Projective scheme of a homogeneous quotient and its standard affine charts

Assume the Axiom of Choice (The Axiom of Choice), inherited here from the affine structure-sheaf construction (The localization construction extends to the structure sheaf on Spec A): its localization data form a sheaf on distinguished opens using the prime-existence and finite-subcover results. The finite chart gluing makes no additional choice.

Let $k$ be a field, let $R=k[x_0,\dots ,x_n]$ carry the standard grading in which every $x_i$ has degree one (Nonnegatively graded rings and modules, homogeneous elements, and twists, homogeneous polynomial and homogeneous ideal), let $I\subseteq R$ be a homogeneous ideal, and put $S=R\mathrm{/}I$ with the induced grading. Write

$S_{+}=(x_0,\dots ,x_n)S={⨁}_{m\ge 1}{S}_m$

for the irrelevant ideal of $S$, generated by the images of the variables.

1. Underlying set. The projective spectrum $\mathrm{Proj}S$ has as its points the homogeneous prime ideals $\mathfrak{p}\subseteq S$ with $S_{+}⊈\mathfrak{p}$; equivalently $x_i\mathrm{\notin }\mathfrak{p}$ for at least one $i$.
2. Standard charts. For each $i$ let $S_{x_i}$ be the localization of $S$ in which the powers of $x_i$ are inverted; it is a graded ring and its degree-zero part $(S_{x_i}{)}_0$ is a $k$-algebra. The standard chart $D_{+}(x_i)$ is the affine scheme $\mathrm{Spec}((S_{x_i}{)}_0)$ with its structure sheaf (Affine schemes and their coordinate rings), whose sections on a distinguished open $D(a)$ are $((S_{x_i}{)}_0{)}_a$. Concretely, $(S_{x_i}{)}_0$ is the image of the subring of fractions of $S$ of total degree zero with denominator a power of $x_i$; in the polynomial case with $I=0$ it is the polynomial ring in the ratios $x_j\mathrm{/}{x}_i$ for $j\mathrm{\ne }i$.
3. Overlaps and gluing. For all $i,j$ the localizations $(S_{x_i}{)}_0$ and $(S_{x_j}{)}_0$ have the common localization $(S_{x_i{x}_j}{)}_0$, obtained from either one by inverting the degree-zero ratio $x_j\mathrm{/}{x}_i$ respectively $x_i\mathrm{/}{x}_j$; these identifications of $\mathrm{Spec}((S_{x_i{x}_j}{)}_0)$ with distinguished open subschemes of the two charts are isomorphisms of locally ringed spaces (A principal localization identifies its spectrum with a distinguished open). They are the transition isomorphisms, and they satisfy the identity and cocycle conditions because both composites are the canonical identification inside the double localization $S_{x_i{x}_j{x}_k}$. The affine schemes $D_{+}(x_i)$ therefore glue to a scheme $\mathrm{Proj}S$ (Gluing affine schemes along compatible open isomorphisms, Schemes), the standard charts forming an open affine cover, with the points of part 1 as its underlying set and the subset $D_{+}(x_i{x}_j)\subseteq D_{+}(x_i)$ as the locus where $x_j\mathrm{/}{x}_i$ is invertible.
4. Conventions. The construction is available over an arbitrary field $k$ and for an arbitrary homogeneous ideal $I$, including a nonradical one; it makes $\mathrm{Proj}S$ a $k$-scheme locally of finite type, with residue fields $\kappa (\mathfrak{p})=S_{(\mathfrak{p})}\mathrm{/}{\mathfrak{m}}_{(\mathfrak{p})}$ that need not equal $k$. Here $T$ is the multiplicative set of homogeneous elements of $S$ outside $\mathfrak{p}$, $S_{(\mathfrak{p})}=(T^{-1}S{)}_0$, and ${\mathfrak{m}}_{(\mathfrak{p})}=(T^{-1}\mathfrak{p}{)}_0$ is its maximal ideal: a degree-zero fraction with numerator outside $\mathfrak{p}$ has its inverse by interchanging numerator and denominator. It is not the same object as the classical projective algebraic set of projective space points, which is defined over an algebraically closed field and is reduced: the classical set records only the $\bar{k}$-points of the reduced subscheme, while $\mathrm{Proj}S$ retains the scheme structure on its charts, including any nilpotents that survive localization. When $k$ is algebraically closed and $I$ is radical, the closed points of $\mathrm{Proj}S$ correspond to the points of $V_{+}(I)$ in the classical sense; that dictionary is not asserted here.

For the empty case: if $S_{+}=0$, that is, if the images of all $x_i$ vanish in $S$, then $S$ has no homogeneous prime avoiding $S_{+}$ and $\mathrm{Proj}S=\mathrm{\varnothing }$; every $D_{+}(x_i)=\mathrm{Spec}0$ is empty, consistently with the gluing. In particular $\mathrm{Proj}(k[x_0,\dots ,x_n]\mathrm{/}(x_0,\dots ,x_n))=\mathrm{\varnothing }$.