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In algebraic geometry, the Proj construction provides a method to associate a scheme to a graded ring, generalizing the classical notion of projective space and enabling the study of projective varieties and schemes in a scheme-theoretic framework.^[1] Specifically, for a graded ring

S = \bigoplus_{n \geq 0} S_n

, the space

\operatorname{Proj} S

consists of all homogeneous prime ideals of

S

that do not contain the irrelevant ideal

S_+ = \bigoplus_{n \geq 1} S_n

, equipped with a Zariski topology generated by distinguished open sets

D_+(f) = \{ p \in \operatorname{Proj} S \mid f \notin p \}

for homogeneous elements

f \in S

of positive degree.^[2] This construction yields a locally ringed space

(\operatorname{Proj} S, \tilde{\mathcal{O}}_S)

, where the structure sheaf

\tilde{\mathcal{O}}_S

D_+(f)

is the degree-zero part of the localization

S_f

, making

\operatorname{Proj} S

a scheme that is affine on each basic open and quasi-compact when

S

is finitely generated.^[3] The intuitive picture behind Proj views it as the "projectivization" of the affine cone

\operatorname{Spec} S

, obtained by removing the vertex (corresponding to the irrelevant ideal) and quotienting by the action of the multiplicative group of scalars, which aligns with lines through the origin in vector spaces.^[2] For instance, when

S = k[x_0, \dots, x_n]

is the polynomial ring over a field

k

with each variable in degree 1,

\operatorname{Proj} S

recovers the projective space

\mathbb{P}^n_k

.^[3] Homogeneous ideals in

S

define closed subschemes of

\operatorname{Proj} S

, allowing the construction of projective varieties as zero loci of homogeneous polynomials, while morphisms between graded rings induce scheme morphisms between their Projs, provided the images respect the irrelevant ideals.^[1] Key properties of the Proj construction include its role in ensuring projective schemes are proper and of finite type over the base ring

S_0

, with quasi-coherent sheaves on

\operatorname{Proj} S

arising naturally from graded

S

-modules via the functor

\tilde{\cdot}

.^[3] This framework, originally developed in the context of Grothendieck's Éléments de géométrie algébrique, underpins much of modern algebraic geometry, including the study of ample line bundles via twisting sheaves

\mathcal{O}(n)

and the gluing of affine charts to form global projective objects.^[1]

Proj of a graded ring

Proj as a set

In algebraic geometry, given a graded ring

S = \bigoplus_{n \geq 0} S_n

, the underlying set of

\Proj S

consists of all homogeneous prime ideals

\mathfrak{p} \subset S

such that

\mathfrak{p} \not\supseteq S_+

, where

S_+ = \bigoplus_{n > 0} S_n

is the irrelevant ideal generated by all elements of positive degree.^[1] A prime ideal

\mathfrak{p}

S

is homogeneous if it is generated by homogeneous elements, or equivalently, if whenever a sum of homogeneous elements lies in

\mathfrak{p}

, each individual homogeneous component also lies in

\mathfrak{p}

.^[1] These homogeneous prime ideals represent the points of

\Proj S

, capturing the projective structure by excluding ideals that contain the entire irrelevant ideal, which would correspond to "degenerate" points not contributing to the projective geometry.^[1] The condition

\mathfrak{p} \not\supseteq S_+

ensures that the primes in

\Proj S

intersect the degree-zero part

S_0

nontrivially in a suitable sense, focusing on ideals relevant to projective quotients.^[1] Homogeneous primes are crucial because they preserve the grading structure of

S

, allowing the Proj construction to model geometric objects like projective varieties where points correspond to 1-dimensional subspaces (lines through the origin) in the vector space associated to the degree-1 component of

S

.^[1] In contrast to the prime spectrum

\Spec S

, which includes all prime ideals of the underlying ungraded ring

S

(without regard to homogeneity or the irrelevant ideal),

\Proj S

restricts to the homogeneous primes excluding those containing

S_+

, thereby emphasizing the projective nature over the full affine structure.^[1] This distinction avoids incorporating irrelevant or non-projective points, such as the generic point of the entire ring if it contains

