alternative lecture notes - Rational points and algebraic cycles

We have
1 = π 2 (EG/G) → π 1 (X) → π 1 ((X × EG)/G) → G → 1
if X is connected. If there is a section, the surjection in the short exact sequence above
has a section. If π 1 (X) is abelian, the obstruction is given by an element in H 2 (G, π 1 (X)).
This specializes to the Grothendieck section obstruction. This motivates looking for an
obstruction in
H 3 (Γ k , π et
2 (X)),
whatever that means.
Plan: Build a functor F from k-varieties to topological spaces with Γ k -action such that
F (Spec k) is contractible (with some Γ k -action, not necessarily free). Then every section
of X → Spec k gives a Γ k -equivariant map F (Spec k) → F (X), which gives an element of
π 0 (F (X) hΓ k . So if π0 (F (X) hΓ k = ∅, then there is no section. In fact, one gets a sequence of
obstructions π 0 (F (X) hΓ k ), H 2 (Γ k , π 1 (F (X)), H 3 (Γ k , π 2 (F (X)).
3. July 9 (Schlank)
What is a limit of diagrams in a category?
Let C be a category. Let A, B be two objects of C. Recall that a product A × B is an
object with maps to A and B such that given any X with maps to A and B, arises by
composition with a unique map X → A × B. More generally, a fiber product of A → D and
B → D is an object X that is universal for maps to A and B commuting with the given
maps. The limit of · · · A 2 → A 1 → A 0 is an object X with compatible maps to all the A i
that is universal. Given a group G acting on X, the object of fixed points X G is an object
with a map to X that is unchanged when composed with any g ∈ G, and that is universal
for this property.
Let D be a small category. A diagram is a functor F : D → C. Given F , the limit lim F is ←−
an object of C equipped with a map α d : lim F → F (d) for all d ∈ D, compatible with each
←−
morphism in D (i.e., if d 1 → d 2 in D, then α d1 and α d2 form a commutative triangle with
F (d 1 → d 2 ), such that for any other X with maps β d : X → F (d) satisfying the compatibility
conditions, there is a unique X → lim ←−
F that when composed with the α d yield β d .
Given E → A and E → B, the colimit A ∪
E
B is an object X with maps A → X and
B → X compatible with the maps from E, such that for any other such X ′ with maps, the
maps arise from the maps to X by composition with a unique map X → X ′ . One gets a
general notion of colimit, by using any functor from a small category.
If D is a small category and C is any category, the functor category C D is a category
whose objects are functors F : D → C and whose morphisms are natural transformations.
A diagram is an object of C D , i.e., a functor from D to C. The limit of a diagram, if it exists,
is an object of C: we want a functor lim: C D → C.
←−
We have const: C → C D sending C 0 to the functor sending each d ∈ D to C 0 and each
morphism of D to id C0 . We want there to be a bijection
functorially for X ∈ C.
Hom C D(const(X), diag) ≃ Hom C (X, lim ←−
diag)
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