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A standard lemma says that for a scheme $X$ of finite type over an algebraically closed field $k$ the set of derivations $\mathcal{O}_{X,x} \to \kappa(x)=k$, is isomorphic to the Zariski tangent space $(\mathfrak{m}/\mathfrak{m}^2)^*$ where $\mathfrak{m}$ is the maximal ideal of $\mathcal{O}_{X,x}$.

The left-to-right inclusion is the natural one, but proving that it is bijective is depedent on one of the forms of the Nullstellensatz from which one deduces that $\mathcal{O}_{X,x} \cong k \oplus \mathfrak{m}$. This already breaks down when $k$ is not algebraically closed and $\kappa(x)$ is a finite extension of $k$.

I wonder if there are other interesting situations when this lemma doesn't work, i.e. when the inclusion $\mathrm{Der}(\mathcal{O}_{X,x},\kappa(x)) \hookrightarrow (\mathfrak{m}/\mathfrak{m}^2)^*$ (as $\kappa(x)$-vector spaces) is proper.

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Actually, the homomorphism $\mathrm{Der}(\mathcal O_{X,x},\kappa(x))\to(\mathfrak m/\mathfrak m^2)^*$ is not always injective. For a characterization of surjectivity in the case of a ground field, see Eisenbud, Commutative Algebra, Corollary 16.13. – user2035 Jan 31 '12 at 19:40
You can take $X = \mathrm{Spec} \mathbb{Z}_p$ and $\mathfrak{m} = (p)$. Then $\mathrm{Der}(\mathbb{Z}_p, \mathbb{F}_p)=0$. Is this the sort of thing you are looking for? – David Speyer Feb 1 '12 at 13:39
@a-fortiory: I see, there is no reason for it to be injective in general. It follows from the conormal sequnce for $\mathcal{O}_{X,x}$, $\mathfrak m$ and $k$ that it is non-injective if $\Omega(\mathcal{O}_{X,x}/\mathfrak{m},k)$ is non-zero, which you can only expect in a non-separable case. – Dima Sustretov Feb 2 '12 at 17:20
@David Speyer: Thank you for the example. I was rather wondering if the failure of this lemma is described by some theory, similarly to the Jacobson's "Galois" theory for fields lying between $k$ and $k^p$ for imperfect fields. Looking at the conormal sequence helps in the ground field case, as a-fortiori pointed out. I wonder if something can be said about some class of schemes not over a field too. – Dima Sustretov Feb 2 '12 at 17:32
Derivations are also nontrivial for non-closed points of varieties. The general case, say $f\colon X\to S$ and $f(x)=s$, may be reduced to the field case by considering the relative Zariski tangent space $\mathfrak m_x/(\mathfrak m_x^2+\mathfrak m_s\mathcal O_{X,x})$ instead and noticing $\Omega_{X_s,x/\kappa(s)}=\Omega_{X,x}\otimes\kappa(s)$. – user2035 Feb 7 '12 at 14:21

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