We will call a topological group complete if is an isomorphism.

The case that we are particularly concerned with is when our group is a ring and we take for our inverse system some ideal and . The topology that this determines is the “-adic topology”. This makes into a topological ring.

If we take the completion , then the continuous ring homomorphism has kernel .

Now we can also do all this with -modules by taking the group to be and the inverse system . The topology determined by this system is called the -topology on M. If we take the completion with respect to this topology (i.e. w.r.t this system), we get which is a topological -module meaning the action is continuous.

Rephrasing the motivating example from last time in this language we see that the -adic integers are formed as the completion of the ring with respect to the -topology where is the ideal .

The other really important example is to form the completion of with respect to the -adic topology. The completion is the ring of formal power series. Recall that by definition the inverse limit are all sequences such that . This just says that each is a polynomial, and it has to agree with the one before it up to the coefficient. So we can write each sequence where is the coefficient on the of the polynomial . And for any power series we get a sequence in this way.

Recall our notion of -filtrations. We had a chain such that , and if equality held for all large , then we called the filtration stable. Well, in our new language, these filtrations are inverse systems of modules, and hence determine a topology on . A few posts ago we used the fact that any stable -filtrations have bounded difference. In this new language, this says precisely that all stable -filtrations determine the same topology on M, moreover this is the -topology.

Lastly, if we convert the Artin-Rees Lemma to this language, we get that if is Noetherian, an ideal, a f.g. -module, and a submodule of , then the -topology on is actually just the subspace topology from the -topology on .

We should probably do some properties of completions next time.

Today we’ll start a new section, but only because it is a tool we need when we come back to the stuff we just finished. We will look at completions.

To motivate the process take a Hausdorff abelian topological group . Suppose there is a countable local basis at 0 (which implies countable basis, since the neighborhoods of 0 determine the entire topology). Since we assumed Hausdorff we have the usual notion of Cauchy sequences, so we can define the completion of to be completion in the usual sense . In particular, if , then .

Now suppose we have a local basis about 0 of subgroups (this rules out ), say . If we are in this situation, then our topology is actually determined by a sequence of subgroups, so we will want to try to define the completion solely in terms of algebra.

Take any Cauchy sequence . If we fix k, then at some , is constant for all . Note that really does depend on . Set the limit .

If we make what we mod out by bigger, namely we go from to , then projection maps . Thus our Cauchy sequence determined a “coherent sequence” , where .

Conversely, we can define a Cauchy sequence corresponding to any coherent sequence by just picking an element in the equivalence class at each step. So we can now define the completion to be the set of coherent sequences with group structure given entry-wise by the quotient group. The standard example here is the -adic integers, where the group is and our fundamental system is . Coherent sequences are where .

Whenever we have in general a sequence of groups and homomorphisms this is called an inverse system. The group of all coherent sequences is called the inverse limit of the system written . Thus our definition of completion can be written succinctly as .

Next time we’ll transfer this to module language and get to a few results.

I’ve decided that these last couple of days I’ll post “standards” or as one professor used to say, “old standbys”. These are quick theorems that are in some sense standard in the literature, and so have at least some positive probability of showing up on a prelim exam.

Old standby 1: The fundamental group of every connected topological group is abelian. (Already a good thing I’m doing this one. I wrote “Lie group” thinking it was only true in this case, but ended up never using smooth structure).

Lemma: Let be a topological space. Let be continuous, and define the following paths in X: , , , and . Then ~ .

This is just annoying to actually write for how little substance this actually has. But note that is a path starting at and ending at , as is . Thus it is possible to be path homotopic. Now the homotopy itself is just deforming the arguments of through , and since is defined and continuous through all that, it is just composing with a continuous function and hence is itself continuous.

Now for the actual problem. Fix , then by the standard trick of left multiplication being a homeomorphism, so WLOG we figure out whether or not is abelian.

Let . Define by (note that this is actual group multiplication whereas in the Lemma the meant path concatenation). Since we are in a topological group, is continuous since it is multiplication. Now by the lemma, ~ . Note where start and end and we get ~ . Thus is abelian.

This set of posts might not be as useful as I thought it would be considering I left out all the parts I didn’t want to fill in, and the point is to sort of force me to go through it before actually taking the test…

Next I think I’ll do is orientable if and only if n is odd.

P.S. Ack. WordPress weirdness strikes again! Who knows a hack to make a ~ without leaving the latex environment?

