Birational invariants, purity and the Gersten conjecture Lectures at ...

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maps are induced by f ∗ :
O Y,f(x) −−→ O X,x
⏐
⏐
↓
↓
O Y,P −−→ O X,ξ = k(X)
Since α belongs to F loc (Y ), there exists an element α f(x) ∈ F (O Y,f(x) ) with image
α ∈ F (k(Y )). The image of that element in F (O Y,P ) may differ from α P , but their
images in F (k(Y )) coincide. Arguing as above, we find that β is the image of α f(x)
in F (k(Y )) under the composite map F (O Y,f(x) ) → F (O Y,P ) → F (k(X)). From the
above diagram we conclude that β is the image of α f(x) under the composite map
F (O Y,f(x) ) → F (O X,x ) → F (k(X)), hence that β comes from F (O X,x ). Since x was
arbitrary, we conclude that β belongs to F loc (X). We thus have a map f ∗ : F loc (Y ) −→
F loc (X), under the sole assumption that X is integral and that Y is integral and that
the specialization property holds for F and the local rings of Y .
It remains to show contravariance, i.e. given morphisms X −→ f
Y −→ g
Z with Y and Z
regular, we need to show that the two maps (g ◦ f) ∗ and f ∗ ◦ g ∗ from F loc (Z) to F loc (X)
coincide. To check this, let ξ ∈ X, resp. η ∈ Y be the generic points of X, resp. Y . Let
P = f(ξ) ∈ Y , Q = g(P ) ∈ Z, R = g(η) ∈ Z . We then have the commutative diagram
of local homomorphisms of local rings :
O X,ξ ←−− O Y,P ←−− O Z,Q
⏐
⏐
↓
↓
O Y,η ←−− O Z,R
Given α ∈ F loc (Z) ⊂ F (k(Z)), we may represent it as an element α Q ∈ F (O Z,Q ), and
this element restricts to an element α R ∈ F (O Z,R ) with image α in F (k(Z)). Applying
functoriality of F on rings to the above commutative diagram, we get :
β ←−− α P ←−− α Q
⏐ ⏐
↓ ↓
α i ←−− α R .