Timeline for Is the Gysin map in etale cohomology compatible with taking function fields?
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| Aug 9, 2017 at 13:30 | comment | added | Jason Starr | 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. | |
| Aug 9, 2017 at 10:42 | history | edited | Bernie | CC BY-SA 3.0 |
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| Aug 9, 2017 at 10:36 | comment | added | Bernie | @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. | |
| Aug 8, 2017 at 11:56 | comment | added | Jason Starr | 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. | |
| Aug 8, 2017 at 11:45 | history | asked | Bernie | CC BY-SA 3.0 |