Geometric Intuition of Prime Ideals

Question

Commutative Algebra is difficult.

Many things are agreeable and I usually can understand the proof (as syntactic manipulation), but could not create one as I don't understand any motivation at all. So I would like your help filling the missing pieces. To me, understanding the definition without understanding why it is defined in certain ways is difficult.

Specifically, (correct me if I'm wrong), I understand that we have curves in some affine space that we could "model" as affine domain, i.e. $R := k[x_1, \dots, x_n]/\mathfrak p$ for some prime ideal. The localization of the ring $R$ at some maximal ideal $\mathfrak m$ is the neighborhood of the point (in the affine domain) corresponding to $\mathfrak m$.

What is a localization at some prime $\mathfrak p$ in this picture? Are we intersecting the curve of $R$ to the curve of $\mathfrak p$? If so, is quotienting with $\mathfrak p$ similar to union?

Thanks for reading and potentially helping out! Appreciate any comments!

Answer 1

I can answer the "graded ring" question partly. In topology, we have groups $H^i(X)$ associated to a topological space $X$ and called cohomology groups. We also have things called "bundles" that are like cartesian products, but possibly with a 'twist' of some sort. Just as a cylinder looks like the cartesian product of a circle, $S^1$, with the real line $\mathbb R$, a Mobius strip looks almost like that, but with a twist. We call each of these a line bundle over a circle.

If we have two bundles $\eta$ and $\phi$ over the same space $X$, we can take a "direct sum" of them (the details don't matter) to get a new bundle over $X$. Topologically, this makes a ton of sense.

To any bundle $\eta$ over a space $X$, there is, for each $i$, a cohomology class $w_i(\eta) \in H^i(X)$, called the $i$th Steiffel-Whitney class. These things are enormously important, for they provide an algebraic tool for telling whether two spaces (or bundles over those spaces) are the same or not.

There's a formula for $w_i(\eta \oplus \phi)$ that looks something like this:

$$ w_i(\eta \oplus \phi) = w_0(\eta) w_i(\phi) +w_1(\eta)w_{i-1}(\phi) + \ldots w_{i}(\eta)w_0(\phi), $$ where here we're using a "product" that takes something in $H^k$ and something in $H^p$ and gives a result in $H^{k+p}$. This formula is nice, but when we allow ourselves to think of the whole graded ring consisting of all the cohomology groups, we can define $$ w(\eta) = w_0(\eta) + w_1(\eta) + ... $$ where each term lives in a different cohomology group in the graded ring. When we do this, the 'Whitney sum formula" becomes $$ w(\eta \oplus \phi) = w(\eta)w(\phi). $$ That alone is enough to make the idea of a graded group seem natural to me.