Okay... I'm really not trying to beat this thing to death, but I have a preliminary exam coming up soon, and I'm very nervous because I still feeling like I'm struggling with what should be the most basic of questions.

I DO know that, if the hyperplane is defined simply by one of the coordinates (for sake of example, say $\displaystyle H=\mathcal{Z}(x_0)$), then the statement is true. On the complement of $\displaystyle H$, we know that the $\displaystyle x_0$ coordinate is nonzero. Since we are in projective space, we may assume that $\displaystyle x_0=1$ (because in $\displaystyle \mathbb{P}^n,(x_0,\ldots,x_n)\equiv (\lambda x_0,\ldots,\lambda x_n)\, \forall \, \lambda\in k\setminus\{0\}$; if we are restricting to points where $\displaystyle x_0\neq 0$, we may as well "normalize" our points by dividing through by $\displaystyle x_0$). In this case, after this normalization, we may define a map

$\displaystyle \varphi:\mathbb{P}^n\setminus H\rightarrow\mathbb{A}^n$

$\displaystyle (1,x_1,\ldots,x_n)\mapsto(x_1\ldots,x_n)\in\mathbb {A}^n$

This turns out to be an isomorphism $\displaystyle \mathbb{P}^n\setminus H\cong\mathbb{A}^n$.

Is there some sort of way I can generalize this to the complement of an arbitrary hyperplane (that is, to a set defined by the equation $\displaystyle a_0x_0+\cdots+a_nx_n\neq 0$,each $\displaystyle a_i\in k$)? Maybe some sort of (projective..) linear change of coordinates (which I don't know how to do...)? Would the mapping, for example, $\displaystyle x_0\mapsto f,x_1\mapsto x_1,\ldots,x_n\mapsto x_n$ be a legal isomorphism on the points of $\displaystyle \mathbb{P}^n$, in which case I could apply the idea above? Or is there something completely different I should be looking at?

I'm sorry for posting this stuff over and over when it seems no one really is able to help anyway... but I'm.. well, don't wanna say *desperate*, but... yea. I really feel like I am understanding the concepts, but I'm not able to apply them at all to solve anything.

EDIT: Am I asking this in the appropriate forum, or would another section be more appropriate? I just figure it would go here since it uses a good deal of abstract algebra...