S_+

. A canonical example arises when

S = k[x_0, \dots, x_n]

is the polynomial ring in

n+1

variables over a field

k

, graded by total degree (with each

x_i

in degree 1). Here,

\Proj S

identifies with the projective space

\mathbb{P}^n_k

, whose points are the lines through the origin in the affine space

\mathbb{A}^{n+1}_k

, corresponding precisely to the homogeneous maximal ideals not containing

S_+ = (x_0, \dots, x_n)

.^[1]

Proj as a topological space

The Proj construction equips the set

\Proj S

, consisting of homogeneous prime ideals of the graded ring

S

not containing the irrelevant ideal

S_+

, with the Zariski topology. This topology is defined such that the basic open sets are the distinguished opens

D_+(f)

for homogeneous elements

f \in S

, given by

D_+(f) = \{ p \in \Proj S \mid f \notin p \}.

These sets form a basis for the topology, and closed sets are complements of finite unions of such

D_+(f)

.^[2] The collection of basic opens

\{D_+(f) \mid f \in S_+\}

covers

\Proj S

. To see this, suppose

f_1, \dots, f_n

generate the irrelevant ideal

S_+

as an ideal. For any prime

p \in \Proj S

, since

p

does not contain

S_+

, at least one

f_i \notin p

, placing

p

D_+(f_i)

. Thus,

\Proj S = \bigcup_{i=1}^n D_+(f_i)

. This covering property ensures the topology is well-defined and nonempty for relevant graded rings.^[2] Each basic open

D_+(f)

carries a natural structure homeomorphic to the spectrum of the degree-zero part of the graded localization

S_{(f)}

. Specifically,

S_{(f)}

is the localization of

S

at the multiplicative set

\{f^n \mid n \geq 0\}

, and

S_{(f)}^{(0)}

denotes its homogeneous elements of degree zero, which form a ring. The map sending primes in

\Spec(S_{(f)}^{(0)})

to their contractions in

\Proj S

induces a homeomorphism

D_+(f) \cong \Spec(S_{(f)}^{(0)})

, with the Zariski topology on the right.^[2] The topology on

\Proj S

arises by gluing these affine open sets along their intersections, inheriting the Zariski topology from the affine schemes

\Spec(S_{(f)}^{(0)})

. Intersections

D_+(f) \cap D_+(g) = D_+(fg)

correspond to the degree-zero spectra of the relevant localizations, ensuring compatibility and that the overall space is a topological space covered by affines. This construction parallels the Zariski topology on affine schemes but adapts to the projective setting via homogeneous localizations.^[2]

Proj as a scheme

To equip the topological space

\operatorname{Proj} S

with the structure of a scheme, one defines a sheaf of rings

\mathcal{O}_{\operatorname{Proj} S}

on its basic open subsets

D_+(f)

, where

S

is a graded ring and

f \in S_d

is a nonzero homogeneous element of positive degree. Specifically, for each such

D_+(f)

, the stalk

\mathcal{O}_{\operatorname{Proj} S}(D_+(f))

consists of the degree-zero elements of the graded localization

S_{(f)}

, denoted S_{(f)}_0. Here,

S_{(f)}

is obtained by formally inverting the powers of

f

, so elements are fractions

g/f^k

with

g \in S_{k d}

and

k \geq 0

, and the degree-zero part comprises those with

\deg g = k d

. This assignment satisfies the sheaf axioms, particularly the gluing condition. On an overlap

D_+(f) \cap D_+(g) = D_+(fg)

, sections from

\mathcal{O}_{\operatorname{Proj} S}(D_+(f))

and

\mathcal{O}_{\operatorname{Proj} S}(D_+(g))

agree via the natural localization map to

(S_{(fg)})_0

, since inverting

f

and then

g

(or vice versa) yields the same ring

S_{(fg)}

and its degree-zero elements, ensuring compatibility. The sheaf

\mathcal{O}_{\operatorname{Proj} S}

is thus a sheaf of

\mathbb{Z}

-algebras, and restricting to the degree-zero subring

S_0

makes it a sheaf of

S_0

-algebras. With this structure sheaf,

\operatorname{Proj} S

becomes a locally ringed space that is a scheme, as the basic opens

D_+(f)

cover

\operatorname{Proj} S

and each is affine, isomorphic to \operatorname{Spec} S_{(f)}_0. Furthermore,

\operatorname{Proj} S

carries a natural structure morphism

\pi: \operatorname{Proj} S \to \operatorname{Spec} S_0

to the spectrum of the degree-zero subring

S_0

, assuming

S_0

is an integral domain or noetherian as needed for the construction. This morphism satisfies a universal property: for any graded

