Let $D\subset X$ be a smooth divisor on a smooth variety over $\mathbb{C}$.

Then we have Gysin maps in étale cohomology $H^2(X\backslash D,\mu_2)\rightarrow H^1(D,\mathbb{Z}/2)$ as well as $H^2(k(X),\mu_2)\rightarrow H^1(k(D),\mathbb{Z}/2)$ for the function fields.

$\textbf{Question:}$ Are these two Gysin maps related by a commutative diagramm of the form

$\require{AMScd} \begin{CD} H^2(X\backslash D,\mu_2) @>{Gysin}>> H^1(D,\mathbb{Z}/2)\\ @V{p}VV @VV{q}V \\ H^2(k(X),\mu_2) @>{Gysin}>> H^1(k(D),\mathbb{Z}/2) \end{CD}$

with $p$ and $q$ injective, where both maps are induced by the inclusion of the generic point of $X\backslash D$ and $D$ resp.

$\textbf{Reference request:}$ Using Jason Starr's comment, this would follow from the functoriality of the Gysin sequence in the pair $(X,D)$. I have seen this mentioned in some articles without proof. Does anybody have a reference (some book / article) where this fact is proven?

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    $\begingroup$ For the generic point $\eta_D$ of $D$, for the DVR $\mathcal{O}_{X,\eta_D}$, there is a morphism of schemes $\text{Spec}(\mathcal{O}_{X,\eta_D}) \to X$ mapping the closed point to $\eta_D$ and mapping the generic point to the generic point of $X$. The pullback of $D$ is the closed point. So this should follow by functoriality of the residue map with respect to the pullback homomorphisms of 'etale cohomology. I bet this is in Grothendieck's three articles on the Brauer group. $\endgroup$ Aug 8, 2017 at 11:56
  • $\begingroup$ @Jason Starr: Thank you. Looking through some papers it seems that the Gysin sequence is functorial in the pair $(X,D)$. But I could not find a proof of this fact, it is only mentioned. Do you know a reference? I could not find it in Grothendieck's articles in Dix Exposes. $\endgroup$
    – Bernie
    Aug 9, 2017 at 10:36
  • $\begingroup$ The place in Dix Exposes where this is discussed is Section 6 of "Le groupe de Brauer III" on p. 133. There is a further reference to SGA 4, Expos'es V and VIII, particularly VIII.6. In Milne's Lectures on Etale Cohomology, see the proof of Theorem 16.1. $\endgroup$ Aug 9, 2017 at 13:30


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