Projective scheme

Intuition [Beginner]

A projective scheme is the global, scheme-theoretic generalisation of projective space. Where the affine scheme $\mathrm{Spec}(R)$ is built from a single commutative ring, the projective scheme $\mathrm{Proj}(S)$ is built from a graded commutative ring, and its points are homogeneous prime ideals — exactly the right structure to model projective varieties.

Why graded rings? Because projective space is built from polynomials with the homogeneity property: a polynomial $f(x_0,\dots ,x_n)$ defines a hypersurface in ${\mathbb{P}}^n$ only if it is homogeneous (the same total degree in every term). The vanishing locus of a non-homogeneous polynomial isn't well-defined in projective space because the polynomial doesn't transform consistently under rescaling.

The Proj construction takes any graded ring $S={⨁}_d{S}_d$ and produces a scheme whose points correspond to homogeneous primes (other than the irrelevant ideal $S_{+}={⨁}_{d>0}{S}_d$). This way every quotient of a polynomial ring by a homogeneous ideal — which is exactly the setting of classical projective varieties — becomes a scheme. The output is the modern algebraic-geometric view of projective varieties.

Visual [Beginner]

A graded ring on one side, the projective scheme (set of homogeneous primes) on the other; affine charts cover it via dehomogenisation.

Worked example [Beginner]

The simplest example: ${\mathbb{P}}_k^n=\mathrm{Proj}(k[x_0,\dots ,x_n])$, the standard projective space we already know. The graded ring $k[x_0,\dots ,x_n]$ has its grading by total degree.

A more interesting example: a plane projective curve. Take a homogeneous polynomial $f(x_0,x_1,x_2)\in k[x_0,x_1,x_2]$ of degree $d$. The quotient ring $S=k[x_0,x_1,x_2]/(f)$ inherits a grading. The projective scheme

$$X=\mathrm{Proj}(k[x_0,x_1,x_2]/(f))\subset {\mathbb{P}}_k^2$$

is a plane projective curve of degree $d$. For $d=2$ and $f=x_0{x}_2-x_1^2$, this is a smooth conic, isomorphic to ${\mathbb{P}}^1$. For $d=3$ and $f=x_0{x}_2^2-x_1^3-x_0^2{x}_1$ (or similar), this is an elliptic curve.

Every smooth projective curve over $\mathbb{C}$ embeds as a closed subscheme of ${\mathbb{P}}^N$ for some $N$ — projective schemes are the fundamental geometric objects of classical algebraic geometry.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $S={⨁}_{d\ge 0}{S}_d$ be a graded commutative ring (with $S_d\cdot S_e\subseteq S_{d+e}$, $1\in S_0$). The projective scheme $\mathrm{Proj}(S)$ is constructed as follows.

Underlying set. Define the irrelevant ideal

$$S_{+}:=\underset{d>0}{⨁}{S}_d.$$

Then

$$\mathrm{Proj}(S):=\{\mathfrak{p}\subset S:\mathfrak{p}\text{ is a homogeneous prime ideal with }{S}_{+}⊈\mathfrak{p}\}.$$

A homogeneous ideal is one generated by homogeneous elements; equivalently, $\mathfrak{p}={⨁}_d(\mathfrak{p}\cap S_d)$.

Topology. The Zariski topology has closed subsets

$$V(I):=\{\mathfrak{p}\in \mathrm{Proj}(S):I\subseteq \mathfrak{p}\}$$

for homogeneous ideals $I\subseteq S$. The distinguished opens are $D(f):=\{\mathfrak{p}:f\notin \mathfrak{p}\}$ for $f\in S_d$ a homogeneous element of positive degree.

Affine cover. The key fact: each $D(f)$ is naturally affine. For $f\in S_d$ (with $d>0$),

$$D(f)\cong \mathrm{Spec}((S[f^{-1}]{)}_0),$$

the spectrum of the degree-zero part of the localisation at $f$. As $f$ ranges over homogeneous elements of positive degree, the $D(f)$ cover $\mathrm{Proj}(S)$, giving an affine cover.

