The Stacks project

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 $\omega _ A^\bullet = \varphi ^!(k[0])$ by Lemma 47.24.3. We conclude by Lemma 47.25.2. $\square$

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