The Stacks project

Lemma 38.12.5. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \in S$. Assume that

  1. $f$ is of finite presentation,

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

  3. $\mathcal{F}$ is flat over $S$ at every point of the fibre $X_ s$.

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

\[ \xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ S & S' \ar[l] } \]

such that $g$ is étale, $X_ s \subset g(X')$, the schemes $X'$, $S'$ are affine, and such that $\Gamma (X', g^*\mathcal{F})$ is a projective $\Gamma (S', \mathcal{O}_{S'})$-module.

Proof. For every point $x \in X_ s$ we can use Proposition 38.12.4 to find a commutative diagram

\[ \xymatrix{ (X, x) \ar[d] & (Y_ x, y_ x) \ar[l]^{g_ x} \ar[d] \\ (S, s) & (S_ x, s_ x) \ar[l] } \]

whose horizontal arrows are elementary étale neighbourhoods such that $Y_ x$, $S_ x$ are affine and such that $\Gamma (Y_ x, g_ x^*\mathcal{F})$ is a projective $\Gamma (S_ x, \mathcal{O}_{S_ x})$-module. In particular $g_ x(Y_ x) \cap X_ s$ is an open neighbourhood of $x$ in $X_ s$. Because $X_ s$ is quasi-compact we can find a finite number of points $x_1, \ldots , x_ n \in X_ s$ such that $X_ s$ is the union of the $g_{x_ i}(Y_{x_ i}) \cap X_ s$. Choose an elementary étale neighbourhood $(S' , s') \to (S, s)$ which dominates each of the neighbourhoods $(S_{x_ i}, s_{x_ i})$, see More on Morphisms, Lemma 37.35.4. We may also assume that $S'$ is affine. Set $X' = \coprod Y_{x_ i} \times _{S_{x_ i}} S'$ and endow it with the obvious morphism $g : X' \to X$. By construction $g(X')$ contains $X_ s$ and

\[ \Gamma (X', g^*\mathcal{F}) = \bigoplus \Gamma (Y_{x_ i}, g_{x_ i}^*\mathcal{F}) \otimes _{\Gamma (S_{x_ i}, \mathcal{O}_{S_{x_ i}})} \Gamma (S', \mathcal{O}_{S'}). \]

This is a projective $\Gamma (S', \mathcal{O}_{S'})$-module, see Algebra, Lemma 10.94.1. $\square$

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Your email address will not be published.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 05KW. Beware of the difference between the letter 'O' and the digit '0'.