The Stacks project

Lemma 29.25.15. Let $f : X \to Y$ be a flat morphism of schemes. Let $V \subset Y$ be a retrocompact open which is scheme theoretically dense. Then $f^{-1}V$ is scheme theoretically dense in $X$.

Proof. We will use the characterization of Lemma 29.7.5. We have to show that for any open $U \subset X$ the map $\mathcal{O}_ X(U) \to \mathcal{O}_ X(U \cap f^{-1}V)$ is injective. It suffices to prove this when $U$ is an affine open which maps into an affine open $W \subset Y$. Say $W = \mathop{\mathrm{Spec}}(A)$ and $U = \mathop{\mathrm{Spec}}(B)$. Then $V \cap W = D(f_1) \cup \ldots \cup D(f_ n)$ for some $f_ i \in A$, see Algebra, Lemma 10.29.1. Thus we have to show that $B \to B_{f_1} \times \ldots \times B_{f_ n}$ is injective. We are given that $A \to A_{f_1} \times \ldots \times A_{f_ n}$ is injective and that $A \to B$ is flat. Since $B_{f_ i} = A_{f_ i} \otimes _ A B$ we win. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 29.25: Flat morphisms

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Your email address will not be published.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 081H. Beware of the difference between the letter 'O' and the digit '0'.