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Archive for November, 2009

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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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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Probably the first important result in algebraic number theory is the following. Let K be a finite field extension of \mathbb{Q}. Let \mathcal{O}_K be the ring of algebraic integers in K.

Theorem: The ideals of \mathcal{O}_K factor uniquely into prime ideals.

This is the “correct” generalization of the fact that \mathbb{Z}, as well as some small extensions of it such as \mathbb{Z}[ \imath ], \mathbb{Z}[\omega], have unique factorization of elements. My goal, in this next series of posts, is to gain some intuition about this result from a geometric perspective, wherein one thinks of a ring as the ring of functions on some space. Setting up this perspective will take some time, and I want to do it slowly. Let’s start with the following simple questions.

  • What is the right notion of prime?
  • Why look at ideals instead of elements?

In this series I will assume the reader is familiar with basic abstract algebra but may not have a strong intuition for it.

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Today I’d like to introduce a totally explicit combinatorial definition of the Schur functions. Let \lambda \vdash n be a partition. A semistandard Young tableau T of shape \lambda is a filling of the Young diagram of \lambda with positive integers that are weakly increasing along rows and strictly increasing along columns. The weight of a tableau T is defined as \mathbf{x}^T = x_1^{T_1} x_2^{T_2} ... where T_i is the total number of times i appears in the tableau.

Definition 4: \displaystyle s_{\lambda}(x_1, x_2, ...) = \sum_T \mathbf{x}^T

where the sum is taken over all semistandard Young tableaux of shape \lambda.

As before we can readily verify that s_{(k)} = h_k, s_{(1^k)} = e_k. This definition will allow us to deduce the Jacobi-Trudi identities for the Schur functions, which describe among other things the action of the fundamental involution \omega. Since I’m trying to emphasize how many different ways there are to define the Schur functions, I’ll call these definitions instead of propositions.

Definition 5: \displaystyle s_{\lambda}= \det(h_{\lambda_i+j-i})_{1 \le i, j \le n}.

Definition 6: \displaystyle s_{\lambda'} = \det(e_{\lambda_i+j-i})_{1 \le i, j \le n}.

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One of my favorite results in algebraic combinatorics is a surprisingly useful lemma which allows a combinatorial interpretation of the determinant of certain integer matrices. One of its more popular uses is to prove an equivalence between three other definitions of the Schur functions (none of which I have given yet), but I find its other applications equally endearing.

Let G be a locally finite directed acyclic graph, i.e. it has a not necessarily finite vertex set V with finitely many edges between each pair of vertices such that no collection of edges forms a cycle. For example, G could be \mathbb{Z}^2 with edges (x, y) \to (x, y + 1) and (x, y) \to (x + 1, y ), which we’ll denote the acyclic plane. Assign a weight w(e) to each edge and assign to a path the product of the weights of its edges. Given two vertices a, b let e(a, b) denote the sum of the weights of the paths from a to b. Hence even if there are infinitely many such paths this sum is well-defined formally, and if there are only finitely many paths between two vertices then setting each weight to 1 gives a well-defined non-negative integer.

Let a_1, ... a_n and b_1, ... b_n be a collection of vertices called sources and vertices called sinks. We are interested in n-tuples of paths, hereafter to be referred to as n-paths, sending each source to a distinct sink. Let \mathbf{M} be the n \times n matrix such that \mathbf{M}_{ij} = e(a_i, b_j). Then the permanent of \mathbf{M} counts the number of n-paths, but this is not interesting as permanents are hard to compute.

A n-path is called non-intersecting if none of the paths that make it up share a vertex; in particular, each a_i is sent to distinct b_i. A non-intersecting path determines a permutation \pi of the vertices; let the sign of a non-intersecting n-path be the sign of this permutation.

Lemma (Lindström, Gessel-Viennot): \det \mathbf{M} is the signed sum of the weights of all non-intersecting n-paths.

Corollary: If the only possible permutation is \pi = 1 (i.e. G is non-permutable), then \det \mathbf{M} is the sum of the weights of all non-intersecting n-paths.

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A few weeks ago on MathOverflow Greg Muller asked, “why do groups and abelian groups feel so different?” The answers were very interesting and came from several different perspectives, but I still don’t feel as if the question was resolved. So I’ll try to synthesize and summarize some of the answers and hopefully something will be clearer in the end.

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The last time we talked about symmetric functions, I asked whether the vector space \mathcal{R} could be turned into an algebra, i.e. equipped with a nice product. It turns out that the induced representation allows us to construct such a product as follows:

Given representations \rho_1, \rho_2 of S_n, S_m, their tensor product \rho_1 \otimes \rho_2 is a representation of the direct product S_n \times S_m in a natural way. Now, S_n \times S_m injects naturally into S_{n+m}, which gives a new representation

\rho = \text{Ind}_{S_n \times S_m}^{S_{n+m}} \rho_1 \otimes \rho_2.

The character of this representation is called the induction product \rho_1 * \rho_2 of the characters of \rho_1, \rho_2, and with this product \mathcal{R} becomes a graded commutative algebra. (Commutativity and associativity are fairly straightforward to verify.) It now remains to answer the first question: does there exist an algebra homomorphism \phi : \Lambda \to \mathcal{R}? And can we describe the inner product on \Lambda coming from the inner product on \mathcal{R}?

To answer this question we’ll introduce perhaps the most important class of symmetric functions, the Schur functions s_{\lambda}.

N.B. I’ll be skipping even more proofs than usual today, partly because they require the development of a lot of machinery I haven’t described and partly because I don’t understand them all that well. Again, good references are Sagan or EC2.

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