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I have the following problem:

Let $X$ be a scheme and $x$ a closed point on it. If $F$ is a sheaf on X with nontrivial stalk $F_x$ at $x$, then one has a canonical surjective morphism $F\rightarrow G$, where $G$ is the structure sheaf of the point $x$. I dont understand how to describe $G$ and where this morphism comes from and why it should be surjective. Perhaps one can explain that in detail for me.

Thank you very much

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    Dear Descartes, $k(x)$ (what I called $\kappa(x)$ above) means the residue field at $x$, i.e. the residue field of the local ring $\mathcal O_{X,x}$. Regards,2011-08-10

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The map exists (under some hypotheses --- e.g. $\mathcal F$ is a coherent sheaf of $\mathcal O_X$-modules), but is not canonical.

We can form the stalk $\mathcal F_x$, which is a module over $\mathcal O_{X,x}$. We can then form the fibre $\mathcal F_x /\mathfrak m_x \mathcal F_x$, which is a vector space over $\kappa(x)$. If the original sheaf $\mathcal F$ was coherent, and $x$ is in its support, then the fibre will be non-zero. Any non-zero vector space over $\kappa(x)$ admits a surjective map to $\kappa(x)$. (If a vector space $V$ is non-zero then the same is true of its dual.)

Choosing such a non-zero map $\mathcal F_x/\mathfrak m_x \mathcal F_x \to \kappa(x)$, we obtain the map $\mathcal F \to \mathcal G$ that you are asking about.

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    Perfect, that's what I wanted. Sorry for the error with "canonical", but in the text it looked as if it was.2011-08-10