Lemma 47.25.3. Let $K/k$ be an extension of fields. Let $A$ be a finite type $k$-algebra. Let $A_ K = A \otimes _ k K$. If $\omega _ A^\bullet $ is a dualizing complex for $A$, then $\omega _ A^\bullet \otimes _ A A_ K$ is a dualizing complex for $A_ K$.
Proof. By the uniqueness of dualizing complexes, it doesn't matter which dualizing complex we pick for $A$; we omit the detailed proof. Denote $\varphi : k \to A$ the algebra structure. We may take $\omega _ A^\bullet = \varphi ^!(k)$ by Lemma 47.24.3. We conclude by Lemma 47.25.2. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.