alternative lecture notes - Rational points and algebraic cycles

The diagonal map must be in W .
(2) For reflexivity, use A ∧ I := A. For symmetry, compose with the involution A ∐ A →
A ∐ A interchanging the factors. Now we prove transitivity. Then we have the commutative
diagram.
A
A ∐ f ∐ g
g ∐ h
A
X
A ∐ A
H 1 H 2
A × I
A × I
Let P be the pushout of the two maps out of A to A × I. From the diagram, we obtain
P → X. We have compatible maps A × I → A, so we get P → A. We want to show
that P → A is a weak equivalence.
Claim: COF and W ∩ COF are preserved by cobase change, i.e., if A → B has the
property, then any pushout C → D arising from A → C also has the property. Dually,
FIB and W ∩ FIB are preserved by base change.
More general claim: Let M be any class of maps. Then ⊥ M is closed under retracts
and cobase change and composition, and M ⊥ is closed under retracts and base change
and composition.
Proof: Composition: Suppose that we have A → C and C → B in ⊥ M.
Cobase change:
∈ ⊥ M
Retracts: equally easy, left as exercise.
A X
g
C X
B
A C X
B
C ∐
B Y
A
We return to the proof of the transitivity.
∈M
A
A ∐ A
A × I
(3)
∅
A
A ∐ A → A × I H → X
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