Lemma 98.27.13. In Situation 98.27.1 the tangent spaces of the functor F are finite dimensional.
Proof. In the proof of Lemma 98.27.12 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 98.8.1. Write W = \mathop{\mathrm{colim}}\nolimits W_ n as in the proof of Lemma 98.27.12. 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 98.8.1. \square
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