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Example 90.11.11. Let us work out the tangent space of Example 90.11.10 when $F : \mathcal{C}_\Lambda \to \textit{Sets}$ is a prorepresentable functor, say $F = \underline{S}|_{\mathcal{C}_\Lambda }$ for $S \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$. Then $F$ commutes with arbitrary limits and thus satisfies the hypotheses of Lemma 90.11.8. We compute

\[ TF = F(k[\epsilon ]) = \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }(S, k[\epsilon ]) = \text{Der}_\Lambda (S, k) \]

and more generally for a finite dimensional $k$-vector space $V$ we have

\[ F(k[V]) = \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }(S, k[V]) = \text{Der}_\Lambda (S, V). \]

Explicitly, a $\Lambda $-algebra map $f : S \to k[V]$ compatible with the augmentations $q : S \to k$ and $k[V] \to k$ corresponds to the derivation $D$ defined by $s \mapsto f(s) - q(s)$. Conversely, a $\Lambda $-derivation $D : S \to V$ corresponds to $f : S \to k[V]$ in $\mathcal{C}_\Lambda $ defined by the rule $f(s) = q(s) + D(s)$. Since these identifications are functorial we see that the $k$-vector spaces structures on $TF$ and $\text{Der}_\Lambda (S, k)$ correspond (see Lemma 90.11.5). It follows that $\dim _ k TF$ is finite by Lemma 90.4.5.

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