The functors of points of $\AA^n$, $\AA^n \setminus \{0\}$ and $\PP^n$

(Affine space)

The functor \(X \mapsto \Gamma(X, \OO_X)^{n}\), sending a morphism \(f \colon X' \to X\) to \((a_i) \mapsto (f^\sharp a_i)\), is represented by \(\AA^n_R = \Spec R[x_1, \ldots, x_n]\), with universal element \((x_1, \ldots, x_n)\).

Over \(R = \ZZ\) this is the functor \(X \mapsto \Gamma(X, \OO_X)^n\) on all schemes, represented by \(\Spec \ZZ[x_1, \ldots, x_n]\): \(n\) variables, not \(n+1\).

(Punctured affine space)

\(\AA^n_R \setminus \{0\}\) denotes the open complement of \(V(x_1, \ldots, x_n)\). It represents the functor sending \(X\) to the set of \((a_1, \ldots, a_n) \in \Gamma(X, \OO_X)^n\) that are unimodular: the \(a_i\) generate the unit ideal sheaf, that is, for every \(x \in X\) some \(a_i(x) \neq 0\) in \(\kappa(x)\).

The condition is pointwise: on \(X = \Spec k[t]\), the pair \((t, 0)\) is not the zero pair, but both entries vanish at \(t = 0\), so it is not a point of \(\AA^2_k \setminus \{0\}\). When \(X = \Spec A\) it says \((a_1, \ldots, a_n) = A\).

(Projective space)

\(\PP^n_R\) represents the functor sending an \(R\)-scheme \(X\) to the set of isomorphism classes of tuples \((\mathcal{L}, s_0, \ldots, s_n)\) with \(\mathcal{L}\) an invertible sheaf on \(X\) and \(s_0, \ldots, s_n \in \Gamma(X, \mathcal{L})\) generating \(\mathcal{L}\), where \((\mathcal{L}, s_i) \cong (\mathcal{L}', s'_i)\) if there is an isomorphism \(\mathcal{L} \to \mathcal{L}'\) carrying each \(s_i\) to \(s'_i\). The universal tuple is \((\OO(1), x_0, \ldots, x_n)\), and a tuple on \(X\) corresponds to the morphism \(f \colon X \to \PP^n_R\) with \(f^* \OO(1) \cong \mathcal{L}\) and \(f^* x_i = s_i\) [@Har10a, Theorem II.7.1]; the construction is the gluing in D-OPENSUBFUNCTOR.

If every invertible sheaf on \(X\) is trivial, for instance \(X = \Spec A\) with \(A\) local, a tuple is a unimodular \((a_0, \ldots, a_n) \in \Gamma(X, \OO_X)^{n+1}\) up to multiplication by a unit of \(\Gamma(X, \OO_X)\). So the map \(\AA^{n+1}_R \setminus \{0\} \to \PP^n_R\) sends \((a_0, \ldots, a_n)\) to \((\OO_X, a_0, \ldots, a_n)\), and over a field \(k\) its \(k\)-points are the classical description of \(\PP^n(k)\) as nonzero \((n+1)\)-tuples up to \(k^\times\).