## 48.26 More on dualizing complexes

Some lemmas which don't fit anywhere else very well.

Lemma 48.26.1. Let $f : X \to Y$ be a morphism of locally Noetherian schemes. Assume

1. $f$ is syntomic and surjective, or

2. $f$ is a surjective flat local complete intersection morphism, or

3. $f$ is a surjective Gorenstein morphism of finite type.

Then $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ is a dualizing complex on $Y$ if and only if $Lf^*K$ is a dualizing complex on $X$.

Proof. Taking affine opens and using Derived Categories of Schemes, Lemma 36.3.5 this translates into Dualizing Complexes, Lemma 47.26.2. $\square$

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).