It is time to pull together some ideas we’ve built up, and show that they actually correlate how we want them to.

Recall that we have a notion of dimension for a ring called the Krull dimension. Review this if necessary, but essentially you just take the sup of the heights of all prime ideals in your ring.

For a topological space, we first define “irreducible.” Irreducible simply means that you can’t express the space as the union of two nonempty closed sets. To familiarize yourself with this definition some more you can try to prove that it is equivalent to the definition that any two non-empty open sets intersect non-trivially, or any non-empty open set is dense. So note right away that irreducible is a pretty rough condition. Almost none of the spaces I usually talk about are irreducible, since there are tons of non-dense open sets. And by the other criterion, any Hausdorff space is reducible.

Moving on, we now can define the dimension of a topological space to be where the ’s are irreducible subspaces.

Naturally, we are now interested to see if the topological notion of dimension on the topological space is the same as the Krull dimension of the ring R.

First, we’ll need a quick lemma:

A subspace is irreducible if and only if is a prime ideal. Note that this is really exactly what we wanted to happen, since prime ideals are points in Spec(R). When we say something is “irreducible,” what we are talking about are the smallest things that cannot be broken apart, i.e. points. I confess I am far oversimplifying this idea of “points” as we will see before the week is over if all goes as planned. (At this point you might want to review spec).

Proof of Lemma: I promise to fill this in later in the week. I just realized that it is an assignment to be turned in on Friday, and not all the readers of this blog that are in my commutative algebra class have this done yet.

Now let’s actually check dimensions. Theorem: . Just write it out now:

where that switching comes from the Lemma. The correspondence is 1-1.

So for some future posts, I want to clarify some more on dimension and what “points” really are. I also want to do the Nullstellensatz and talk about why it is so important, but I may not. I’ll do some hunting to see what other math bloggers have done on it. I’m pretty sure it has been posted on extensively already, in which I’ll just point people in those directions and add some things that I personally find interesting.

Let’s think about what is going on in a different way. So now let’s think of elements of the ring as functions with domain . We define the value of the function at a point in our space to be the residue class in . This looks weird at first, since the image space depends on the point that you are evaluating the function.

Before worrying about that too much, let’s see if we can get this notion to match up with what we did yesterday. We have the nice property that if and only if . (Remember that even though we think of f as a function, it is really an element of the ring).

Define for any subset of the ring S the zero set: . Now from what I just noted in the previous paragraph, we get that these are just precisely the elements of that contain S, i.e. the closed sets of the Zariski topology. Thus we can define our basis for the Zariski topology to be the collection of .

We also will want what is “an inverse” to the zero set. We want the ideal that vanishes on a subset of Spec. So given , define . Now this isn’t really an inverse, but we get close in the following sense:

If is an ideal, then . Taking the ideal of the zero set is the radical of the ideal. And the radical has two equivalent definitions: .

If we take the ideal and zero set in the other order we get that : the closure in the Zariski topology.

We can abstract one step further and put a sheaf on . Note that for any we have that is a multiplicative set, so we can localize at it. Since I haven’t talked at all about sheaves, I’m not sure if I want to go any further with this, so maybe I’ll do some more examples next time and possibly start to scratch this surface.

So I’ve built up localization starting there, and I’ve built up the theory of prime ideals scattered throughout, but ending here. I also just assume the basics of topology in my posts, so we are in the perfect position to talk about a very fascinating construction and incredibly useful tool that combines all these things.

Warning: I have just started learning about this stuff, so it could be riddled with confusion or error. Luckily, I’m just posting the basics which some readers probably know like the back of their hand and will hopefully point out problems.

Of course what I’m referring to is Spec. As usual let’s assume that R is a commutative ring with 1 (I don’t think we need the 1). Then . So we have the collection of all (proper) prime ideals of the ring. Other than prime ideals being my favorite type of ideal, this seems to be useless right now.

Let’s put a topology on our set now (the “points” of our space are prime ideals). Let be any ideal. Define . Then we define the closed sets of the topology to be the family of all such sets, i.e. are the closed sets. This is known as the Zariski topology.

To check that these really satisfy the right axioms, (I won’t go through it, but) note that , , and (The last is probably the least trivial, but they all follow in a straightforward from definition way).

