Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $\pi: X \to Y$ be a morphism of sites (I need this for the ├ętale topology) and $Z \hookrightarrow Y$ be a closed subscheme.

Since an injective sheaf is flabby (in the sense that $H^i(U,F) = 0$, $i > 0$ for all $U$ of a site), and the pushforward of a flabby sheaf is flabby, it follows from the long exact localisation sequence that $H^i_Z(Y,\pi_*F) = 0$ for $i > 1$.

Now my question is: Does this also hold for $i = 1$? (Equivalently, is $(\pi_*F)(Y) \to (\pi_*F)(U)$ surjective for all $U$ and $F$ injective?

For the Zariski topology, this can be proved as follows: Let $U \hookrightarrow X$ be open, $F$ an injective sheaf on $X$. Then one has an exact sequence $0 \to j_!\mathcal{O}_U \to \mathcal{O}_X$ and $F(X) = \mathrm{Hom}(\mathcal{O}_X, F) \to \mathrm{Hom}(j_!\mathcal{O}_U, F) = \mathrm{Hom}(\mathcal{O}_U, F|_U) = F(U)$. Here, the middle arrow is surjective.)

share|cite|improve this question
It seems to the me that the étale case should reduced to the Zariski case, because the pushforward from the étale site to the Zariski site is also flabby. Am I missing something? – Angelo Apr 20 '13 at 9:35
A variation on Angelo's question: where does the argument in the last paragraph use the Zariski topology? All you need is that $\pi_* F(Y) \to \pi_* F(Y - Z)$ is surjective, which is the same as asking that $F(X) \to F(X - f^{-1} Z)$ be surjective, which is what the last paragraph shows. – anon Apr 20 '13 at 17:12
Thanks to both of you. – Timo Keller Apr 22 '13 at 14:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.