The sheaf of differential 1-forms

Let $X$ be a separated $S$ -scheme. For each $n\in \mathbb{N}$ , there exists a natural projection map.

$$\pi_n:\mathcal J_{n+1} \to \mathcal J_{n}$$

Let us restrict ourselves to the case where $n=0$ . $\mathcal J_{0}$ is equal to $\mathcal O_X$ and $\pi_0$ splits in a natural way.

$$\pi_0: \mathcal J_1 \to \mathcal O_X \qquad ($$split$$)$$

which yields an decomposition

$$\mathcal J_1 \cong \mathcal O_X \oplus \Omega_{X/S}^1$$

for a unique quasi coherent sheaf $\Omega_{X/S}^1$ .

DEFINITION 9.15   The sheaf $\Omega_{X/S}^1$ is called the sheaf of 1-forms on $X$ relative to $S$ .