Zariski tangent space of a scheme as the vector space of derivations

2012-01-31T17:41:07Z

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.

Zariski tangent space of a scheme as the vector space of derivations - MathOverflow

Zariski tangent space of a scheme as the vector space of derivations