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## Localization and the strong Nullstellensatz

A basic idea in topology and analysis is to study a space by restricting attention to arbitrarily small neighborhoods of a point. It is desirable, therefore, to have a notion of looking at small neighborhoods of a point which can be stated in entirely ring-theoretic terms. More generally, we’d like to have a way to ignore some points and only think about others. The tool that allows us to do this is called localization, and it offers a conceptual proof of the strong Nullstellensatz from the weak Nullstellensatz, which, as you’ll recall, is the tool that allows us to describe the category of affine varieties as the opposite of a category of algebras.

## The ideal-variety correspondence

I guess I didn’t plan this very well! Instead of completing one series I ended one and am right in the middle of another. Well, I’d really like to continue this series, but seeing as how finals are coming up I probably won’t be able to maintain the one-a-day pace. So I’ll just stop tagging MaBloWriMo.

Let’s summarize the story so far. $R$ is a commutative ring, and $X = \text{MaxSpec } R$ is the set of maximal ideals of $R$ endowed with the Zariski topology, where the sets $V(f) = \{ x \in X | f \in m_x \}$ are a basis for the closed sets. Sometimes we will refer to the closed sets as varieties, although this is mildly misleading. Here $x$ denotes an element of $X$, while $m_x$ denotes the corresponding ideal as a subset of $R$; the difference is more obvious when we’re working with polynomial rings, but it’s good to observe it in general.

We think of elements of $R$ as functions on $X$ as follows: the “value” of $f$ at $x$ is just the image of $f$ in the residue field $R/m_x$, and we say that $f$ vanishes at $x$ if this image is zero, i.e. if $f \in m_x$. (As we have seen, in nice cases the residue fields are all the same.)

For any subset $J \subseteq R$ the set $V(J) = \{ m | J \subseteq m \}$ is an intersection of closed sets and is therefore itself closed, and it is called the variety defined by $J$ (although note that we can suppose WLOG that $J$ is an ideal). In the other direction, for any subset $V \subseteq X$ the set $I(V) = \{ f | \forall x \in V, f \in m_x \}$ is the ideal of “functions vanishing on $V$” (again, note that we can suppose WLOG that $V$ is closed).

A natural question presents itself.

Question: What is $I(V(-))$? What is $V(I(-))$?

In other words, how close are $I, V$ to being inverses?

The idea of the weak Nullstellensatz is quite simple: the polynomial ring $\mathbb{C}[x_1, ... x_n]$ has evaluation homomorphisms $e_a : \mathbb{C}[x_1, ... x_n] \to \mathbb{C}$ sending $x_i \to a_i$ for some point $a = (a_1, ... a_n) \in \mathbb{C}^n$, so we can think of it as a ring of functions on $\mathbb{C}^n$. The ideal of functions $m_a$ vanishing at $a$ is maximal, so a natural question given our discussion yesterday is whether these exhaust the set of maximal ideals of $\mathbb{C}^n$. It turns out that the answer is “yes,” and there are a lot of ways to prove it. Below I’ll describe the proof presented in Artin, which has the virtue of being quite short but the disadvantage of not generalizing. Then we’ll discuss how the Nullstellensatz allows us to describe the maximal spectra of finitely-generated $\mathbb{C}$-algebras.