Originally I had asked this on mathstack exchange but seems possibly appropriate for overflow as well. 


Let $k$ be a field. For the vector space $k^d$, a line in $k^d$ is a one dimensional subspace of $k^d$. 


A *supplemented line bundle* of a free module $A^{d}$ over a commutative ring $A$ is a projective submodule $L$ of rank one with the extra condition that there exists a surjection $A^d \to L$


What exactly motivates this definition? 


Naively, I would have guessed it simply to be a rank one free submodule of $A^{d+1}$. 



Why are supplemented line bundles the correct generalization of "lines"?
Supplemented line bundles are the things which correspond to maps $$Spec(A)\to \mathbb P^{d-1}$$
from the scheme $Spec(A)$ to the projective space of dimension $d-1$ (defined over $\mathbb Z$).