Structure sheaf. The structure sheaf ${\mathcal{O}}_{\mathrm{Proj}(S)}$ on the affine cover is given by

$${\mathcal{O}}_{\mathrm{Proj}(S)}(D(f))=(S[f^{-1}]{)}_0.$$

This makes $\mathrm{Proj}(S)$ a locally ringed space; the affine charts identify it with affine schemes locally, hence with a scheme globally.

Examples.

1. $\mathrm{Proj}(k[x_0,\dots ,x_n])={\mathbb{P}}_k^n$, the standard projective space.
2. $\mathrm{Proj}(k[x_0,\dots ,x_n]/I)$ for a homogeneous ideal $I$: a closed subscheme of ${\mathbb{P}}_k^n$, the projective variety defined by $I$.
3. $\mathrm{Proj}({⨁}_{d\ge 0}k[x_0,x_1,x_2{]}_{ad})={\mathbb{P}}_k^2$ via the Veronese embedding of degree $a$.
4. $\mathrm{Proj}(k[x_0,x_1,x_2,x_3]/(x_0{x}_3-x_1{x}_2))$: a smooth quadric surface in ${\mathbb{P}}^3$, isomorphic to ${\mathbb{P}}^1\times {\mathbb{P}}^1$.

Twisting sheaves. For each $d\in \mathbb{Z}$, the twisting sheaf $\mathcal{O}(d)$ on $\mathrm{Proj}(S)$ is the sheaf associated to the graded $S$-module $S(d)$ (degree shift by $d$). Global sections:

$$H^0(\mathrm{Proj}(S);\mathcal{O}(d))=S_d\text{(degree-}d\text{ component, when S is generated by S1 over S0)}.$$

Properties.

• $\mathrm{Proj}(S)$ is quasi-compact and Noetherian if $S$ is finitely generated over $S_0$.
• Closed subschemes of ${\mathbb{P}}_k^n$ correspond to homogeneous ideals of $k[x_0,\dots ,x_n]$.
• A morphism of graded rings (degree-preserving) $S\to T$ induces a morphism of schemes $\mathrm{Proj}(T)⇢\mathrm{Proj}(S)$ — defined where the morphism doesn't kill the irrelevant ideal.

Veronese and Segre embeddings. Important explicit projective embeddings:

• Veronese $v_d:{\mathbb{P}}^n\to {\mathbb{P}}^N$ with $N=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)-1$: $[x_0:\cdots :x_n]\mapsto [\text{all degree-}d\text{ monomials}]$. The image is a smooth projective variety.
• Segre $\sigma :{\mathbb{P}}^m\times {\mathbb{P}}^n\to {\mathbb{P}}^{(m+1)(n+1)-1}$: $([x],[y])\mapsto [x_i{y}_j]$. Realises products of projective spaces inside larger projective spaces.

These embeddings are foundational: they let you reduce many problems about projective varieties to study of subvarieties of ${\mathbb{P}}^N$.

Key theorem with proof [Intermediate+]

Theorem (every projective scheme is a closed subscheme of some ${\mathbb{P}}_k^n$). Let $X=\mathrm{Proj}(S)$ for $S$ a finitely-generated graded $k$-algebra with $S_0=k$ and $S$ generated by $S_1$ over $S_0$. Then $X$ is isomorphic to a closed subscheme of ${\mathbb{P}}_k^N$, where $N+1={\dim }_k{S}_1$.

Proof sketch. Pick a basis $f_0,f_1,\dots ,f_N$ of the degree-1 component $S_1$ as a $k$-vector space. The $f_i$ generate $S$ as a graded $k$-algebra (by hypothesis). Define a graded $k$-algebra homomorphism

$$\phi :k[x_0,\dots ,x_N]\to S,x_i\mapsto f_i.$$

This is degree-preserving (each $x_i$ has degree 1 and goes to $f_i\in S_1$). Surjectivity follows because $f_i$'s generate $S$. The kernel $I=\ker (\phi )\subset k[x_0,\dots ,x_N]$ is a homogeneous ideal, giving an isomorphism

$$S\cong k[x_0,\dots ,x_N]/I.$$

Applying Proj:

$$X=\mathrm{Proj}(S)\cong \mathrm{Proj}(k[x_0,\dots ,x_N]/I)=V_{+}(I)\subset {\mathbb{P}}_k^N,$$

where $V_{+}(I)\subset {\mathbb{P}}_k^N$ is the closed subscheme cut out by the homogeneous ideal $I$. $\square$

This theorem is foundational: it identifies projective schemes (over a field) with closed subschemes of projective space. Combined with the Hilbert basis theorem (every ideal is finitely generated), it shows that projective schemes are exactly the schemes of the form $V_{+}(f_1,\dots ,f_r)\subset {\mathbb{P}}^N$ for finitely many homogeneous polynomials. Every such object is a projective variety in the classical sense.

