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?

isthe 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