Lemma 90.12.2 (06IH)—The Stacks project

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Table of contents
Part 6: Deformation Theory
Chapter 90: Formal Deformation Theory
Section 90.12: Tangent spaces of predeformation categories
Lemma 90.12.2 (cite)

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Lemma 90.12.2. Let $\mathcal{F}$ be a predeformation category such that $\overline{\mathcal{F}}$ satisfies (S2)¹. Then $T \mathcal{F}$ has a natural $k$ -vector space structure. For any finite dimensional vector space $V$ we have $\overline{\mathcal{F}}(k[V]) = T\mathcal{F} \otimes _ k V$ functorially in $V$ .

Proof. Let us write $F = \overline{\mathcal{F}} : \mathcal{C}_\Lambda \to \textit{Sets}$ . This is a predeformation functor and $F$ satisfies (S2). By Lemma 90.10.4 (and the translation of Remark 90.10.3) we see that

$F(A \times _ k k[V]) \longrightarrow F(A) \times F(k[V])$

is a bijection for every finite dimensional vector space $V$ and every $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_\Lambda )$ . In particular, if $A = k[W]$ then we see that $F(k[W] \times _ k k[V]) = F(k[W]) \times F(k[V])$ . In other words, the hypotheses of Lemma 90.11.8 hold and we see that $TF = T \mathcal{F}$ has a natural $k$ -vector space structure. The final assertion follows from Lemma 90.11.15. $\square$

[1] For example if

$\mathcal{F}$ satisfies (S2), see Lemma 90.10.5.

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