Examples:

1. If our ring is a field k, then the spectrum is a point.

2.Another common example would be . In other words, the prime ideals can just be identified with the prime number that generates them (and we have (0) as a special circumstance). So open sets are subsets of that are missing finitely many prime numbers. So we see that the Zariski topology is not Hausdorff (and rarely is). It will, however, always be compact.

3. Possibly the most important examples are the ones dealing with polynomial rings. In the nicest case, when k is an algebraically closed field, we have that since the prime ideals are just multiples of linear polynomials, we have the bijection of sending any to the prime ideal generated by (and we still have that pesky “zero” floating around that we’ll talk about later).

Last for today is that Spec is a contravariant functor from rings to topological spaces. We’ve basically done everything we need, since we see how it takes a ring object to a Top object. Also if we have a ring hom , then define in the obvious way, i.e. .

I promised some localization and we should be able to get to that next time, but there is just so much going on here that it is nearly impossible to exhaust (well, from my perspective as a newbie to the topic).

I’m going to do a change in plan.

Galois Theory: Let F be a field. In some sense the “universal” Galois group is where is the algebraic closure, since given any algebraic extension we have that . In fact, there is a bijective correspondence between subgroups of the Galois group and algebraic extensions (this is just loosely speaking to show a connection later on, I’m not being careful about finiteness and things). In this case the we have an inverse corrolation. As the fields get bigger, the groups get smaller.

Covering spaces: For suggestive notation, let’s denote to mean Y is a covering of X. Then if X has sufficiently nice conditions, we have that there is a universal cover with covering map . Then we have that where is the group of “deck transformations,” i.e. the automorphisms such that . Now any other cover will “sit below” the universal one, in that the covering will have a factoring . Moreover . Just as in the Galois case, there is a bijective correspondence between conjugacy classes of subgroups of and isomorphism classes of coverings. This time in a sense it is not reversed, though it depends on how you want to look at it.

I found the similarities of these two situations very strange. There must be something deeper. All field extensions are in bijective correspondence to subgroups of the Galois group of the “largest one,” and all (iso classes of) coverings are in bijective correspondence with (up to conjugacy) subgroups of the fundamental group.

It turns out that after some hunting, there is a huge deep field called “the theory of descent” or something similar. It all looks so fascinating, but it is just too far astray from what I’m studying for me to actually learn right now. I thought I could dip a toe in or something and report back my findings, but there doesn’t seem to be any good introductions to the subject or any hope for quickly seeing some of the ideas. So, after a few days of hunting, I’m changing my plans and am going to look for something new to go on about (possibly back to the prime and localization that I built up, then left for dead?).

Actually, if anyone knows of a place to learn some of this stuff, it would be greatly appreciated if you let me know!

I officially promise this is my last post on Hilbert’s Theorem 90, but because of that it is going to go really fast for those who have not seen group cohomology (it is really cool, so I couldn’t pass it up).

An abelian group is a G-module (G a group) if for all and there is a unique element satisfying two conditions: , and for all and .

Just check any algebra text or here for more information on modules.

Now define an n-cochain of G over A to be a a function of n variables from G into A. If it is just an element of A. is the set of all n-cochains, and can be made into a group by the operation .

We can also get from to (which is what people who know about cohomology were hoping for), by the function :

.

OK. That looks bad, but really in some sense it is the natural choice. I’ll leave it to you to check that this is both a homomorphism and that (i.e. we have a chain complex).

Now if we label them

and

. Then we form . We call the elements of the n-cocycles and the elements of the n-coboundaries.

So if you don’t like that, we can scratch it now, since in HT90 we only care about , so let’s take a closer look at that. We can completely classify what the elements of look like. For any and any we need . Which is to say that .

Now let’s classify what looks like. Well, if , then for some . So for some . Well, I think you might be able to see the previous formulation of the theorem coming from unravelling these definitions.

Theorem statement: If is a finite Galois extension and , then is trivial.

Proof: Let a be a cocycle. Then let by . As in the last post, is not 0 by linear independence. So let such that and set .

Then for any we have

. Now we use that is a cocycle (in the kernel) to continue the equality as

.

Aha, so is a coboundary! Thus every cocycle is a coboundary, so the quotient is trivial.

Test your understanding by now trying to prove the other formulation as a corollary to this (remember you assume that G is cyclic in that version and have to relate it back to the norm).

