Notation: Let $X$ be an $S$-scheme and let $F:=X\times_{S}X$. We have the diagonal map $\Delta:X\rightarrow F$. Since this is an immersion, it splits as $X\xrightarrow{\delta} W\xrightarrow{i} F$, where $\delta$ is a closed immersion and $W$ is an open set.
Hartshorne defines $\Omega_{X/S}$ as $\delta^*(\mathcal{I}/\mathcal{I}^2)$ where $\mathcal{I}$ is the ideal-sheaf of the closed immersion $\delta$ and observes that this is independent of the choice of $W$. My question is:
Why not define $\Omega_{X/S}$ as $\Delta^*(\mathcal{J}/\mathcal{J}^2)$, where $\mathcal{J}=Ker(\mathcal{O}_F\rightarrow \Delta_*(\mathcal{O}_X)$? I seem to be able to verify that this is equivalent to the above definition.