The Stacks project

Lemma 107.7.4. Let $X$ be a proper scheme over a field $k$ of dimension $\leq 1$. Then properties (3)(a), (b), (c) are also equivalent to $\text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X) = 0$.

Proof. In the discussion above we have seen that $G = \mathit{Aut}(X)$ is a group scheme over $\mathop{\mathrm{Spec}}(k)$ which is finite type and separated; this uses Lemma 107.5.1 and More on Groupoids in Spaces, Lemma 77.10.2. Then $G$ is unramified over $k$ if and only if $\Omega _{G/k} = 0$ (Morphisms, Lemma 29.34.2). By Groupoids, Lemma 39.6.3 the vanishing holds if $T_{G/k, e} = 0$, where $T_{G/k, e}$ is the tangent space to $G$ at the identity element $e \in G(k)$, see Varieties, Definition 33.16.3 and the formula in Varieties, Lemma 33.16.4. Since $\kappa (e) = k$ the tangent space is defined in terms of morphisms $\alpha : \mathop{\mathrm{Spec}}(k[\epsilon ]) \to G = \mathit{Aut}(X)$ whose restriction to $\mathop{\mathrm{Spec}}(k)$ is $e$. It follows that it suffices to show any automorphism

\[ \alpha : X \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k[\epsilon ]) \longrightarrow X \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k[\epsilon ]) \]

over $\mathop{\mathrm{Spec}}(k[\epsilon ])$ whose restriction to $\mathop{\mathrm{Spec}}(k)$ is $\text{id}_ X$. Such automorphisms are called infinitesimal automorphisms.

The infinitesimal automorphisms of $X$ correspond $1$-to-$1$ with derivations of $\mathcal{O}_ X$ over $k$. This follows from More on Morphisms, Lemmas 37.9.1 and 37.9.2 (we only need the first one as we don't care about the reverse direction; also, please look at More on Morphisms, Remark 37.9.7 for an elucidation). For a different argument proving this equality we refer the reader to Deformation Problems, Lemma 91.9.3. $\square$


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