The purpose of this post is to describe Sullivan’s proof of the Adams conjecture via algebraic geometry; the conjecture and its motivation were described in the previous post (from where the notation is taken). The classical reference is Sullivan’s paper “Genetics of homotopy theory and the Adams conjecture,” and the MIT notes on “Geometric topology.”

1. First step: completion at a prime

Sullivan’s proof of the Adams conjecture  is based on interpreting the Adams operations via a surprising Galois symmetry in the (profinitely completed) homotopy types of classifying spaces. Let’s work in the complex case for simplicity. Our goal is to show that the composite

$\displaystyle BU(n) \stackrel{\psi^k - 1}{\rightarrow } BU \stackrel{J}{\rightarrow} B \mathrm{gl}_1(S)[1/k]$

is nullhomotopic. (The map $J$ was defined in the previous post.)

Since the homotopy groups of ${B \mathrm{gl}_1(S)[1/k]}$ are finite, it will follow (by the Milnor exact sequence) that we can let ${n \rightarrow \infty}$ and conclude that the map

$\displaystyle BU \stackrel{\psi^k - 1}{\rightarrow } BU \stackrel{J}{\rightarrow} B \mathrm{gl}_1(S)[1/k]$

is nullhomotopic (i.e., there are no phantom maps into a spectrum with finite homotopy groups).

Using again the finiteness of the homotopy groups of ${B \mathrm{gl}_1(S)[1/k]}$, we can get a splitting

$\displaystyle B \mathrm{gl}_1(S)[1/k] = \prod_{p \nmid k} \widehat{ B \mathrm{gl}_1(S)}_p$

into the respective profinite completions. There is a well-behaved theory of profinite completions for connective spectra, or for sufficiently nice (e.g. simply connected with finitely generated homology) spaces, which will be the subject of a different post. (more…)

I’d like to take a quick (one-post) break from simplicial methods. This summer, I will be studying étale cohomology and the proofs of the Weil conjectures through the HCRP program. I have currently been going through the basic computations in étale cohomology, and, to help myself understand one point better, would like to mention a very pretty and elementary argument I recently learned from Johan de Jong’s course notes on the subject (which are a chapter in the stacks project).

1. Motivation via étale cohomology

When doing the basic computations of the étale cohomology of curves, one of the important steps is the computation of the sheaf ${\mathcal{O}_X^*}$ (that is, the multiplicative group of units), and in doing this one needs to know the cohomology of the generic point. That is, one needs to compute

$\displaystyle H^*(X_{et}, \mathcal{O}_X^*)$

where ${X = \mathrm{Spec} K}$ for ${K}$ a field of transcendence degree one over the algebraically closed ground field, and the “et” subscript means étale cohomology. Now, ultimately, whenever you have the étale cohomology of a field, it turns out to be the same as Galois cohomology. In other words, if ${X = \mathrm{Spec} K}$, then the small étale site of ${X}$ is equivalent to the site of continuous ${G = \mathrm{Gal}(K^{sep}/K)}$-sets, and consequently the category of abelian sheaves on this site turns out to be equivalent to the category of continuous ${G}$-modules. Taking the étale cohomology of this sheaf then turns out to be the same as taking the group cohomology of the associated ${G}$-module. So, if you’re interested in étale cohomology, then you’re interested in Galois cohomology. In particular, you are interested in things like group cohomologies of the form

$\displaystyle H^2(\mathrm{Gal}(K^{sep}/K), (K^{sep})^*).$ (more…)