I am currently in the midst of an algebraic geometry course. Recently we learned the definition of a projective morphism of schemes and that all projective morphisms are proper. However after carefully thinking through some examples I arrived at the following problem which has been baffling me for some hours.
It seems to me that we can define an open immersion of affine $n$-space into Proj $k(x_0, x_1, \cdots x_n)$ by taking the image of said immersion to be the principal open set $D_+(x_i)$ which is of course part of its affine cover. Clearly this is wrong though as that then implies the structure morphism from affine space to $k$ to be projective, thence proper and thus that the structure map is finite which is clearly ridiculous.
This is such a natural question that the fact it does not seem to have been asked before seems worrying to me and I suspect that there may be a problem with my mental definition of open immersion that I cannot find. Any help would be appreciated. Thank you.