The Stacks project

Lemma 107.23.1. Let $S$ be a scheme and $s \in S$ a point. Let $f : X \to S$ and $g : Y \to S$ be families of curves. Let $c : X \to Y$ be a morphism over $S$. If $c_{s, *}\mathcal{O}_{X_ s} = \mathcal{O}_{Y_ s}$ and $R^1c_{s, *}\mathcal{O}_{X_ s} = 0$, then after replacing $S$ by an open neighbourhood of $s$ we have $\mathcal{O}_ Y = c_*\mathcal{O}_ X$ and $R^1c_*\mathcal{O}_ X = 0$ and this remains true after base change by any morphism $S' \to S$.

Proof. Let $(U, u) \to (S, s)$ be an étale neighbourhood such that $\mathcal{O}_{Y_ U} = (X_ U \to Y_ U)_*\mathcal{O}_{X_ U}$ and $R^1(X_ U \to Y_ U)_*\mathcal{O}_{X_ U} = 0$ and the same is true after base change by $U' \to U$. Then we replace $S$ by the open image of $U \to S$. Given $S' \to S$ we set $U' = U \times _ S S'$ and we obtain étale coverings $\{ U' \to S'\} $ and $\{ Y_{U'} \to Y_{S'}\} $. Thus the truth of the statement for the base change of $c$ by $S' \to S$ follows from the truth of the statement for the base change of $X_ U \to Y_ U$ by $U' \to U$. In other words, the question is local in the étale topology on $S$. Thus by Lemma 107.4.3 we may assume $X$ and $Y$ are schemes. By More on Morphisms, Lemma 37.64.7 there exists an open subscheme $V \subset Y$ containing $Y_ s$ such that $c_*\mathcal{O}_ X|_ V = \mathcal{O}_ V$ and $R^1c_*\mathcal{O}_ X|_ V = 0$ and such that this remains true after any base change by $S' \to S$. Since $g : Y \to S$ is proper, we can find an open neighbourhood $U \subset S$ of $s$ such that $g^{-1}(U) \subset V$. Then $U$ works. $\square$

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Your email address will not be published.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 0E88. Beware of the difference between the letter 'O' and the digit '0'.