Let $K$ be an algebraically closed field, and $V$ a finite dimensional vector space over it, $x$ a point in $V -\{ 0\}$. What does the tangent space of $V -\{ 0\}$ at $x$ look like?
When $X \in \mathbb{A}^n$ is a closed subset, with $\mathscr{I}(X) =
Further, in order to prove
The canonical morphism $\phi: V- \{0 \} \rightarrow \mathbb{P}(V)$ is separable,
do I have to prove $d \phi_x : \mathscr{T}(V- \{0 \})_x \rightarrow \mathscr{T}(\mathbb{P}(V))_y$ is surjective for a point $x \in V- \{0 \}$ and $y = \phi (x) \in \mathbb{P}(V)$?
If I have to prove the surjetivity of differential, how can I identify the tangent spaces and the differential?
(Page 42 on Linear Algebraic Groups, GTM21, written by James E. Humphreys)
Thanks a lot.