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 $\theta _1 + \theta _2$.
$\square$

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