Lemma 29.31.7. 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

$0 \to \mathcal{J}/\mathcal{J}^2 \to \mathcal{O}_ W/\mathcal{J}^2 \to \Delta _*\mathcal{O}_ X \to 0.$

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 29.30.1 and Lemma 29.30.2).

$0 \to \mathcal{C}_{X/X \times _ S X} \to \Delta ^{-1}(\mathcal{O}_ W/\mathcal{J}^2) \to \mathcal{O}_ X \to 0$

This places us in the situation of Modules, Lemma 17.26.11. 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

$s_ i : \mathcal{O}_ X \to \Delta ^{-1}(\mathcal{O}_ W/\mathcal{J}^2).$

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 17.26.11. 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

$\Omega _{X/S} \longrightarrow \mathcal{C}_{X/X \times _ S X},\quad f\text{d}g \longmapsto fs_2(g) - fs_1(g)$

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

$0 \to J/J^2 \to (A \otimes _ R A)/J^2 \to A \to 0$

The sections $s_ i$ correspond to the ring maps

$A \longrightarrow (A \otimes _ R A)/J^2,\quad s_1 : a \mapsto a \otimes 1,\quad s_2 : a \mapsto 1 \otimes a.$

By Lemma 29.30.2 we have $\Gamma (U, \mathcal{C}_{X/X \times _ S X}) = J/J^2$ and by Lemma 29.31.5 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 10.130.13. $\square$

Comment #476 by Nuno on

"Note that the sheaf of rings $\mathcal{O}_U/\mathcal{J}^2$..." Here U should be W.

Comment #1210 by Mohamed Hashi on

The word 'open' is repeated in the third sentence of the proof.

Comment #4684 by Andy on

I don't understand why this proof is so long if we are using Lemma 00RW. Let $U$ be affine open of $X$ mapping in to affine subset $V$ of $Y$, we can consider the kahler differential on $U$ induced from $U \to U \times_V U$, which is clearly the same as the kahler differential here restricted to $U$, because of the local definition of the kahler differential.

So by lemma 00RW, this kahler differential is also the conormal sheaf of $S \otimes_R S \to S$, where $V \cong \mathrm{Spec}(S)$ and $U \cong \mathrm{Spec}(R)$. I guess what is left is to show that conormal sheaves restrict well.

The conormal sheaf is given by the pullback of $\mathscr{I}/\mathscr{I^2}$ on some maximal subscheme where $X \to X \times_Y X$ is closed immersion. So we are in the following situation.

We have closed immersion $i: X \to Y$ and $i|_{f^(U)} : f^{-1}(U) \to U$ and we just need to show that the conormal sheaf for $i$ restricts to the conormal sheaf of $i|_{f^{-1}(U)}$, but this is completely clear because everything defined is local. What am I missing?

Comment #4685 by on

The proof is the same as what you are saying but also carefully constructs a global map between the two modules first before reducing to an affine local case. If you don't have a global map first, then you have to show that the local identifications glue, which is something we try to avoid as much as possible.

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