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I am trying to solve the exercice 1.9 page 173 in Liu's Algebraic Geometry and I cannot find my way in question c).

Let $\mathcal{F}$ be a coherent sheaf on a locally Noetherian scheme $X$. We define the annihilator $Ann(\mathcal{F})$ of $\mathcal{F}$ to be

$$Ann(\mathcal{F})(U) := \{f\in \mathcal{O}_X(U) \ | \ \forall x\in U, \ \forall m_x\in \mathcal{F}_x, f\cdot m_x=0\} $$

for any open set $U\subset X$. (Liu defines it differently, to be the kernel of the sheaf morphism $ \alpha : \mathcal{O}_X\rightarrow \mathcal{Hom}_{\mathcal{O}_X}(\mathcal{F},\mathcal{F})$)

We need to prove that for any affine open set $U$, elements in $Ann(\mathcal{F})(U)$ are the one in $\mathcal{O}_X(U)$ killing global sections of $\mathcal{F}$. Explicitely this means

$$ Ann(\mathcal{F})(U)= \{f\in \mathcal{O}_X(U) \ | \ \forall m\in \mathcal{F}(U), f\cdot m=0\} $$

I can prove that if an element $f\in \mathcal{O}_X(U)$ kills all the germs then using sheaf property it kills the global sections. I am missing the other way, i.e. that if an element $f\in \mathcal{O}_X(U)$ kills all the global sections then it kills all the germs. I guess here comes the coherency of $\mathcal{F}$.

Any idea?

BinAcker
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    You should post the full context of the question. – user113102 Aug 23 '19 at 16:19
  • @user113102 done – BinAcker Aug 25 '19 at 09:55
  • Well, you've pretty much done the hardest part, if $f$ kills an element $s\in \mathcal{F}(U)$, then it certainly kills its germ $s_x$, but every element in the stalk $\mathcal{F}_x$ is of the form $s_x$ for $s$ a section of $\mathcal{F}(U)$ as $\mathcal{F}$ is quasi coherent and $U$ is affine. – Ahr Aug 26 '19 at 12:00
  • @Ahr I know this is not true if $\mathcal{F}$ is only quasi-coherent, where does it appear in your explaination? – BinAcker Aug 26 '19 at 14:39
  • Well it's because if $\mathcal{F}$ is not coherent then $Ann(\mathcal{F})$ might not be quasi-coherent. But in any case the two $O_X(U)$-modules in your question should be the same. – Ahr Aug 26 '19 at 14:55

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