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I have the following question (should be easy for those who know something about the field): On page 92 (97 of the old edition) of Mumford's book "Abelian varieties", the author talks about an isomorphism $\frac{\mathfrak{M}_x}{\mathfrak{M}_x^{2}}\rightarrow\Omega_{X,x}\otimes_{\mathcal{O}_{X,x}}k$,

where $\left(X,\mathcal{O}_{X}\right)$ is a scheme of finite type over an (algebraically closed) field $k$. In order for that morphism to make sense, $k$ has to be a $\mathcal{O}_{X,x}$-module. Can someone please explain to me how $\mathcal{O}_{X,x}$ acts on $k$?

Thanks in advance,

jadahue

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    $\begingroup$ You have the homomorphism ${\mathcal O}_{X,x}\to k$ (evaluating a function at $x$). This one does define the action of ${\mathcal O}_{X,x}$ on $k$. $\endgroup$
    – Sasha Anan'in
    Commented Feb 4, 2014 at 17:44
  • $\begingroup$ If this doesn't look obvious to you, Hartshorne has a user-friendly introduction to differentials: Chap. II, §8. $\endgroup$
    – abx
    Commented Feb 4, 2014 at 18:43

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