Definition of the sheaf of differential

algebraic-geometry

There is a lot of question about this subject but I can't get any comprehensive explanation for me.

In his section II.8 Hartshorne defines the sheaf of differential for $f:X\to Y$: he considers the diagonal morphism $\Delta:X\to X\times_Y X$ and says that it is a closed immersion in $W\subset X\times_Y X$. Telling $\mathscr{I}$ the ideal sheaf of $\Delta(X)$ in an open $W$ (union of the $U\times_Y U$ with the $U$ affines) he defines the sheaf of differentials $\Omega$ to be $\Delta^*(\mathscr{I}/\mathscr{I}^2)$.

My question is: why can we consider $\mathscr{I}/\mathscr{I}^2$ as a $\mathcal{O}_{X\times_Y X}$-module? it is a $\mathcal{O}_W$-module (ideal in fact), it is the kernel of $j^*:\mathcal{O}_W\to j_*\mathcal{O}_X$ with $j:X\to W$ induced by $\Delta$.

Should we consider in fact $i_*(\mathscr{I}/\mathscr{I}^2)$ where $i:W\to X\times_Y X$ is the canonical open immersion?

Best Answer

Let's start by reproducing the text.

Let $f:X\to Y$ be a morphism of schemes. We consider the diagonal morphism $\Delta: X\to X\times_YX$. It follows from the proof of (4.2) that $\Delta$ gives 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 $W\subset X\times_YX$.

Definition. Let $\mathscr{I}$ be the sheaf of ideals of $\Delta(X)$ in $W$. Then we can define the sheaf of relative differentials of $X$ over $Y$ to be the sheaf $\Omega_{X/Y}=\Delta^*(\mathscr{I/I}^2)$ on $X$.

Hartshorne is not claiming that $\mathscr{J/J}^2$ is a sheaf on $X\times_Y X$. Instead, he's using $\Delta$ to refer to the map which is the same as $\Delta:X\to X\times_YX$ but with the codomain restricted to to $W$ (which we can talk about since $\Delta$ lands in $W$ by definition of $W$).

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