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. $\square$

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