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.
Posts Tagged ‘Nullstellensatz’
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. is a commutative ring, and is the set of maximal ideals of endowed with the Zariski topology, where the sets are a basis for the closed sets. Sometimes we will refer to the closed sets as varieties, although this is mildly misleading. Here denotes an element of , while denotes the corresponding ideal as a subset of ; 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 as functions on as follows: the “value” of at is just the image of in the residue field , and we say that vanishes at if this image is zero, i.e. if . (As we have seen, in nice cases the residue fields are all the same.)
For any subset the set is an intersection of closed sets and is therefore itself closed, and it is called the variety defined by (although note that we can suppose WLOG that is an ideal). In the other direction, for any subset the set is the ideal of “functions vanishing on ” (again, note that we can suppose WLOG that is closed).
A natural question presents itself.
Question: What is ? What is ?
In other words, how close are to being inverses?
Hilbert’s Nullstellensatz is a basic but foundational theorem in commutative algebra that has been discussed on the blogosphere repeatedly, but thematically now is the appropriate time to say something about it.
The idea of the weak Nullstellensatz is quite simple: the polynomial ring has evaluation homomorphisms sending for some point , so we can think of it as a ring of functions on . The ideal of functions vanishing at is maximal, so a natural question given our discussion yesterday is whether these exhaust the set of maximal ideals of . 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 -algebras.
See also the relevant post at Rigorous Trivialities.