Lemma 88.12.2. 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 88.10.4 (and the translation of Remark 88.10.3) 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 88.11.8 hold and we see that $TF = T \mathcal{F}$ has a natural $k$-vector space structure. The final assertion follows from Lemma 88.11.15. $\square$

[1] For example if $\mathcal{F}$ satisfies (S2), see Lemma 88.10.5.

There are also:

• 2 comment(s) on Section 88.12: Tangent spaces of predeformation categories

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).