Lemma 29.32.7. Let $f : X \to S$ be a morphism of schemes. There is a canonical isomorphism between $\Omega _{X/S}$ and the conormal sheaf of the diagonal morphism $\Delta _{X/S} : X \longrightarrow X \times _ S X$.

**Proof.**
We first establish the existence of a couple of “global” sheaves and global maps of sheaves, and further down we describe the constructions over some affine opens.

Recall that $\Delta = \Delta _{X/S} : X \to X \times _ S X$ is an immersion, see Schemes, Lemma 26.21.2. Let $\mathcal{J}$ be the ideal sheaf of the immersion which lives over some open subscheme $W$ of $X \times _ S X$ such that $\Delta (X) \subset W$ is closed. Let us take the one that was found in the proof of Schemes, Lemma 26.21.2. Note that the sheaf of rings $\mathcal{O}_ W/\mathcal{J}^2$ is supported on $\Delta (X)$. Moreover it sits in a short exact sequence of sheaves

Using $\Delta ^{-1}$ we can think of this as a surjection of sheaves of $f^{-1}\mathcal{O}_ S$-algebras with kernel the conormal sheaf of $\Delta $ (see Definition 29.31.1 and Lemma 29.31.2).

This places us in the situation of Modules, Lemma 17.27.11. The projection morphisms $p_ i : X \times _ S X \to X$, $i = 1, 2$ induce maps of sheaves of rings $(p_ i)^\sharp : (p_ i)^{-1}\mathcal{O}_ X \to \mathcal{O}_{X \times _ S X}$. We may restrict to $W$ and quotient by $\mathcal{J}^2$ to get $(p_ i)^{-1}\mathcal{O}_ X \to \mathcal{O}_ W/\mathcal{J}^2$. Since $\Delta ^{-1}p_ i^{-1}\mathcal{O}_ X = \mathcal{O}_ X$ we get maps

Both $s_1$ and $s_2$ are sections to the map $\Delta ^{-1}(\mathcal{O}_ W/\mathcal{J}^2) \to \mathcal{O}_ X$, as in Modules, Lemma 17.27.11. Thus we get an $S$-derivation $\text{d} = s_2 - s_1 : \mathcal{O}_ X \to \mathcal{C}_{X/X \times _ S X}$. By the universal property of the module of differentials we find a unique $\mathcal{O}_ X$-linear map

To see the map is an isomorphism, let us work this out over suitable affine opens. We can cover $X$ by affine opens $\mathop{\mathrm{Spec}}(A) = U \subset X$ whose image is contained in an affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$. According to the proof of Schemes, Lemma 26.21.2 $U \times _ V U \subset X \times _ S X$ is an affine open contained in the open $W$ mentioned above. Also $U \times _ V U = \mathop{\mathrm{Spec}}(A \otimes _ R A)$. The sheaf $\mathcal{J}$ corresponds to the ideal $J = \mathop{\mathrm{Ker}}(A \otimes _ R A \to A)$. The short exact sequence to the short exact sequence of $A \otimes _ R A$-modules

The sections $s_ i$ correspond to the ring maps

By Lemma 29.31.2 we have $\Gamma (U, \mathcal{C}_{X/X \times _ S X}) = J/J^2$ and by Lemma 29.32.5 we have $\Gamma (U, \Omega _{X/S}) = \Omega _{A/R}$. The map above is the map $a \text{d}b \mapsto a \otimes b - ab \otimes 1$ which is shown to be an isomorphism in Algebra, Lemma 10.131.13. $\square$

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