sheaf of differentials

《Algebraic Geometry》 · 第 175 页 · Chapter II，Section 8，Definition

The diagonal morphism $\Delta$ induces an isomorphism of $X$ onto its image $\Delta(X)$, which is a locally closed subscheme of $X\times_YX$, i.e., a closed subscheme of an open subset $W$ of $X\times_YX$. Let $\mathcal J$ be the sheaf of ideals of $\Delta(X)$ in $W$. Then we define the sheaf of relative differentials of $X$ over $Y$ to be the sheaf $\Omega_{X/Y}=\Delta^*(\mathcal J/\mathcal J^2)$ on $X$.