I am trying to understand the relative dualising sheaf of a morphism of varieties in general but also in the special case of covers of the projective line by a elliptic curve, let us say of the form $Y=Spec(R) \to X=Spec(S)$ with $R=k[x,y]/(y^2=x(x-1)(x-\lambda))$ and $S=k[x]$ with the morphism coming from the inclusion.
In the literature I have found two different global descriptions of the dualising sheaf of a morphism by
- $\mathcal{Hom}_{f^{-1}\mathcal{O}_X}(f^{-1} \Omega_X, \Omega_Y)$
- $(f^*\Omega_X)' \otimes \Omega_Y$
and the local description as
- the coherent module associated to $\Omega_{R/S}$.
I would like to know how these concepts are related, so in the best case what an explicit isomorphism is. Especially, I am interested in what the image of the "canonical" element mapping $d \varphi \to d(\varphi \circ f)$ in the first case in concept $2$ and $3$ is.
This whole thing is quite confusing for me, as in my case I think $\Omega_X$ is just associated to the $k[x]$ module generated by $dx$ and $\Omega_Y$ is the $R$-module generated by $dx, dy$ modulo the relation $r=2ydy-(3x^2-2(\lambda+1)x-\lambda)dx$.
But then description $2$ gives me the module associated to $R[dy,dx,(dx)^{-1}]/r$ while concept $3$ gives me (as $dx=0$) just $R[dy]/2ydy$, and in none of these cases I am able to describe the "canonical element". So I would be grateful for some clarification.