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

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Let α ∈ H 2 ét (Y, µ n) be the image of 1 ∈ Z/n under the composite map
Z/n ≃ HP 4 (X, µ ⊗2
n ) → HY 4 (X, µ ⊗2
n ) ≃ Hét(Y, 2 µ n ).
The point P defines a divisor on the curve Y . Let [P ] ∈ Pic(Y ) be its class. Checking
through the definitions ([SGA4 1/2], Cycle) one sees that the image of [P ] under the map
Pic(Y ) → H 2 ét (Y, µ n) deduced from the Kummer sequence is none other than α.
On the smooth irreducible curve Y , we may find a rational function f ∈ k(Y ) ∗ with
divisor div Y (f) = P + ∑ i=1,...,n m iP i where the P i are closed points of Y , distinct from
P and the m i are integers. If we now define X 1 to be X with the closed points P i deleted,
and Y 1 to be Y with these closed points deleted, we find that the map
H 4 P (X, µ ⊗2
n
) = HP 4 (X 1 , µ ⊗2
n ) → HY 4 1
(X 1 , µ ⊗2
n )
is zero – the first equality follows from the excision property for cohomology with support
([Mi80] III 1.27 p.92 or [CT/Oj92], Prop. 4.4).
In particular, this proves that any element of H 3 (X − P, µ ⊗2
n ), once restricted to
H 3 (X 1 − Y, µ ⊗2
n ), comes from H 3 (X 1 , µ ⊗2
n ).
Passing over to the cohomology of the local ring of X at P , one gets the following
perhaps more striking statement. Let X = Spec(A) be the local ring of a k-surface at a
smooth k-point P . Let Y = Spec(A/f) ⊂ X be defined by a regular parameter f ∈ A.
An element γ ∈ H 3 (X − P, µ ⊗2
n ) need not lift to an element in H 3 (X, µ ⊗2
n ). However, its
image in H 3 (X − Y, µ ⊗2
n ) does lift to an element of H 3 (X, µ ⊗2
n ).
What the Gersten conjecture postulates is that what we have just observed is a general
phenomenon.
Let X be a regular noetherian scheme of finite Krull dimension. Following Bloch and
Ogus [Bl/Og74], we shall say that the Gersten conjecture holds for étale cohomology over
X if the following key property holds :
LOCAL ACYCLICITY (Gersten’s conjecture for étale cohomology). — Let n > 0 be an
integer invertible on X and let j be an integer. Let S be a finite set of points in an affine
open set of X. Let Y ⊂ X be a closed subset of codimension at least p + 1 in X. Then
there exists a closed subset Z ⊂ X, with Y ⊂ Z and codim(Z) = p and an open set
U ⊂ X containing S such that the composite map
is zero.
H i Y (X, µ ⊗j
n
) → HZ(X, i µ ⊗j
n ) → HZ∩U i (U, µ ⊗j
n )
§ 3.6 The Gersten conjecture for étale cohomology : discrete valuation rings
Let X = Spec(A), A a discrete valuation ring with residue field κ and quotient field
K, let Y = Spec(κ) ⊂ X be the closed point and η = Spec(K) ⊂ X be the generic point.
Let n > 0 be an integer invertible in A.