The Stacks project

Lemma 37.9.8. Let $S$ be a scheme. Let $X \subset X'$ be a first order thickening over $S$. Let $Y$ be a scheme over $S$. Let $a', b' : X' \to Y$ be two morphisms over $S$ with $a = a'|_ X = b'|_ X$. This gives rise to a commutative diagram

\[ \xymatrix{ X \ar[r] \ar[d]_ a & X' \ar[d]^{(b', a')} \\ Y \ar[r]^-{\Delta _{Y/S}} & Y \times _ S Y } \]

Since the horizontal arrows are immersions with conormal sheaves $\mathcal{C}_{X/X'}$ and $\Omega _{Y/S}$, by Morphisms, Lemma 29.31.3, we obtain a map $\theta : a^*\Omega _{Y/S} \to \mathcal{C}_{X/X'}$. Then this $\theta $ and the derivation $D$ of Lemma 37.9.1 are related by Equation (

Proof. Omitted. Hint: The equality may be checked on affine opens where it comes from the following computation. If $f$ is a local section of $\mathcal{O}_ Y$, then $1 \otimes f - f \otimes 1$ is a local section of $\mathcal{C}_{Y/(Y \times _ S Y)}$ corresponding to $\text{d}_{Y/S}(f)$. It is mapped to the local section $(a')^\sharp (f) - (b')^\sharp (f) = D(f)$ of $\mathcal{C}_{X/X'}$. In other words, $\theta (\text{d}_{Y/S}(f)) = D(f)$. $\square$

Comments (0)

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04FL. Beware of the difference between the letter 'O' and the digit '0'.