# ag.algebraic geometry – When is the relative dualizing sheaf trivializable?

Let $$varphi: X to Y$$ be a finite locally free morphism of schemes. Then we have a right adjoint $$varphi^!$$ to the pushforward $$varphi_*$$ of quasicoherent sheaves along $$varphi$$. The relative dualizing sheaf $$omega_{X/Y}$$ of $$varphi$$ is defined to be $$varphi^! mathcal{O}_Y$$. My question is:

Are there any conditions on $$X$$, $$Y$$, or $$varphi$$, which guarantee that $$omega_{X/Y}$$ is isomorphic to $$mathcal{O}_X$$?

If $$varphi$$ is of degree $$d$$, we may construct the norm map $$Nm_{varphi}: pi_* mathcal{O}_X to mathcal{O}_Y$$, and so under these circumstances we have a global section of $$omega_{X/Y}$$ corresponding to the norm by adjunction. Can this map ever be an isomorphism?