# Restriction of the sheaf of relative differentials

Let $$f:X\rightarrow C$$ be a morphism, where $$C$$ is a smooth curve, and let $$\Omega_f$$ be the sheaf of relative differentials.

For $$t\in C$$ let $$i_t:X_t = f^{-1}(t)\rightarrow X$$ be the inclusion of the fiber of $$f$$ over $$t$$. Does there exist an isomorphism $$i_t^{*}\Omega_f \cong \Omega_{X_t}$$ ?

• Yes. This EGA IV, 16.4.5 (there might exist a lighter reference). Note that this has not much to do with the dualizing sheaf. – abx Jan 11 at 19:26
• Hartshorne Proposition II.8.10. – Frank Jan 12 at 9:09
• Thanks a lot to both of you! – gxg Jan 12 at 12:44