Pushforward of sheaf of relative differentials in family of elliptic curves

Question

Update: never mind, This question has been asked before here Let $f:E \to S$ be an elliptic curve (precise definition is given below from Hida’s book Geometric Modular Forms and Elliptic Curves). Is $f_*\Omega_{E/S} \cong \mathcal{O}_S$? I would think so by Grothendieck-Serre duality and (2.15) in the image below, but few pages later Hida says (the push forward in question) is invertible on S, i.e locally free on S - while I seem to think it is globally free.