Bridge. The construction here builds toward 04.05.05 (ample and very ample line bundle), where the same data is upgraded, and the symmetry side is taken up in 04.06.02 (coherent sheaf). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has AlgebraicGeometry.Proj, AlgebraicGeometry.ProjectiveSpectrum, and surrounding API.

[object Promise]

Advanced results [Master]

Functorial Proj. The Proj construction is functorial for graded ring homomorphisms in negative degrees: a graded ring map $S\to T$ that doesn't kill the irrelevant ideal induces a morphism $\mathrm{Proj}(T)\to \mathrm{Proj}(S)$. Properly understood as a partial functor.

Proj of a graded module. For a graded $S$-module $M$, define the coherent sheaf $\tilde{M}$ on $\mathrm{Proj}(S)$ by $\tilde{M}(D(f))=(M[f^{-1}]{)}_0$. This gives the equivalence

$$\widetilde{(\cdot )}:\mathbf{grMod}(S)/(\text{torsion})\stackrel{\sim }{\to }\mathbf{Coh}(\mathrm{Proj}(S))$$

between coherent sheaves on $\mathrm{Proj}(S)$ and graded modules modulo torsion (Serre's theorem).

Castelnuovo-Mumford regularity. A coherent sheaf $\mathcal{F}$ on ${\mathbb{P}}^n$ has regularity $\mathrm{reg}(\mathcal{F})$, the smallest $m$ such that $H^i({\mathbb{P}}^n;\mathcal{F}(m-i))=0$ for all $i\ge 1$. Bounds on regularity give effective control on global sections of twists.

Hilbert scheme. The moduli space ${\mathrm{Hilb}}^P({\mathbb{P}}^n)$ of closed subschemes of ${\mathbb{P}}^n$ with given Hilbert polynomial $P$ is itself a projective scheme. Hilbert schemes are foundational objects in moduli theory, their geometry studied extensively by Mumford, Hartshorne, Grothendieck.

Severi-Brauer varieties. Over non-algebraically-closed fields, "twisted forms" of ${\mathbb{P}}^n$ exist — these are the Severi-Brauer varieties, parametrised by the Brauer group of the base field.

Toric projective schemes. The toric varieties associated to projective polytopes are projective. The Cox ring construction gives a uniform Proj-like description.

Kontsevich moduli of stable maps. Refining Hilbert schemes, $\overline{{\mathcal{M}}_{g,n}}({\mathbb{P}}^N,\beta )$ — the moduli of stable maps from genus-$g$, $n$-marked curves to ${\mathbb{P}}^N$ representing class $\beta$ — is a projective scheme central to Gromov-Witten theory and mirror symmetry.

$\mathrm{Proj}$ vs $\mathrm{Spec}$. A graded ring $S$ has both: $\mathrm{Spec}(S)$ is the affine cone (the zero locus of $S_{+}$ removed), and $\mathrm{Proj}(S)$ is its projectivisation. The relation: $\mathrm{Spec}(S)\setminus V(S_{+})\to \mathrm{Proj}(S)$ is a ${\mathbb{G}}_m$-torsor.

Synthesis. This construction generalises the pattern fixed in 04.01.01 (sheaf), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Detailed proofs of: every projective $k$-scheme embeds in some ${\mathbb{P}}_k^N$; Serre's theorem on coherent cohomology vanishing; the equivalence between graded modules and coherent sheaves via $\widetilde{(\cdot )}$; the Hilbert polynomial / regularity theory — these are deferred to companion units. The key theorem of the section is given in the formal-definition section above.

Connections [Master]

• Affine scheme 04.02.02 — projective schemes are covered by affine schemes; both are core scheme constructions.

