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 $i_t^{*}\Omega_f \cong \Omega_{X_t}$ ?

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    $\begingroup$ 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. $\endgroup$ – abx Jan 11 at 19:26
  • $\begingroup$ Hartshorne Proposition II.8.10. $\endgroup$ – Frank Jan 12 at 9:09
  • $\begingroup$ Thanks a lot to both of you! $\endgroup$ – gxg Jan 12 at 12:44

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