Projective space represents invertible quotients (${\mathbb{P}}^n(T)=\{{\mathcal{O}}_T^{n+1}\twoheadrightarrow \mathcal{L}\}/\cong$)

With the quotient convention, projective space represents the functor of surjections from a rank-$n+1$ free sheaf to a line bundle, modulo isomorphisms of the target compatible with the map. On an affine base the target may be a nontrivial rank-one projective module, so a tuple of elements generating the unit ideal is only the description when that target is trivial. Where the image of the $i$th basis vector generates the target, divide the other images by it to get the usual affine coordinates. Infinitesimal variations of the quotient form $\mathcal{H}om(\ker q,\mathcal{L})$ and give the Euler sequence.