The Stacks project

Remark 29.32.17. Let $X \to S$ be a morphism of schemes. According to Lemma 29.32.11 we have

\[ \Omega _{X \times _ S X/S} = \text{pr}_1^*\Omega _{X/S} \oplus \text{pr}_2^*\Omega _{X/S} \]

On the other hand, the diagonal morphism $\Delta : X \to X \times _ S X$ is an immersion, which locally has a left inverse. Hence by Lemma 29.32.16 we obtain a canonical short exact sequence

\[ 0 \to \mathcal{C}_{X/X \times _ S X} \to \Omega _{X/S} \oplus \Omega _{X/S} \to \Omega _{X/S} \to 0 \]

Note that the right arrow is $(1, 1)$ which is indeed a split surjection. On the other hand, by Lemma 29.32.7 we have an identification $\Omega _{X/S} = \mathcal{C}_{X/X \times _ S X}$. Because we chose $\text{d}_{X/S}(f) = s_2(f) - s_1(f)$ in this identification it turns out that the left arrow is the map $(-1, 1)$1.

[1] Namely, the local section $\text{d}_{X/S}(f) = 1 \otimes f - f \otimes 1$ of the ideal sheaf of $\Delta $ maps via $\text{d}_{X \times _ S X/X}$ to the local section $1 \otimes 1 \otimes 1 \otimes f - 1 \otimes f \otimes 1 \otimes 1 -1 \otimes 1 \otimes f \otimes 1 + f \otimes 1 \otimes 1 \otimes 1 = \text{pr}_2^*\text{d}_{X/S}(f) - \text{pr}_1^*\text{d}_{X/S}(f)$.

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