S

-algebra

T^\bullet

(with

T_0 = S_0

) and a homomorphism

\operatorname{Spec} T_0 \to \operatorname{Spec} S_0

, there exists a unique

\pi_T: \operatorname{Proj} T \to \operatorname{Proj} S

over

\operatorname{Spec} S_0

such that the induced map on degree-zero parts is compatible. This property characterizes

\operatorname{Proj} S

as the relative Proj over the base

\operatorname{Spec} S_0

, making it a scheme over that base.

Sheaf associated to a graded module

Given a graded ring

S = \bigoplus_{n \geq 0} S_n

and a graded

S

-module

M = \bigoplus_{n \in \mathbb{Z}} M_n

, the sheaf

\widetilde{M}

associated to $M$ on $X = \Proj S$ is constructed as a sheaf of $\mathcal{O}_X$ -modules that generalizes the structure sheaf $\mathcal{O}_X = \widetilde{S}$

.^[1] For each basic open $D_+(f) \subseteq X$ , where $f \in S_d$ is homogeneous of positive degree $d > 0$ , the sections are given by $\Gamma(D_+(f), \widetilde{M}) = M_{(f)}$

)=M(f), where $M_{(f)}$ denotes the degree-zero elements in the localization of $M$ at the multiplicative set $\{1, f, f^2, \dots \}$ .^[1] Explicitly, these sections consist of elements of the form $m / f^k$ with $m \in M_{kd}$ and $k \geq 0$ , forming an $S_{(f)}$ -module, where $S_{(f)}$ is defined analogously.^[1] To define $\widetilde{M}$

, first associate the presheaf on the basis of standard opens $\{D_+(f)\}$ by setting $\widetilde{M}(D_+(f)) = M_{(f)}$

(D+(f))=M(f), with restriction maps $M_{(f)} \to M_{(fg)}$ induced by the canonical localization homomorphism for $g$ homogeneous.^[1] This presheaf satisfies the sheaf axiom on the basis because the Čech complex for any cover $D_+(f) = \bigcup_i D_+(f g_i)$ is exact: $0 \to M_{(f)} \to \bigoplus_i M_{(f g_i)} \to \bigoplus_{i < j} M_{(f g_i g_j)} \to \cdots$ is exact, as localization is exact and the degrees match appropriately.^[1] Thus, $\widetilde{M}$

is already a sheaf on the basis, and it extends uniquely to a sheaf of $\mathcal{O}_X$ -modules on all opens of $X$ by the sheaf property of $\mathcal{O}_X$ .^[1] The sheaf $\widetilde{M}$

is quasi-coherent on $X$ , meaning that for every affine open $U \subseteq X$ , the restriction $\widetilde{M}|_U$

∣U is the $\widetilde{N}$

-image of some $N$ under the tilde functor on $\Spec \Gamma(U, \mathcal{O}_X)$ .^[1] Specifically, on each $D_+(f) \cong \Spec S_{(f)}$ , we have $\widetilde{M}|_{D_+(f)} \cong \widetilde{M_{(f)}}$

∣D+(f)≅M(f)

, where $M_{(f)}$ is viewed as an $S_{(f)}$ -module.^[4] If $S$ is Noetherian and $M$ is finitely generated, then $\widetilde{M}$

is coherent.^[5] Under the finiteness condition that $X = \bigcup_{f \in S_1} D_+(f)$ , the global sections satisfy $\Gamma(X, \widetilde{M}) = M_0$

)=M0, with the canonical map $M_0 \to \Gamma(X, \widetilde{M})$

) being an isomorphism.^[6] This identifies the degree-zero part of $M$ with the untwisted global sections of the associated sheaf.^[6]

