Example 89.11.14. Suppose that $f : R \to S$ is a ring map in $\widehat{\mathcal{C}}_\Lambda$. Set $F = \underline{R}|_{\mathcal{C}_\Lambda }$ and $G = \underline{S}|_{\mathcal{C}_\Lambda }$. The ring map $f$ induces a transformation of functors $G \to F$. By Lemma 89.11.13 we get a $k$-linear map $TG \to TF$. This is the map

$TG = \text{Der}_\Lambda (S, k) \longrightarrow \text{Der}_\Lambda (R, k) = TF$

as follows from the canonical identifications $F(k[V]) = \text{Der}_\Lambda (R, V)$ and $G(k[V]) = \text{Der}_\Lambda (S, V)$ of Example 89.11.11 and the rule for computing the map on tangent spaces.

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