The Stacks project

Proof. In the proof of Lemma 97.27.12 we have seen that $F(V) = U'(V) \amalg W(V)$ if $V$ is the spectrum of an Artinian local ring. The tangent spaces are computed entirely from evaluations of $F$ on such schemes over $S$. Hence it suffices to prove that the tangent spaces of the functors $U'$ and $W$ are finite dimensional. For $U'$ this follows from Lemma 97.8.1. Write $W = \mathop{\mathrm{colim}}\nolimits W_ n$ as in the proof of Lemma 97.27.12. Then we see that the tangent spaces of $W$ are equal to the tangent spaces of $W_2$, as to get at the tangent space we only need to evaluate $W$ on spectra of Artinian local rings $(A, \mathfrak m)$ with $\mathfrak m^2 = 0$. Then again we see that the tangent spaces of $W_2$ have finite dimension by Lemma 97.8.1. $\square$