The Stacks project

Lemma 88.29.4. With notation and assumptions as in Situation 88.29.1. Assume $\mathcal{F}$ is a deformation category. Then there is a canonical $l$-vector space isomorphism

\[ \text{Inf}(\mathcal{F}) \otimes _ k l \longrightarrow \text{Inf}(\mathcal{F}_{l/k}) \]

of infinitesimal automorphism spaces.

Proof. Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$ and denote $x_{l, 0}$ the corresponding object of $\mathcal{F}_{l/k}$ over $l$. Recall that $\text{Inf}(\mathcal{F}) = \text{Inf}_{x_0}(\mathcal{F})$ and $\text{Inf}(\mathcal{F}_{l/k}) = \text{Inf}_{x_{l, 0}}(\mathcal{F}_{l/k})$, see Remark 88.19.4. Recall that the vector space structure on $\text{Inf}_{x_0}(\mathcal{F})$ comes from identifying it with the tangent space of the functor $\mathit{Aut}(x_0)$ which is defined on the category $\mathcal{C}_{k, k}$ of Artinian local $k$-algebras with residue field $k$. Similarly, $\text{Inf}_{x_{l, 0}}(\mathcal{F}_{l/k})$ is the tangent space of $\mathit{Aut}(x_{l, 0})$ which is defined on the category $\mathcal{C}_{l, l}$ of Artinian local $l$-algebras with residue field $l$. Unwinding the definitions we see that $\mathit{Aut}(x_{l, 0})$ is the restriction of $\mathit{Aut}(x_0)_{l/k}$ (which lives on $\mathcal{C}_{k, l}$) to $\mathcal{C}_{l, l}$. Since there is no difference between the tangent space of $\mathit{Aut}(x_0)_{l/k}$ seen as a functor on $\mathcal{C}_{k, l}$ or $\mathcal{C}_{l, l}$, the lemma follows from Lemma 88.29.3 and the fact that $\mathit{Aut}(x_0)$ satisfies (RS) by Lemma 88.19.6 (whence we have (S2) by Lemma 88.16.6). $\square$


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