Serre twisting sheaf

In the context of the Proj construction for a graded ring

S = \bigoplus_{d \geq 0} S_d

, the Serre twisting sheaf

\mathcal{O}_{\operatorname{Proj} S}(n)

is defined as the sheaf

\widetilde{S(n)}

S(n)

associated to the graded $S$ -module $S(n)$ , where the grading on $S(n)$ is given by $S(n)_k = S_{n+k}$ for all $k \geq 0$ .^[7] This construction specializes the general association of sheaves to graded modules, twisting the structure sheaf $\mathcal{O}_{\operatorname{Proj} S}$ by $n$ degrees to account for the homogeneous nature of the coordinates.^[8] Locally on the distinguished open sets $D_+(x_i)$ in $\operatorname{Proj} S$ , where $x_i$ is a homogeneous element of positive degree (typically degree 1 in standard gradings), the twisting sheaf $\mathcal{O}_{\operatorname{Proj} S}(n)$ is isomorphic to the structure sheaf $\mathcal{O}_{\operatorname{Proj} S}$ . This isomorphism is induced by multiplication by $x_i^{-n}$ in the localization $S_{(x_i)}$ , which preserves degrees because $\deg(x_i) = 1$ ensures the inverse has the appropriate negative degree to shift back.^[7] Such local trivializations highlight the line bundle structure of the twisting sheaves, enabling the inversion of homogeneous coordinates in the ringed space. For the specific case of projective space $\mathbb{P}^n = \operatorname{Proj} k[x_0, \dots, x_n]$ over a field $k$ , Serre's theorem states that the global sections are $\Gamma(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(n)) = S_n$ , the vector space of homogeneous polynomials of degree $n$ in $n+1$ variables.^[8] Moreover, the higher cohomology groups vanish: $H^i(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(n)) = 0$ for all $i > 0$ .^[8] These properties underscore the role of twisting sheaves in computing cohomology and realizing projective modules on $\operatorname{Proj} S$ .

Projective space

\mathbb{P}^n

over a field

k

is constructed as

\operatorname{Proj} k[x_0, \dots, x_n]

, where the polynomial ring is graded by total degree.^[9] The points of

\mathbb{P}^n

correspond to equivalence classes

[x_0 : \dots : x_n]

(n+1)

-tuples in

k^{n+1} \setminus \{0\}

, where two tuples are equivalent if one is a nonzero scalar multiple of the other.^[10] These homogeneous coordinates provide a classical description of the points, aligning with the scheme-theoretic points as homogeneous prime ideals not containing the irrelevant ideal

(x_0, \dots, x_n)

.^[9] The space

\mathbb{P}^n

is covered by

n+1

standard affine charts

U_i = D_+(x_i)

, each isomorphic to the affine space

\mathbb{A}^n_k

. On

U_i

, a point

[x_0 : \dots : x_n]

with

x_i \neq 0

maps to the affine coordinates

(x_0 / x_i, \dots, \hat{x_i}/x_i, \dots, x_n / x_i)

, where the hat indicates omission.^[10] The transition functions between charts

U_i

and

U_j

(for

i \neq j

) are given by

x_k / x_i = (x_k / x_j) \cdot (x_j / x_i)

on the overlap

U_i \cap U_j

, which are invertible regular functions ensuring the gluing is algebraic. The sheaf

\mathcal{O}_{\mathbb{P}^n}(n)

serves as the

n

-th power of the tautological line bundle

\mathcal{O}_{\mathbb{P}^n}(1)

, whose global sections are generated by the homogeneous linear forms

x_0, \dots, x_n

, corresponding to the homogeneous coordinates. This twisting sheaf, as defined in prior sections, associates to degree-

n

homogeneous polynomials the sections over

\mathbb{P}^n

.^[9] As a scheme,

\mathbb{P}^n

is integral of dimension

n

, being irreducible and reduced over an algebraically closed field

k

.^[9] This dimension follows from the affine charts each having dimension

n

, with the covering being a scheme-theoretic open cover.^[10]

Examples of Proj

Proj of polynomial rings

In algebraic geometry over a field

k

, the Proj construction applied to quotients of polynomial rings by homogeneous ideals provides a standard way to obtain projective varieties as closed subschemes of projective space. Consider the polynomial ring

