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

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: