The Stacks project

Remark 98.21.7 (Compatibility with previous tangent spaces). Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$ has (RS*). Let $k$ be a field of finite type over $S$ and let $x_0$ be an object of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(k)$. Then we have equalities of $k$-vector spaces

\[ T\mathcal{F}_{\mathcal{X}, k, x_0} = T_{x_0}(k) \quad \text{and}\quad \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) = \text{Inf}_{x_0}(k) \]

where the spaces on the left hand side of the equality signs are given in ( and ( and the spaces on the right hand side are given by Lemma 98.21.2.

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