The Stacks project

Lemma 39.6.4. Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Let $s \in S$. Then the composition

\[ T_{G/S, e(s)} \oplus T_{G/S, e(s)} = T_{G \times _ S G/S, (e(s), e(s))} \rightarrow T_{G/S, e(s)} \]

is addition of tangent vectors. Here the $=$ comes from Varieties, Lemma 33.16.7 and the right arrow is induced from $m : G \times _ S G \to G$ via Varieties, Lemma 33.16.6.

Proof. We will use Varieties, Equation (33.16.3.1) and work with tangent vectors in fibres. An element $\theta $ in the first factor $T_{G_ s/s, e(s)}$ is the image of $\theta $ via the map $T_{G_ s/s, e(s)} \to T_{G_ s \times G_ s/s, (e(s), e(s))}$ coming from $(1, e) : G_ s \to G_ s \times G_ s$. Since $m \circ (1, e) = 1$ we see that $\theta $ maps to $\theta $ by functoriality. Since the map is linear we see that $(\theta _1, \theta _2)$ maps to $\theta _1 + \theta _2$. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 39.6: Properties of group schemes

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Your email address will not be published.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 0BF5. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 39.6.4 as pdf