The Stacks project

Lemma 38.12.8. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in X$ with image $s \in S$. Assume that

  1. $f$ is locally of finite presentation,

  2. $\mathcal{F}$ is of finite type, and

  3. $\mathcal{F}$ is flat at $x$ over $S$.

Then there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram of pointed schemes

\[ \xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^ g \ar[d] \\ (S, s) & (\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}), s') \ar[l] } \]

such that $X' \to X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is étale, $\kappa (x) = \kappa (x')$, the scheme $X'$ is affine of finite presentation over $\mathcal{O}_{S', s'}$, the sheaf $g^*\mathcal{F}$ is of finite presentation over $\mathcal{O}_{X'}$, and such that $\Gamma (X', g^*\mathcal{F})$ is a free $\mathcal{O}_{S', s'}$-module.

Proof. To prove the lemma we may replace $(S, s)$ by any elementary étale neighbourhood, and we may also replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$. Hence by Proposition 38.10.3 we may assume that $\mathcal{F}$ is finitely presented and flat over $S$ in a neighbourhood of $x$. In this case the result follows from Proposition 38.12.4 because Algebra, Theorem 10.85.4 assures us that projective $=$ free over a local ring. $\square$

Comments (0)

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05KY. Beware of the difference between the letter 'O' and the digit '0'.