# Sheaf of Kähler Differentials is Invertible in Dense Open Subset

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?

• $\Omega ^2_{S/B}$ is zero. – abx Jun 30 at 4:45
• I suspect that the OP meant $\omega_{S/B}$... – Sándor Kovács Jun 30 at 23:05
• @Karl_Peter: try to use Nakayama's lemma. – Sándor Kovács Jun 30 at 23:07
• If the generic fiber is smooth, there are only finitely many singular fibers $F_i$. $\omega _{S/B}$ is invertible in $S\smallsetminus \cup F_i$. – abx Jul 1 at 14:35
• ...and this is based on a little Nakayama argument as Sándor states: Concretely if $x \in S_\eta$ then since $f^{\#}m_{\eta} \subset m_x \subset O_{S,x}$ then since $(\omega_{S/B} \vert _{S_{\eta}})_x= \omega_{S/B} \otimes O_{S,x}/m_{\eta}$ is invertible then also $\omega_{S/B} \otimes O_{S,x}$ (by Nakayama) and then there exist an open neighbourhood around $x$ where $\omega_{S/B}$ is also open. – Karl_Peter Jul 1 at 14:56