R = k[x_0, \dots, x_n]

graded by total degree, so that

\operatorname{Proj} R \cong \mathbb{P}^n_k

. For a homogeneous ideal

I \subset R

, the quotient

S = R/I

is a graded ring, and

\operatorname{Proj} S

identifies with the closed subscheme

V(I) \subset \mathbb{P}^n_k

defined by the vanishing of the generators of

I

. This correspondence holds because the homogeneous ideal sheaf associated to

I

determines the subscheme structure on the zero set of

I

.^[9]^[11] The points of

\operatorname{Proj} S

consist of the homogeneous prime ideals

\mathfrak{p} \subset S

that do not contain the irrelevant ideal

S_+ = \oplus_{d \geq 1} S_d

. These primes

\mathfrak{p}

arise as quotients

\mathfrak{q}/I

, where

\mathfrak{q} \subset R

is a homogeneous prime ideal containing

I

but not the irrelevant ideal

R_+

. Geometrically, such points correspond to irreducible subvarieties of

\mathbb{P}^n_k

contained in

V(I)

, excluding the irrelevant components at infinity. This set-theoretic description aligns with the scheme-theoretic structure, where the topology on

\operatorname{Proj} S

is induced from that of

\mathbb{P}^n_k

.^[1] A representative example is the quadric curve

\operatorname{Proj} k[x,y,z]/(xy - z^2)

, which defines a smooth conic in

\mathbb{P}^2_k

. This scheme is isomorphic to

\mathbb{P}^1_k

, as it parameterizes lines in the plane via the embedding given by the Veronese map of degree 2, and the relation

xy = z^2

enforces the quadratic form without singularities over algebraically closed fields of characteristic not 2.^[2] When

I

is saturated—that is,

I = I : R_+^\infty = \{ f \in R \mid f \cdot R_+^m \subset I

for some

m \gg 0\}

—the canonical morphism

\operatorname{Proj} S \to \mathbb{P}^n_k

is a closed immersion. Saturation ensures that the ideal sheaf of the subscheme is quasi-coherent and properly defines the closed embedding without extraneous components supported on the irrelevant ideal. In this case,

\operatorname{Proj} S

inherits the reduced induced structure from

\mathbb{P}^n_k

along

V(I)

.^[12]

Projective hypersurfaces

A projective hypersurface

X

over a field

k

is constructed as

\operatorname{Proj} k[x_0, \dots, x_n]/(f)

, where

f \in k[x_0, \dots, x_n]

is a homogeneous polynomial of degree

d \geq 1

.^[13] This scheme-theoretic definition embeds

X

as a closed subscheme of the projective space

\mathbb{P}^n_k

, capturing the zero locus of

f

in a way that resolves ambiguities in classical varieties by incorporating the homogeneous structure.^[13] For instance, when

f

is irreducible,

X

is an integral hypersurface of dimension

n-1

. The canonical sheaf

\omega_X

of a smooth hypersurface

X \subset \mathbb{P}^n

of degree

d

is given by the adjunction formula

\omega_X = \mathcal{O}_{\mathbb{P}^n}(d - n - 1)|_X

.^[14] This follows from the general adjunction principle for a Cartier divisor

D

on a smooth variety

Y

, where

\omega_D = (\omega_Y \otimes \mathcal{O}_Y(D))|_D

, applied with

Y = \mathbb{P}^n

and

\omega_{\mathbb{P}^n} = \mathcal{O}_{\mathbb{P}^n}(-n-1)

.^[14] The twisting by degree

d

reflects how the hypersurface inherits and adjusts the negativity of the ambient space's canonical bundle. For curves, consider the case

n=2

, where a smooth plane curve

X \subset \mathbb{P}^2

of degree

d

has genus

g = \frac{(d-1)(d-2)}{2}

.^[15] This formula arises from the degree-genus relation for irreducible plane curves, computable via the adjunction formula or Riemann-Hurwitz theorem applied to the normalization, and it quantifies the topological complexity increasing quadratically with degree.^[15] For example, a smooth cubic (

d=3

) has genus 1, corresponding to an elliptic curve. Singularities on a hypersurface

X = V(f) \subset \mathbb{P}^n

occur at points

p \in X

where the differential

df_p = 0

, meaning

f(p) = 0

and all partial derivatives

\partial f / \partial x_i (p) = 0

for

i=0, \dots, n

.^[16] In the Proj construction, these singular points are detected scheme-theoretically as points where the ideal

(f, \partial f / \partial x_0, \dots, \partial f / \partial x_n)

has positive-dimensional support, potentially reducing the dimension or altering cohomology compared to the smooth case.^[16] The Proj functor thus highlights such loci by examining the homogeneous coordinate ring's Jacobian ideal.

