= Projective space represents invertible quotients
{title2=$\mathbb P^n(T)=\{\mathcal O_T^{n+1}\twoheadrightarrow\mathcal L\}/\cong$}
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>.
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