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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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$\def\sO{\mathcal{O}} \def\sF{\mathcal{F}} \def\ann{\operatorname{Ann}} \def\p{\mathfrak{p}}$The annihilator sheaf $\ann_{\sO_X}\sF$ is defined over an arbitrary ringed space $X$ and for any sheaf of $\sO_X$-modules $\sF$, see Tag 0H2G. For each $x\in X$, we have an inclusion of ideals of $\sO_{X,x}$ $$ \tag{1}\label{1} (\ann_{\sO_X}\sF)_x\subset\ann_{\sO_{X,x}}\sF_x, $$ that in general is not an equality.

Noetherianity assumptions don't play any role. Specifically:

Lemma. Let $X$ be a scheme.

  1. Let $\sF$ be a finite quasi-coherent sheaf of $\sO_X$-modules. Then $\ann_{\sO_X}\sF$ is quasi-coherent.

  2. Suppose $X=\operatorname{Spec}A$ is affine and let $M$ be an $A$-module. Then we have a containment of ideal sheaves of $\sO_X$ $$\tag{2}\label{2} \widetilde{\ann_AM}\subset\ann_{\sO_X}\widetilde{M} $$ that is an equality if $M$ is $A$-finite.

The containment \eqref{2} is not an equality in general, and $\ann_{\sO_X}\sF$ isn't quasi-coherent in general either [ref].

Proof. The first point follows from the second one, so we just show the latter. Since $S^{-1}\ann_AM\subset\ann_{S^{-1}A}S^{-1}M$, in particular for $f\in A$ we have that every element of $(\ann_AM)[f^{-1}]$ kills every section of $\widetilde{M}$ over $D(f)$. Thus, we have a containment $$ \widetilde{\ann_AM}\subset\ann_{\sO_X}\widetilde{M} $$ of ideal sheaves of $\sO_X$. Taking stalks at $x\in X$, we get containments $$ \label{3}\tag{3} (\ann_AM)_x\subset(\ann_{\sO_X}\widetilde{M})_x\subset\ann_{\sO_{X,x}}\widetilde{M}_x, $$ the first containment comes from \eqref{2} and the latter is \eqref{1}. If $M$ is finite, then the first term equals the third one (i.e, $(\ann_AM)_\p=\ann_{A_\p}M_\p$, where $\p\subset A$ is the prime associated to $x$ [ref]), so \eqref{3} becomes an equality and thus \eqref{2} too. $\square$

Elías Guisado Villalgordo
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