I’m going to try to clean up some stuff from last time. I’ve realized that I was pretty sloppy. Given a classical field theory we can get to a QFT. To do this we just take the algebra to be the universal *-algebra generated by the classical fields. Now to get the inner product to behave the way we want it, take any state such that . Now we just define the inner product by and the module . If you want more details on why this works, it is known as the Gelfand-Naimark-Segal Construction.

Now we need to somehow get on the algebra . In physics people call this the Feynman integral. So we get something that is probably familiar to people that have taken a quantum mechanics class, . Here we run into another of those math vs physics problems. Since our space could be infinite dimensional, we don’t have a well-defined notion of what a measure is there. We can’t even define a Radon measure and do our trick of using bump functions.

Let’s beat the problem this time by ignoring the how to define on things we don’t care about. Look at that Feynman integral. We need to define the integral of things that look like . We create what is called the Feynman measure, which is just a measure on this particular space. Now although this measure is not unique, it equates to picking a renormalization scheme, so we have a group acting on the space of F. measures and Lagrangians that preserve the QFT. This is essentially what gives rise to what are known as “anomalies.”

Well, I don’t think I want to go into the nitty-gritty of the specifics of everything. This is a pretty rough idea that I just threw out there in case people were wondering. If you’ve followed this and would like some more details, just comment. Otherwise, my guess is that for the most part people don’t really care and so I’ll move on to something else.

Edited: Well, darn. I’m sure I’m going to keep remembering things I left out since I took such a scatter brained approach to this. Well, at the absolute very least I should include the Wightman axioms. We want to extrapolate from what has been presented so far to get what the axioms of a QFT are.

1) We’ve seen this explicitly along with the rationale for it. We want to be generated by where f is a classical field with compact support.

2) The inner product is positive definite, for pretty intuitive reasons.

3) We have Lorentz and translation invariance. This is also intuitive since we want QFT to work with relativistically.

4)  An operator that pushes things forward in time is positive. This amounts to an operator that increases energy is positive ( for any ).

5) We have a “locality” condition. This amounts to if the supports of f and g are spacelike separated (the commutator is 0).

6) There is a vacuum vector. This amounts to something being fixed by the Lorentz group.

There are some other minor axioms as well. Remember these things kind of vary depending on the situation. We are still not sure of the correct formalism. I should also emphasize that everything I’ve done so far is the free Hermition scalar QFT. OK. Hopefully that will cover me for now.

Let’s actually try to make some progress on QFT today. There are three parts to make a minimal definition. First, you need a module D over a *-commutative ring. So to get a few definitions on the table. A *-ring, R, is pretty easy. You just have a ring with an antiautomorphism and involutive mapping . This means that (i) , (ii) , (iii) , and (iv) . So if you’ve seen rings, this shouldn’t be out of grasp. An example would be complex numbers with complex conjugation. A <a href=”http://en.wikipedia.org/wiki/Module_(mathematics)”>module</a> is basically a generalization of a vector space.

The second part is a Hermitian inner product . So recall that Hermitian just means that it is self-adjoint. You could think of this as when you express the operator as a matrix the conjugate transpose is itself again. Lots of operators satisfy this, like the differential operator. Essentially the property Hermitian is in place, because if something is obsevable then it is Hermitian.

The last part is that we need a *-algebra, , of operators acting on D. Let’s jump out to a bigger picture for a second. The details here are sort of the details of getting around a problem. What we really want is basic. We want a Hilbert space H and an operator satisfying the axioms we want. So our field , and our operator defined at each as (an operator on H). The problem we are skirting is one of how to get around when x and y get arbitrarily close (an uncertainty problem as you might guess).

So we do the standard trick of “smoothing out the singularities.” Instead of points we will use bump functions. A bump function on M is just a smooth function with compact support. We redefine the operator then to be . Here is why I jumped out to the big picture we are skirting around. is generated by .

Some examples will be instructive. Let G be a group and D an orthogonal representation. Then is the group-ring of G, with “*” as . Or we could let L be a Lie algebra acting on a vector space D with an invariant symmetric inner product. The algebra can be with . Or we could take as any -algebra or von Neumann algebra and D any Hilbert space that is a *-representation.

These three examples should make us notice something. These are not things physicists typically work with (unless they are doing mathematical foundations of QFT or something). So despite having a definition in place, we might need to make some restrictions or correlations to what computations are being made down the road. These three examples are QFT’s, but that is sort of weird, since we usually speak of “QFT” and not “a QFT” or “this QFT” as if there is only one.