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Consider a general scheme $S$ and the projective scheme $\mathbb{P}^n_S$ over $S$.

Question: Is it possible to construct it as a quotient of an (open? affine?) subscheme $W_n\subset \mathbb{A}^{n+1}_S$ modulo the action of the multiplicative group $\mathbb{G}_m$?

I understand that, if $S=\operatorname{Spec}(K)$, $K$ a field, then one can take $W_n=\mathbb{A}^{n+1}\setminus \{(0,\dots,0)\}$, but over a more general affine scheme, say $S=\operatorname{Spec}(R)$, $R$ a ring, my guess was that the points of the (hypothetical) scheme $W_n$ should be something like $$W_n(R)=\{(a_0,\dots,a_n)\in R^{n+1} \mid a_0R+\cdots +a_nR=R\}$$ and I am not sure if this functor is representable by an scheme.

May be my confusion is to expect that for a general ring one would have $$\mathbb{P}^n(R)\cong W_n(R)/\mathbb{G}_m(R).$$

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    $\begingroup$ It can be done. See 19.2 here – but there's not much detail, and the formula there has to be interpreted in topos language. Your idea is almost correct – the quotient defines a presheaf which will need to be sheafified with respect to the Zariski topology. $\endgroup$
    – Zhen Lin
    Oct 1, 2020 at 8:53
  • $\begingroup$ @ZhenLin So you say one can define "presheaf" in Big Zarisky site: $$R\mapsto W_n(R):=\{(a_0,\dots,a_n)\in R^{n+1} \mid a_0R+\cdots +a_nR=R\}$$ and then shheafified one gets an scheme, and then $\mathbb{P}^n$ is just the quotient? $\endgroup$
    – Nulhomologous
    Oct 1, 2020 at 9:33
  • $\begingroup$ Your $W_n$, or at least its sheafification, is a scheme: it can be covered by the $n + 1$ affine schemes defined by $R \mapsto \{ (a_0, \ldots, a_n) \in R^{n+1} : a_j \text{ is invertible in } R \}$. $\endgroup$
    – Zhen Lin
    Oct 1, 2020 at 22:38

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