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I have the question concerning one part of the proof of proper base change in etale cohomology. At one point during the proof we have the following setup and the statement: Let $X_0$ be scheme proper over separably closed field such that $dim X_0 \le 1$. Then $Pic X_0 \rightarrow Pic X_0^{red}$ is surjective.

My question is about the proof of this fact. Lei Fu gives different proof (not so difficult), but I do not see where is the mistake in the following. Since $f:X_0^{red} \rightarrow X_0$ is finite radiciel and surjective, the canonical map $H^q(X_0, F) \rightarrow H^q(X_0^{red}, f^{*}F)$ is an isomorphism for any sheaf $F$. However we have $Pic X_0=H^1(X_0, G_m)$ which should give the statement.

Marko markovictangens
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  • I don't know the fact you're citing about finite surjective maps, or what radiciel means, but what is $f^*F$? Usually this notation is reserved for $\mathcal{O}_X$ modules, but you seem to be using it for a sheaf of abelian groups with no $\mathcal{O}$-module structure. – hunter Mar 13 '25 at 01:56
  • @hunter https://en.wikipedia.org/wiki/Radicial_morphism – peter a g Mar 13 '25 at 02:40
  • great, i stand by my argument though – hunter Mar 13 '25 at 03:36
  • Cohomology and pullback are calculated on etale site, as for the reference for the statement on isomophism this is Lei Fu Theorem 5.7.1. – Marko markovictangens Mar 13 '25 at 05:23
  • @Markomarkovictangens does the operation $f^*$ make sense for an arbitrary sheaf on the étale site, or only a sheaf of $\mathcal{O}$-modules? – hunter Mar 14 '25 at 02:34
  • It makes sense, please see https://stacks.math.columbia.edu/tag/03PZ . In Stacks it is caled $f^{-1}$, however some standard references such as Lei Fu and Tamme use other notation. I am not sure what notation does SGA use. – Marko markovictangens Mar 14 '25 at 13:13
  • I claim that the theorem you are citing about finite surjective radiciel morphism only makes sense for sheaves of $\mathcal{O}$-modules (and should be, in Stacks notation, about the about the functor $f^*$, not about the functor $f^{-1}$). – hunter Mar 14 '25 at 13:48
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    I disagree, it makes perfect sense for sheaves of abelian groups, $A$ modules ... I repeat this is on etale site, which is not ringed. – Marko markovictangens Mar 14 '25 at 23:42
  • @hunter: I think the OP is correct, see Stacks Proposition 03SI. But is this result introduced in their source at this point, or is it using a “lower-tech solution” for some reason? – Aphelli Mar 15 '25 at 22:57
  • @Aphelli the proof is via calculation on stalks, so we only need to know stalks of higher pushforward which is done via limits. Idk if this is low or high tech though. Tamme uses limits, Lei Fu uses strict henselization in a tricky way. – Marko markovictangens Mar 16 '25 at 00:04
  • (I also agree with OP on f* now and should have said so!) – hunter Mar 16 '25 at 01:04

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In general the natural morphism $f^\ast\mathcal{O}^*_{X_{\text{ét}}} \to \mathcal{O}^*_{X_\text{red, ét}}$ is not an isomorphism. Take, for example, $X = \operatorname{Spec}\mathbb{C}[t]/(t^2)$, then clearly $X_{\text{red}} \cong \operatorname{Spec}\mathbb{C}$ and one can show (note that $\mathbb{C}[t]/(t^2)$ is complete hence strictly henselian, so taking the stalk at the closed point is equivalent to taking the global sections) \begin{align*} & \Gamma(f^\ast\mathcal{O}^*_{X_{\text{ét}}}, X_\text{red}) \cong (f^\ast\mathcal{O}^*_{X_{\text{ét}}})_{(0)} \cong \mathcal{O}^*_{X_{\text{ét}}, (t)} \cong \Gamma(\mathcal{O}^*_{X_{\text{ét}}}, X) \cong \mathbb{C}^\ast + t\mathbb{C},\\ &\Gamma(\mathcal{O}^*_{X_{\text{red, ét}}}, X_\text{red}) \cong \mathbb{C}^\ast \end{align*} which are not isomorphic.

T. Asai
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