Coherent algebraic sheaf on a closed subvariety

Question

I am reading Serre's Algebraic Coherent Sheaves. I can't see why it holds the remark at the end of chapter 39 (page 48 in the link):

"Let $\mathcal{G}$ be a coherent algebraic sheaf on V which is zero outside W (a closed subvariety of V). The annihilator of $\mathcal{G}$ does not necessarily contain $\mathcal{I}(W)$; all we can say is that it contains a power of $\mathcal{I}(W)$".

Could someone please help me?

Answer 1

Here is an example illustrating Serre's Remark:

Let $V=\mathbb A^1_k, \mathcal O(V)=k[x]$ and let $W=\{0\}$, the origin.
The coherent sheaf $\mathcal I(W)$ is asoociated to the ideal $(x)\subset k[x]$.
Now consider the coherent sheaf $\mathcal G:=\mathcal O_V/\mathcal I(W)^2$ on $V$ associated to the $k [x]$-module $k[x]/(x^2)=:k[\epsilon]$.
It has supp $\mathcal G=W$ as support and nevertheless it is not a coherent $\mathcal O_W$-module.
Actually it is not even a sheaf of $\mathcal O_W$-modules for the same reason that $k[\epsilon]=k[x]/(x ^2)$ is not a $k[x]/(x)$-module.