Tensor product of dualizing complexes over a field

TL;DR

This paper investigates conditions under which the tensor product of dualizing complexes over a field yields a dualizing complex on the product scheme, linking this to the local noetherian property and finite Krull dimension.

Contribution

It establishes a precise criterion for when the tensor product of dualizing complexes over a field produces a dualizing complex on the product scheme.

Findings

Tensor product of dualizing complexes is dualizing if and only if the product scheme is locally noetherian of finite Krull dimension.

Derived completion of the tensor product also yields a dualizing complex under the same conditions.

The result applies to both schemes and formal schemes over a field.

Abstract

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