On Hartshorne's book, $\Omega_{X/Y}$ is given by $\Delta^*(\mathcal{F}/\mathcal{F}^2)$, where $\Delta:X\to X\times_YX$ is the diagonal embedding, $\mathcal{F}$ is the sheaf of ideals of $\Delta(X)$. (I just immigate the definition from schemes to varieties (not in the sense of schemes))

By here, let $f:X\to Y$ be a morphism of varieties. The $\Omega_{X/Y}$ is defined to be $\Omega_{\mathcal{O}_X/f^{-1}\mathcal{O}_Y}$.

I wonder if these two definitions of $\Omega_{X/Y}$ over varieties are equivalent?

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    $\begingroup$ Offtopic: Any reasonable definition of $\Omega^1_{X/Y}$ should satisfy the universal property $\mathrm{Hom}_{\mathcal{O}_X}(\Omega^1_{X/Y},M) \cong \mathrm{Der}_{\mathcal{O}_Y}(\mathcal{O}_X,M)$. And actually, this universal property is the definition, which works for all morphisms $X \to Y$ of ringed spaces, and what is usually called a definition is then merely a proof of the existence of $\Omega^1_{X/Y}$, very much like tensor products are defined via their universal property and can be constructed in several ways. Notice that Hartshorne's existence proof is the most cumbersome one. $\endgroup$ – Martin Brandenburg Feb 4 at 0:47
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    $\begingroup$ This also gives a hint why the two definitions agree: since they satisfy the same universal property. $\endgroup$ – Martin Brandenburg Feb 4 at 1:01
  • $\begingroup$ @MartinBrandenburg sorry can you explian how you see they satisfy the same universal property. I can't see it clearly. $\endgroup$ – 6666 Feb 8 at 20:59

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