The Stacks project

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. $\square$


Comments (6)

Comment #476 by Nuno on

"Note that the sheaf of rings ..." 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 be affine open of mapping in to affine subset of , we can consider the kahler differential on induced from , which is clearly the same as the kahler differential here restricted to , because of the local definition of the kahler differential.

So by lemma 00RW, this kahler differential is also the conormal sheaf of , where and . I guess what is left is to show that conormal sheaves restrict well.

The conormal sheaf is given by the pullback of on some maximal subscheme where is closed immersion. So we are in the following situation.

We have closed immersion and and we just need to show that the conormal sheaf for restricts to the conormal sheaf of , 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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