# Definition of the $\Omega_{X/Y}$ over varieties

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?

• 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. – Martin Brandenburg Feb 4 at 0:47
• This also gives a hint why the two definitions agree: since they satisfy the same universal property. – Martin Brandenburg Feb 4 at 1:01
• @MartinBrandenburg sorry can you explian how you see they satisfy the same universal property. I can't see it clearly. – 6666 Feb 8 at 20:59