alternative lecture notes - Rational points and algebraic cycles

A functorial obstruction is given by specifying a subset X(A) obs between X(k) and X(A)
for each X such that for any morphism f : X → Y , we have f(X(A) obs ) ⊆ Y (A) obs .
Examples: X(A) Br , X(A) desc , X(A) fin , X(A) fin-Ab , X(A) conn , . . . .
Suppose that we have f : Y → X a G-torsor over X, where G is a finite étale group.
Given x 0 ∈ X(k), let c(x 0 ) be the class of f −1 (x 0 ) in H 1 (k, G). Then we can twist to obtain
f c(x 0
: Y c(x0) → X. We obtain
X(k) =
∐
f σ (Y σ (k)) ⊆ ⋃ f σ (Y σ (A)) ⊆ X(A).
σ∈H 1 (k,G)
Define X f (A) := ⋃ f σ (Y σ (A)). Given an obstruction obs, define
X f,obs (A) := ⋃ f σ (Y σ (A) obs ) ⊆ X(A) obs .
For example, taking obs = Br gives the étale-Brauer obstruction set, which in our notation
is X(A) fin,Br .
X(A) fin
X(A) fin-Ab
X(A)
X(k)
X(A) fin,Br
X(A) Br
Let p: E → B be a fiber bundle with nonempty connected fiber F (the spaces are CWcomplexes,
simplicial sets, or manifolds): i.e., B is covered by open sets U such that p −1 (U) ≃
F × B over B. We can lift any point of B to a point of E. After lifting points, given a
path in B connecting two points, we can lift it to a path between their lifts (first lift without
worrying about the end of the path, and then modify a small piece of the end so that it ends
where it should).
We may assume that B is triangulated so that each simplex is contained in some U. Let
be a solid triangle; then a section of F × → is a map → F . We are given the
section above the boundary of the triangle; to fill in the lift, the obstruction lives in π 1 (F ).
We get a 2-cochain in C 2 (B, π 1 (F )). But this is not exactly the obstruction to the existence
of a section, because we could also change the choices made along the way.
C 1 (B, π 1 (F )) → C 2 (B, π 1 (F )) d → C 3 (B, π 1 (F ))
The kernel modulo image at the center is the definition of cohomology H 2 (B, π 1 (F )). In fact,
the 2-cochain we constructed above lies in ker d. Also, changing the choice of lifts of paths
corresponds to modifying it by an element of C 1 (B, π 1 (F )), and its differential measures the
change in our 2-cochain. Thus we obtain a canonical obstruction lying in H 2 (B, π 1 (F )).
After lifting the 2-skeleton, to lift the 3-skeleton, we obtain an obstruction in H 3 (B, π 2 (F )).
Thus we get a sequence of obstructions, in H 2 (B, π 1 (F )), H 3 (B, π 2 (F )), H 4 (B, π 3 (F )),
. . . , each one defined if the previous ones vanish (and one makes a choice).
We would like to do the same for X → Spec Q, and even for arbitrary morphisms of
schemes X → S.
Let Sec(E → B) be the space of all sections, which is a topological space. We would
like to compute the homotopy groups π ∗ Sec(E → B). These will be computed by a second
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