Weighted projective space

The weighted projective space

\mathbb{P}(a_0, \dots, a_n)

, where

a_0, \dots, a_n

are positive integers, is defined as

\operatorname{Proj} S

, with

S = k[x_0, \dots, x_n]

the polynomial ring over an algebraically closed field

k

graded by

\deg(x_i) = a_i

for each

i

.^[17] This generalizes the classical projective space

\mathbb{P}^n = \operatorname{Proj} k[x_0, \dots, x_n]

by incorporating non-standard degrees on the generators, leading to a scheme that is typically singular unless all

a_i = 1

. The irrelevant ideal is

S_+ = \bigoplus_{d > 0} S_d

, and points of

\mathbb{P}(a_0, \dots, a_n)

correspond to homogeneous prime ideals of

S

not containing

S_+

.^[17] The topology and structure sheaf on

\mathbb{P}(a_0, \dots, a_n)

follow the standard Proj construction, with distinguished open sets

D_+(x_i)

forming an affine cover. Specifically,

D_+(x_i) \cong \operatorname{Spec} (S_{(x_i)})_0

, where

S_{(x_i)}

is the localization of

S

at the multiplicative set

\{x_i^d \mid d \geq 0\}

and

(S_{(x_i)})_0

denotes the degree-zero elements.^[18] These degree-zero elements form the ring

k\left[ \frac{x_0}{x_i^{a_0 / a_i}}, \dots, \frac{x_{i-1}}{x_i^{a_{i-1}/a_i}}, \frac{x_{i+1}}{x_i^{a_{i+1}/a_i}}, \dots, \frac{x_n}{x_i^{a_n / a_i}} \right]

, which is the invariant subring under the weighted

\mathbb{C}^*

-action scaled by the exponents; this yields a weighted affine space structure on each

D_+(x_i)

.^[19] The intersections

D_+(x_i) \cap D_+(x_j)

are covered by further localizations, ensuring the scheme glues properly as in the unweighted case.^[18] As an orbifold,

\mathbb{P}(a_0, \dots, a_n)

arises as the geometric quotient

[\mathbb{A}^{n+1} \setminus \{0\}] / \mathbb{C}^*

under the weighted action

\lambda \cdot (z_0, \dots, z_n) = (\lambda^{a_0} z_0, \dots, \lambda^{a_n} z_n)

for

\lambda \in \mathbb{C}^*

, resulting in singularities along coordinate subspaces.^[17] In the stacky sense, it is the quotient stack with stabilizers: generic points have trivial stabilizer, but along the locus where only

x_i \neq 0

, the stabilizer is the finite group

\mathbb{Z}/a_i\mathbb{Z}

(the

a_i

-th roots of unity), imparting an orbifold structure with stacky points at these origins. This orbifold perspective highlights the non-smooth nature, where singularities are quotient singularities modeled by cyclic groups.^[17] A representative example is

\mathbb{P}(1,1,2)

, which is isomorphic to the quotient

\mathbb{P}^2 / (\mathbb{Z}/2\mathbb{Z})

under the action

[x:y:z] \mapsto [-x : -y : z]

, a

\mathbb{Z}/2\mathbb{Z}

-action fixing the line at infinity

z=0

.^[20] This space has a singularity at the point

[0:0:1]

, corresponding to the stacky point with stabilizer

\mathbb{Z}/2\mathbb{Z}

, and its open sets include

D_+(z) \cong \mathbb{A}^2

(unweighted) and

D_+(x) \cong \operatorname{Spec} k[u, v]

with

\deg u = \deg v = 1/2

in the localized ring, adjusted to integers via Veronese subrings.^[20]

Proj of bigraded rings

In algebraic geometry, the Proj construction extends naturally to bigraded rings, providing a framework for realizing biprojective varieties as schemes. Let

