Let $f:S→B$ be an elliptic fibration from an integral surface $S$ to integral curve $B$

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Here I use following definitions:

A surface (resp. curve) is a $2$ -dim (resp. $1$-dim) proper k scheme over fixed field $k$.

Fibration has two properties: 1. $O_B = f_*O_S$ 2. all fibers of f are geometrically connected

Futhermore a fibration is elliptic if the generic fiber $S_{\eta}=f^{-1}(\eta)$ is an elliptic curve (over $k(\eta)$.

Denote by $i_S: S_{\eta} \to S$ the canonical immersion. Here I'm ot sure to 100% but I guess that for the structure sheaf holds $O_{S_{\eta}}= O_S \otimes_k k(\eta)$.

Now the QUESTION:

Since $S_{\eta}$ is elliptic curve and therefore smooth the restriction of the Kähler differentials $\omega_{S/B} \vert _{S_{\eta}}$ is invertible.

My question is how to see that there exist open neighbourhood $U \subset S$ of $S_{\eta}$ such that the restriction $\omega_{S/B} \vert _U$ is still invertible?