2
$\begingroup$

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

$\endgroup$
  • 1
    $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.