The Stacks project

Remark 97.21.7 (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 (97.8.0.1) and (97.8.0.2) and the spaces on the right hand side are given by Lemma 97.21.2.


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