Are the affine-space $A_{n}$ as a group-action object vs affine-space $A_{n}$ as subspace (open set complement of a projective hyperplane: coordinate projective hyperplane, etc.) of a projective space $P_{n+1}$ (quotient space) rationale approaches mutually compatible?
Another way of stating the problem perhaps would be: is the approach of an affine-subspace as given by means of a faithful and transitive group action $A \times V \rightarrow{A}$ supportive of (or at least not incompatible with) the projective-space subspace approach (removing an hyperplane from $P_{n+1}$)?
Thanks in advance.