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Archive for the ‘math.AC’ Category

I have to admit I’ve been using somewhat unconventional definitions. The usual definition of an affine variety is as an irreducible Zariski-closed subset of \text{MaxSpec } k[x_1, ... x_n] \simeq \mathbb{A}^n(k), affine n-space over an algebraically closed field k. A generic Zariski-closed subset is usually referred to instead as an algebraic set (although some authors also call these varieties), and the terminology does not apply to non-algebraically closed fields. The additional difficulty that arises in the non-algebraically-closed case is that it’s harder to think about points. For example, \text{MaxSpec } \mathbb{R}[x] has two types of points corresponding to the two types of irreducible polynomials: the usual points (x - a), a \in \mathbb{R} on the real line and additional points (x^2 - 2ax + (a^2 + b^2)), a, b \in \mathbb{R}. These points can be thought of as orbits of the action of \text{Gal}(\mathbb{C}/\mathbb{R}) on \mathbb{C}, hence \text{MaxSpec } \mathbb{R}[x] can be thought of as the quotient of \text{MaxSpec } \mathbb{C}[x] by this group action. This picture generalizes.

Anyway, for convenience let’s stick to k = \mathbb{C}. In this case, and more generally in the algebraically closed case, there is a reasonably simple description of what the category of affine varieties looks like, but first we have to describe what the morphisms look like and then we have to take the strong Nullstellensatz on faith, since we haven’t proven it yet.

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Functoriality

I wanted to talk about the geometric interpretation of localization, but before I do so I should talk more generally about the relationship between ring homomorphisms on the one hand and continuous functions between spectra on the other. This relationship is of utmost importance, for example if we want to have any notion of when two varieties are isomorphic, and so it’s worth describing carefully.

The geometric picture is perhaps clearest in the case where X is a compact Hausdorff space and C(X) = \text{Hom}_{\text{Top}}(X, \mathbb{R}) is its ring of functions. From this definition it follows that C is a contravariant functor from the category \text{CHaus} of compact Hausdorff spaces to the category \mathbb{R}\text{-Alg} of \mathbb{R}-algebras (which we are assuming have identities). Explicitly, a continuous function

f : X \to Y

between compact Hausdorff spaces is sent to an \mathbb{R}-algebra homomorphism

C(f) : C(Y) \to C(X)

in the obvious way: a continuous function Y \to \mathbb{R} is sent to a continuous function X \xrightarrow{f} Y \to \mathbb{R}. The contravariance may look weird if you’re not used to it, but it’s perfectly natural in the case that f is an embedding because then one may identify C(X) with the restriction of C(Y) to the image of f. This restriction takes the form of a homomorphism C(Y) \to C(X) whose kernel is the set of functions which are zero on f(X), so it exhibits C(X) as a quotient of C(Y) .

Question: Does every \mathbb{R}-algebra homomorphism C(Y) \to C(X) come from a continuous function X \to Y?

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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?

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Irreducible components

If it wasn’t clear, in this discussion all rings are assumed commutative.

Given a variety like xy = 0 we’d like to know if there’s a natural way to decompose it into its “components” x = 0, y = 0. These aren’t its connected components in the topological sense, but in any reasonable sense the two parts are unrelated except possibly where they intersect. It turns out that the Noetherian condition is a natural way to answer this question. In fact, we will see that the Noetherian condition allows us to write \text{MaxSpec } R uniquely as a union of a finite number of “components” which have a natural property that is stronger than connectedness.

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Let’s think more about what an abstract theory of unique factorization of primes has to look like. One fundamental property it has to satisfy is that factorizations should be finite. Another way of saying this is that the process of writing elements as products of other elements (up to units) should end in a finite set of irreducible elements at some point. This condition is clearly not satisfied by sufficiently “large” commutative rings such as \mathbb{C}[x, x^{ \frac{1}{2} }, x^{ \frac{1}{3} }, ... ], the ring of fractional polynomials.

Since we know we should think about ideals instead of numbers, let’s recast the problem in a different way: because we can write x^{r} = x^{ \frac{r}{2} } x^{ \frac{r}{2} } for any r, the ascending chain of ideals (x) \subset (x^{ \frac{1}{2} }) \subset (x^{ \frac{1}{4} }) \subset ... never terminates. In any reasonable theory of factorization writing f = f_1 g_1 and then comparing the ideals (f) \subset (f_1), then repeating this process to obtain a chain of ideals (f) \subset (f_1) \subset (f_2) \subset ... , should eventually stabilize at a prime. This leads to the following definition.

Definition: A commutative ring R is Noetherian if every ascending chain of ideals stabilizes.

Akhil’s posts at Delta Epsilons here and here describe the basic properties of Noetherian rings well, including the proof of the following.

Hilbert’s Basis Theorem: If R is a Noetherian ring, so is R[x].

Today we’ll discuss what the Noetherian condition means in terms of the topology of \text{MaxSpec}. The answer turns out to be quite nice.

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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 \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.

See also the relevant post at Rigorous Trivialities.

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An analyst thinks of the ring \mathbb{C}[ t] of polynomials as a useful tool because, on intervals, it is dense in the continuous functions \mathbb{R} \to \mathbb{C} in the uniform topology. If we want to understand the relationship between \mathbb{Z} and polynomial rings in a more general context, it might pay off to expand our scope from polynomial rings to more general types of well-behaved rings.

The rings we’ll be considering today are the commutative rings C( X) = \text{Hom}_{\text{Top}}(X, \mathbb{R}) of real-valued continuous functions X \to \mathbb{R} on a topological space X with pointwise addition and multiplication. It turns out that one can fruitfully interpret ring-theoretic properties of this ring in terms of topological properties of X, and in certain particularly nice cases one can completely recover the space X. Although the relevance of these rings to number theory seems questionable, the goal here is to build geometric intuition. You can consider this post an extended solution to Exercise 26 in Chapter 1 of Atiyah-Macdonald.

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