S = \bigoplus_{i,j \geq 0} S_{i,j}

be a bigraded ring over a field

k

, where each

S_{i,j}

is the component of bidegree

(i,j)

. The underlying topological space of

\Proj S

is defined as the union

\bigcup_{(m,n) \neq (0,0)} D_+(f)

, where the union runs over all bihomogeneous elements

f \in S_{m,n}

and each

D_+(f)

is the basic open set consisting of homogeneous prime ideals of

S

not containing

f

. This space is equipped with a scheme structure via the sheaf of bihomogeneous localizations, analogous to the singly graded case but respecting the bidegree decomposition. The irrelevant ideal in the bigraded setting is the ideal generated by all components

S_{i,j}

with positive bidegrees, specifically

\bigoplus_{(i,j) : i > 0 \text{ or } j > 0} S_{i,j}

. Homogeneous prime ideals containing this irrelevant ideal are excluded from

\Proj S

, ensuring the space captures only the "projective" loci in both gradings. This ideal plays a role similar to the maximal irrelevant ideal in the standard Proj, but adapted to the positive orthant of

\mathbb{Z}^2

. A concrete example illustrates how

\Proj S

yields a product space. Consider the bigraded polynomial ring

S = k[x_0, x_1, t]

with bidegrees

\deg x_0 = (1,0)

\deg x_1 = (1,0)

, and

\deg t = (0,0)

. The space

\Proj S

is isomorphic to

\mathbb{P}^1 \times \mathbb{A}^1

, where the first factor arises from the (1,0)-grading on

x_0, x_1

, and the second is the affine line from

t

; the opens

D_+(x_0)

and

D_+(x_1)

cover affine charts isomorphic to

\mathbb{A}^1_{x_1 / x_0} \times \mathbb{A}^1_t

and

\mathbb{A}^1_{x_0 / x_1} \times \mathbb{A}^1_t

, respectively, patching to the product. In the toric case, when

S

is a monomial bigraded ring generated by monomials corresponding to a fan in

\mathbb{Z}^2

\Proj S

realizes a toric projective variety. For instance, the bigraded ring associated to the fan for

\mathbb{P}^1 \times \mathbb{P}^1

has variables with bidegrees

(1,0)

and

(0,1)

, yielding the product as a toric variety embedded via the Segre map. More generally, such constructions relate to simplicial toric varieties, where the multi-grading encodes the class group action, and Castelnuovo-Mumford regularity bounds provide information on generators of ideals defining subvarieties.^[21]

Global Proj construction

Setup and assumptions

The global Proj construction provides a relative analogue of the classical Proj functor applied to a

\mathbb{Z}_{\geq 0}

-graded ring over a field, where the base scheme is

\Spec k

. In the relative setting, the base is an arbitrary scheme

X

, over which we define a morphism

\underline{\Proj}_X(\mathcal{A}) \to X

.^[22] The primary input is a sheaf

\mathcal{A}

\mathbb{Z}_{\geq 0}

-graded

\mathcal{O}_X

-algebras that is quasi-coherent as an

\mathcal{O}_X

-module. Specifically,

\mathcal{A} = \bigoplus_{n \geq 0} \mathcal{A}_n

where each

\mathcal{A}_n

is a quasi-coherent

\mathcal{O}_X

-module, the multiplication maps

\mathcal{A}_m \otimes_{\mathcal{O}_X} \mathcal{A}_n \to \mathcal{A}_{m+n}

are

\mathcal{O}_X

-linear, and the zeroth graded piece satisfies

\mathcal{A}_0 = \mathcal{O}_X

.^[23]^[22] This ensures that

\mathcal{A}

acts as an algebra over the structure sheaf, compatible with the grading. Additionally,

\mathcal{A}

is assumed to be locally finitely generated over

\mathcal{O}_X

in each degree: for every

n \geq 0

, there exists an affine open covering

\{U_i \to X\}

X

such that

\Gamma(U_i, \mathcal{A}_n)

is a finitely generated

\Gamma(U_i, \mathcal{O}_X)

-module for each

i

. This finiteness condition guarantees that the resulting relative Proj is a scheme of finite type over

