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Let $i:X \hookrightarrow Y$ be a closed subscheme. Assume $X$ and $Y$ are projective schemes. Let $\mathcal{F}$ be an $\mathcal{O}_Y$-module. Under what condition on $\mathcal{F}$ or $X, Y$ can we conclude that the natural morphism from the global sections $\Gamma(Y,\mathcal{F})$ to $\Gamma(X,i^*\mathcal{F})$ is surjective? Here by $i^*\mathcal{F}$ is the pullback of $\mathcal{F}$ by $i$.

Chen
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    In the case where $Y$ is projective space and $\mathcal{F} = \mathcal{O}(n)$ for $n \ge 0$ then we need $X$ to be projectively normal. In particular if $X$ is a complete intersection this holds. This is an exercise in Hartshorne. So if $X$ and $Y$ are both projectively normal and $\mathcal{F}$ as before it should hold too. – Nate Aug 29 '13 at 17:38

1 Answers1

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Using $\Gamma(X,i^* F) = \Gamma(Y,i_* i^* F)$ and $i_* i^* F = F/IF$, where $I$ is the ideal defining $i$, we see that a sufficient condition is $H^1(Y,IF)=0$.

Martin Brandenburg
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  • @Brandenburg: Does this mean that if $\mathcal{F}$ is supported on $X$, then we have the surjectivity above or is it necessary in addition that $Y$ be reduced? – Chen Aug 30 '13 at 09:25
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    When $F$ is supported on $X$ and $F$ is of finite type, then $i_* i^* F = F$. – Martin Brandenburg Aug 30 '13 at 17:51