Lemma 38.19.4. Let $f : X \to S$ be a morphism which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module which is of finite type. Let $x \in X$ with $s = f(x) \in S$. If $\mathcal{F}$ is flat at $x$ over $S$ there exists an affine elementary étale neighbourhood $(S', s') \to (S, s)$ and an affine open $U' \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ which contains $x' = (x, s')$ such that $\Gamma (U', \mathcal{F}|_{U'})$ is a free $\mathcal{O}_{S', s'}$-module.
Proof. The question is Zariski local on $X$ and $S$. Hence we may assume that $X$ and $S$ are affine. Then we can find a closed immersion $i : X \to \mathbf{A}^ n_ S$ over $S$. It is clear that it suffices to prove the lemma for the sheaf $i_*\mathcal{F}$ on $\mathbf{A}^ n_ S$ and the point $i(x)$. In this way we reduce to the case where $X \to S$ is of finite presentation. After replacing $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ and $X$ by an open of $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ we may assume that $\mathcal{F}$ is of finite presentation, see Proposition 38.10.3. In this case we may appeal to Lemma 38.19.3 and Algebra, Theorem 10.85.4 to conclude. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)