really satisfy the right axioms, (I won’t go through it, but) note that , , and (The last is probably the least trivial, but they all follow in a straightforward from definition way).
1. If our ring is a field k, then the spectrum is a point.
2.Another common example would be . In other words, the prime ideals can just be identified with the prime number that generates them (and we have (0) as a special circumstance). So open sets are subsets of that are missing finitely many prime numbers. So we see that the Zariski topology is not Hausdorff (and rarely is). It will, however, always be compact.
3. Possibly the most important examples are the ones dealing with polynomial rings. In the nicest case, when k is an algebraically closed field, we have that since the prime ideals are just multiples of linear polynomials, we have the bijection of sending any to the prime ideal generated by (and we still have that pesky “zero” floating around that we’ll talk about later).
Last for today is that Spec is a contravariant functor from rings to topological spaces. We’ve basically done everything we need, since we see how it takes a ring object to a Top object. Also if we have a ring hom , then define in the obvious way, i.e. .
I promised some localization and we should be able to get to that next time, but there is just so much going on here that it is nearly impossible to exhaust (well, from my perspective as a newbie to the topic).