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

.

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^2_{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^2_{S/B} \vert _U$ is still invertible?