The Stacks project

Lemma 33.16.7. Let $X$, $Y$ be schemes over a base $S$. Let $x \in X$ and $y \in Y$ with the same image point $s \in S$ such that $\kappa (s) = \kappa (x)$ and $\kappa (s) = \kappa (y)$. There is a canonical isomorphism

\[ T_{X \times _ S Y/S, (x, y)} = T_{X/S, x} \oplus T_{Y/S, y} \]

The map from left to right is induced by the maps on tangent spaces coming from the projections $X \times _ S Y \to X$ and $X \times _ S Y \to Y$. The map from right to left is induced by the maps $1 \times y : X_ s \to X_ s \times _ s Y_ s$ and $x \times 1 : Y_ s \to X_ s \times _ s Y_ s$ via the identification (33.16.3.1) of tangent spaces with tangent spaces of fibres.

Proof. The direct sum decomposition follows from Morphisms, Lemma 29.32.11 via Lemma 33.16.5. Compatibility with the maps comes from Lemma 33.16.6. $\square$

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Your email address will not be published.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 0BEB. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 33.16.7 as pdf