Lemma 48.25.1. Let $X$ be a locally Noetherian scheme over the field $k$. Let $k'/k$ be a finitely generated field extension. Let $x \in X$ be a point, and let $x' \in X_{k'}$ be a point lying over $x$. Then we have
If $X$ is locally of finite type over $k$, the same holds for any field extension $k'/k$.
Proof. In both cases the ring map $\mathcal{O}_{X, x} \to \mathcal{O}_{X_{k'}, x'}$ is a faithfully flat local homomorphism of Noetherian local rings. Thus if $\mathcal{O}_{X_{k'}, x'}$ is Gorenstein, then so is $\mathcal{O}_{X, x}$ by Dualizing Complexes, Lemma 47.21.8. To go up, we use Dualizing Complexes, Lemma 47.21.8 as well. Thus we have to show that
is Gorenstein. Note that in the first case $k \to k'$ is finitely generated and in the second case $k \to \kappa (x)$ is finitely generated. Hence this follows as property (A) holds for Gorenstein, see Dualizing Complexes, Lemma 47.23.1. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)