The Stacks project

Lemma 33.5.1. Let $K/k$ be an extension of fields. Let $X$ be scheme over $k$ and set $Y = X_ K$. If $y \in Y$ with image $x \in X$, then

  1. $\mathcal{O}_{X, x} \to \mathcal{O}_{Y, y}$ is a faithfully flat local ring homomorphism,

  2. with $\mathfrak p_0 = \mathop{\mathrm{Ker}}(\kappa (x) \otimes _ k K \to \kappa (y))$ we have $\kappa (y) = \kappa (\mathfrak p_0)$,

  3. $\mathcal{O}_{Y, y} = (\mathcal{O}_{X, x} \otimes _ k K)_\mathfrak p$ where $\mathfrak p \subset \mathcal{O}_{X, x} \otimes _ k K$ is the inverse image of $\mathfrak p_0$.

  4. we have $\mathcal{O}_{Y, y}/\mathfrak m_ x\mathcal{O}_{Y, y} = (\kappa (x) \otimes _ k K)_{\mathfrak p_0}$

Proof. We may assume $X = \mathop{\mathrm{Spec}}(A)$ is affine. Then $Y = \mathop{\mathrm{Spec}}(A \otimes _ k K)$. Since $K$ is flat over $k$, we see that $A \to A \otimes _ k K$ is flat. Hence $Y \to X$ is flat and we get the first statement if we also use Algebra, Lemma 10.39.17. The second statement follows from Schemes, Lemma 26.17.5. Now $y$ corresponds to a prime ideal $\mathfrak q \subset A \otimes _ k K$ and $x$ to $\mathfrak r = A \cap \mathfrak q$. Then $\mathfrak p_0$ is the kernel of the induced map $\kappa (\mathfrak r) \otimes _ k K \to \kappa (\mathfrak q)$. The map on local rings is

\[ A_\mathfrak r \longrightarrow (A \otimes _ k K)_\mathfrak q \]

We can factor this map through $A_\mathfrak r \otimes _ k K = (A \otimes _ k K)_{\mathfrak r}$ to get

\[ A_\mathfrak r \longrightarrow A_\mathfrak r \otimes _ k K \longrightarrow (A \otimes _ k K)_\mathfrak q \]

and then the second arrow is a localization at some prime. This prime ideal is the inverse image of $\mathfrak p_0$ (details omitted) and this proves (3). To see (4) use (3) and that localization and $- \otimes _ k K$ are exact functors. $\square$

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Your email address will not be published.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 0C4Y. Beware of the difference between the letter 'O' and the digit '0'.