Tensor product of dualizing complexes over a field

Abstract

Let $\mathbb{k} $ be a field, and let $X$ and $Y$ be two locally noetherian $\mathbb{k} $-schemes (respectively, $\mathbb{k} $-formal schemes) with dualizing complexes $R_X$ and $R_Y$, respectively. We show that $R_X \boxtimes _{\mathbb{k} } R_Y$ (respectively, its derived completion) is a dualizing complex over $X\times _{\mathbb{k} } Y$ if and only if $X\times _{\mathbb{k} } Y$ is locally noetherian of finite Krull dimension.