• Scheme 04.02.01 — projective schemes are a key subclass of schemes; many natural geometric objects are projective.

• Projective space 04.07.01 — the fundamental example ${\mathbb{P}}_k^n=\mathrm{Proj}(k[x_0,\dots ,x_n])$.

• Sheaf 04.01.01 — the structure sheaf and twisting sheaves are foundational.

• Sheaf cohomology 04.03.01 — Serre's theorem on coherent cohomology of projective schemes.

• Quasi-coherent sheaf 04.06.01 and Coherent sheaf 04.06.02 — coherent sheaves on $\mathrm{Proj}(S)$ correspond to graded $S$-modules modulo torsion.

• Riemann-Roch theorem for curves 04.04.01 — most natural application: smooth projective curves are projective schemes of dimension 1.

• Ample line bundle 04.05.05 — ampleness characterises which line bundles induce projective embeddings.

• Moduli of curves 04.10.01 — moduli problems often produce projective schemes (Hilbert schemes, Kontsevich schemes).

Historical & philosophical context [Master]

Classical algebraic geometry (1850s–1950s) focused on projective varieties as zero sets of homogeneous polynomial systems in ${\mathbb{P}}^n$. Pioneers from Cayley to Severi to Zariski developed an enormous body of techniques, but the foundational language was unsystematic — different texts used different definitions, and rigour was sometimes lacking.

The reformulation came in two stages:

1. Cartan-Serre 1953–55. Henri Cartan's seminars and Jean-Pierre Serre's Faisceaux Algébriques Cohérents (FAC, 1955) introduced the sheaf-theoretic perspective: a projective variety is a topological space equipped with a sheaf of regular functions, and coherent sheaves on it carry the essential geometric information. Serre's vanishing theorem (every coherent sheaf has a high enough twist with vanishing higher cohomology) was a foundational achievement.

2. Grothendieck-Dieudonné 1960–67. The Éléments de Géométrie Algébrique (EGA) introduced schemes in full generality, with $\mathrm{Spec}$ and $\mathrm{Proj}$ as the foundational constructions. Grothendieck's relative perspective (${\mathrm{Proj}}_S$ over an arbitrary base $S$, projective morphisms, projective bundles) made projective schemes the natural setting for relative algebraic geometry.

The Proj construction is the algebraic-geometric formulation of projective space and projective varieties, working uniformly over arbitrary base rings (including $\mathbb{Z}$, finite fields, $p$-adic rings, formal rings, etc.). This generality is what makes modern arithmetic algebraic geometry possible.

The 20th century's deepest theorems — Weil conjectures (Deligne 1974), Mordell conjecture (Faltings 1983), Wiles's modularity theorem (1995), Lafforgue / Drinfeld / V. Lafforgue Langlands (1990s–2010s) — all live on projective schemes over arithmetic bases. The Proj construction is the conceptual foundation.

Today projective schemes are studied through their cohomology, Hilbert polynomials, intersection theory, motivic invariants, and moduli problems. The reach of projective schemes extends from classical curve and surface theory to mirror symmetry, geometric Langlands, and string-theoretic predictions. The single phrase "Proj of a graded ring" generates an essentially endless geometric landscape, just as $\mathrm{Spec}$ does for affine geometry.

Bibliography [Master]

• Hartshorne, Algebraic Geometry — §II.2, §II.5 are the canonical introduction.
• Vakil, The Rising Sea: Foundations of Algebraic Geometry — §4–§5, modern pedagogical treatment.
• Grothendieck-Dieudonné, Éléments de Géométrie Algébrique II (EGA II, 1961) — the foundational reference for projective morphisms and Proj.
• Eisenbud & Harris, The Geometry of Schemes — Ch. III gives geometrically-oriented Proj.
• Mumford, The Red Book of Varieties and Schemes — classical introduction with strong emphasis on projective.
• Liu, Algebraic Geometry and Arithmetic Curves — strong on arithmetic projective schemes.
• Lazarsfeld, Positivity in Algebraic Geometry I and II — comprehensive treatment of ample line bundles and projective embeddings.
• Serre, Faisceaux Algébriques Cohérents (FAC, 1955) — the original sheaf-theoretic foundation.