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 ( 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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