The purpose of this post is to show the existence of a phantom map ${\phi: \Sigma^\infty \mathbb{CP}^\infty \rightarrow \Sigma^\infty S^3}$. To say that is a phantom map means that ${\phi}$ is nonzero, but ${\phi|_{\Sigma^\infty \mathbb{CP}^n}}$ is nontrivial for each ${n}$: in particular, ${\phi}$ induces the zero map in any homology theory. (This is essentially following a paper of Brayton Gray, “Spaces of the same $n$-type for all $n$.”)

In order to do this, we note that ${\Sigma^\infty \mathbb{CP}^\infty }$ is the homotopy colimit of the spectra ${\Sigma^\infty \mathbb{CP}^n}$, and consequently we have a Milnor exact sequence

$\displaystyle 0 \rightarrow \varprojlim^1 [ \Sigma \Sigma^\infty \mathbb{CP}^n, S^3] \rightarrow [\Sigma^\infty \mathbb{CP}^\infty, S^3] \rightarrow \varprojlim [\Sigma^\infty \mathbb{CP}^n, S^3] \rightarrow 0.$

For convenience, we have abbreviated ${\Sigma^\infty S^3}$ to simply ${S^3}$.

Our goal will be to show that the ${\varprojlim^1}$ term is nonzero. It will then follow that we have a “phantom” map ${\Sigma^\infty \mathbb{CP}^\infty \rightarrow S^3}$, as desired. (more…)

(Well, it looks like I should stop making promises on this blog. There hasn’t been a single post about spectra yet. I hope that will change before next semester.)

So, today I am going to talk about the formal function theorem. This is more or less a statement that the properties of taking completions and taking cohomologies are isomorphic for proper schemes. As we will see, it is the basic ingredient in the proof of the baby form of Zariski’s main theorem. In fact, this is a very important point: the formal function theorem allows one to make a comparison with the cohomology of a given sheaf over the entire space and its cohomology over an “infinitesimal neighborhood” of a given closed subset. Now localization always commutes with cohomology on non-pathological schemes. However, taking such “infinitesimal neighborhoods” is generally too fine a job for localization. This is why the formal function theorem is such a big deal.

I will give the argument following EGA III here, which is more general than that of Hartshorne (who only handles the case of a projective scheme). The form that I will state today is actually rather plain and down-to-earth. In fact, one can jazz it up a little by introducing formal schemes; perhaps this is worth discussion next time. (more…)

Today I’ll discuss completions in their algebraic context. All this is really a version of Cauchy’s construction of the real numbers, but it’s also useful in algebra, since one can study a ring through its completions (e.g. in algebraic number theory, as I hope to get to soon).

Generalities on Completions

Suppose we have a filtered abelian group ${G}$ with a descending filtration of subgroups ${\{G_i\}}$. Because of this, we can consider “Cauchy sequences” and “convergence:”

Definition 1

The sequence ${\{x_i\} \subset G}$, ${i \in \mathbb{N}}$ is Cauchy if for each ${A}$, there exists ${N}$ large enough that

$\displaystyle i,j > N \quad \mathrm{implies} \quad x_i - x_j \in G_A.$

The sequence ${\{y_i\} \subset G}$ converges to ${y}$ if for each ${A}$, there exists ${N}$ large enough that

$\displaystyle i>A \quad \mathrm{implies} \quad x_i -y \in G_A.$ (more…)