The Stacks project

Lemma 89.29.3. With notation and assumptions as in Situation 89.29.1. Assume $\mathcal{F}$ is a predeformation category and $\overline{\mathcal{F}}$ satisfies (S2). Then there is a canonical $l$-vector space isomorphism

\[ T\mathcal{F} \otimes _ k l \longrightarrow T\mathcal{F}_{l/k} \]

of tangent spaces.

Proof. By Lemma 89.29.2 we may replace $\mathcal{F}$ by $\overline{\mathcal{F}}$. Moreover we see that $T\mathcal{F}$, resp. $T\mathcal{F}_{l/k}$ has a canonical $k$-vector space structure, resp. $l$-vector space structure, see Lemma 89.12.2. Then

\[ T\mathcal{F}_{l/k} = \mathcal{F}_{l/k}(l[\epsilon ]) = \mathcal{F}(k[l\epsilon ]) = T\mathcal{F} \otimes _ k l \]

the last equality by Lemma 89.12.2. More generally, given a finite dimensional $l$-vector space $V$ we have

\[ \mathcal{F}_{l/k}(l[V]) = \mathcal{F}(k[V_ k]) = T\mathcal{F} \otimes _ k V_ k \]

where $V_ k$ denotes $V$ seen as a $k$-vector space. We conclude that the functors $V \mapsto \mathcal{F}_{l/k}(l[V])$ and $V \mapsto T\mathcal{F} \otimes _ k V_ k$ are canonically identified as functors to the category of sets. By Lemma 89.11.4 we see there is at most one way to turn either functor into an $l$-linear functor. Hence the isomorphisms are compatible with the $l$-vector space structures and we win. $\square$

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