Lemma 29.34.16. Let $f : X \to Y$, $g : Y \to S$ be morphisms of schemes. Assume $f$ is smooth. Then
\[ 0 \to f^*\Omega _{Y/S} \to \Omega _{X/S} \to \Omega _{X/Y} \to 0 \]
(see Lemma 29.32.9) is short exact.
Proof. The algebraic version of this lemma is the following: Given ring maps $A \to B \to C$ with $B \to C$ smooth, then the sequence
\[ 0 \to C \otimes _ B \Omega _{B/A} \to \Omega _{C/A} \to \Omega _{C/B} \to 0 \]
of Algebra, Lemma 10.131.7 is exact. This is Algebra, Lemma 10.139.1. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: