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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.

$\displaystyle \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.

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

which yields an decomposition

$\displaystyle \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$ .



Subsections

2007-12-11