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 $A^{d+1}$.