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?