alternative lecture notes - Rational points and algebraic cycles

If G is a group, then we have the categories GSet of G-sets and SSet G = GSet ∆op
of
simplicial G-sets. We had adjunctions:
const: SSet
where the limit takes fixed points, and
GSet ∆op : lim
colim: GSet ∆op SSet : const
where the colimit takes G-orbits. We can turn these two adjunctions into Quillen adjunctions
by endowing GSet ∆op
= (Set ∆op ) G with the injective resp. projective model structure. In
the injective (projective) model structure, the weak equivalences and cofibrations (weak
equivalences and fibrations) are the weak equivalences and cofibrations (weak equivalences
and fibrations) on the underlying simplicial sets.
The adjunctions persist when the role of the discrete group G is taken over by a profinite
group Γ, with ΓSet ∆op interpreted as in Definition 12.1. In the first case, when Γ = Gal(k/k),
we saw in Example 11.13 that a close variant of the injective model structure could be found
and gave a Quillen adjunction. The argument given there actually works for every profinite
group Γ.) We are less lucky with the second adjunction.
Proposition 12.2. If Γ is infinite, there is no projective model structure on ΓSet ∆op .
Proof. The basic idea is that there are not enough cofibrant objects. Suppose a projective
model structure does exist. Claim: a cofibrant object X has a free Γ-action. Take a continuous
finite quotient Γ ↠ G. Let EG be a Kan and contractible simplicial G-set with free
G-action (such an object exists); so EG ∼ ↠ ∗ is a trivial fibration. Inflating the G-action on
EG gives a Γ-action. Now ∅ ↩→ X has the left lifting property in the diagram below:
∅ EG
∼
X ∗
Since the G-action on EG was free, all stabilizers on X are contained in the kernel of
Γ ↠ G. But G was arbitrary, so the Γ-action on X is free. But such an X does not exist in
ΓSet ∆op .
□
We need some sort of substitute for the projective model structure, and in particular the
cofibrant replacement. For this, we turn to the Pro-category. Given a category C with
subcategories W and F that “behave as weak equivalences and fibrations” (precise definition
below), the cofibrant replacement of an object X should be a morphism ˜X ∼ ↠ X that is
initial among morphisms
X ′ ∼
↠ X.
However, these morphisms do not describe a filtered system, so even in the Pro-category we
need to perform additional tricks.
Definition 12.3. Let C be a category with finite limits. Let W, F ⊂ C be subcategories
containing all isomorphisms. We will say that (C, W, F ) is a weak fibration category if:
(1) W has the 2-out-of-3 property;
(2) W, F are closed under pull-backs;
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