Zariski tangent space of a scheme as the vector space of derivations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T13:09:44Z http://mathoverflow.net/feeds/question/87150 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87150/zariski-tangent-space-of-a-scheme-as-the-vector-space-of-derivations Zariski tangent space of a scheme as the vector space of derivations Dima Sustretov 2012-01-31T17:41:07Z 2012-02-01T11:17:36Z <p>A standard lemma says that for a scheme $X$ of finite type over an algebraically closed field $k$ the set of derivations <code>$\mathcal{O}_{X,x} \to \kappa(x)=k$</code>, is isomorphic to the Zariski tangent space <code>$(\mathfrak{m}/\mathfrak{m}^2)^*$</code> where $\mathfrak{m}$ is the maximal ideal of $\mathcal{O}_{X,x}$.</p> <p>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 <code>$\mathcal{O}_{X,x} \cong k \oplus \mathfrak{m}$</code>. This already breaks down when $k$ is not algebraically closed and $\kappa(x)$ is a finite extension of $k$.</p> <p>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.</p>