X

when

\mathcal{A}

is generated by

\mathcal{A}_1

.^[1]^[24] The irrelevant ideal is the subsheaf

\mathcal{J} \subset \mathcal{A}

generated by the positive-degree part

\mathcal{A}_{>0} = \bigoplus_{n > 0} \mathcal{A}_n

; it is assumed to be quasi-coherent as an

\mathcal{O}_X

-module. This ideal plays the role of excluding the "irrelevant" locus in the construction, analogous to the maximal graded ideal in the absolute case, and its quasi-coherence ensures compatibility with the base scheme's topology.^[1]^[22] Under these assumptions, the global Proj

\underline{\Proj}_X(\mathcal{A})

is defined as a scheme over

X

, representing the functor of graded

\mathcal{A}

-modules up to twisting.^[22]

Definition and construction

The global Proj construction, denoted

\underline{\text{Proj}}_X(\mathcal{A})

X(\mathcal{A})

, defines a scheme over a base scheme

X

from a quasi-coherent sheaf of graded

\mathcal{O}_X

-algebras

\mathcal{A} = \bigoplus_{n \geq 0} \mathcal{A}_n

, where

\mathcal{A}_0 = \mathcal{O}_X

and the sheaf is generated in degree 1 by a quasi-coherent subsheaf

\mathcal{A}_1

.^[22] The underlying topological space consists of points that are pairs

(x, \mathfrak{p})

, where

x

is a point of

X

and

\mathfrak{p}

is a homogeneous prime ideal of the stalk

\mathcal{A}_x

that does not contain the irrelevant ideal

J_x = \bigoplus_{n \geq 1} \mathcal{A}_{x,n}

.^[22] Over each point

x \in X

, the fiber

\underline{\text{Proj}}_X(\mathcal{A})_x

is homeomorphic to the classical

\text{Proj}(\mathcal{A}_x)

, ensuring that the construction is fiberwise the standard Proj of the graded stalk ring.^[22] The topology on

X(\mathcal{A})

is generated by basic open subsets of the form

D_+(\overline{f})

, where

\overline{f}

is the image of a homogeneous section

f \in \Gamma(U, \mathcal{A}_n)

for some affine open

U \subset X

and

n \geq 1

.^[22] Specifically,

D_+(\overline{f}) = \{ (x, \mathfrak{p}) \mid x \in U, \, f_x \notin \mathfrak{p} \}

, and these form a basis for the Zariski topology relative to

X

.^[22] This relative topology ensures compatibility with the base scheme

X

, as the projection

\pi: X(\mathcal{A}) \to X

given by

\pi(x, \mathfrak{p}) = x

is continuous and identifies the fibers appropriately.^[22] The structure sheaf

\mathcal{O}_{X(\mathcal{A})}

is defined on these basic opens by

\mathcal{O}_{X(\mathcal{A})}(D_+(\overline{f})) = (\mathcal{A}_{(f)})_0

, the degree-zero part of the sheaf of graded algebras obtained by localizing

\mathcal{A}|_U

at the multiplicative system generated by

f

\Gamma(U, \mathcal{A}_n)

.^[22] These local definitions glue compatibly over intersections of basic opens because the localizations agree on degree-zero sections, yielding a sheaf of

\mathcal{O}_X

-algebras on

X(\mathcal{A})

that restricts to the structure sheaf of the classical Proj on each fiber.^[22] The resulting ringed space

(X(\mathcal{A}), \mathcal{O}_{X(\mathcal{A})})

is a scheme over

X

via the morphism

\pi

, which is locally of finite type if

\mathcal{A}_1

is finitely presented.^[22] The projection

\pi: X(\mathcal{A}) \to X

is a proper morphism provided that

\mathcal{A}

is finitely generated as a graded

\mathcal{O}_X

-algebra, meaning there exist finitely many global sections of

\mathcal{A}_1

that generate

\mathcal{A}

locally in the graded sense.^[22] This properness follows from the fact that over affine opens of

X

X(\mathcal{A})

is an open subscheme of a finite projective bundle, and properness glues under these conditions.^[22]

Twisting sheaf on Global Proj

In the context of the Global Proj construction, let

X

be a scheme and

\mathcal{A}

a quasi-coherent sheaf of

\mathbb{Z}_{\geq 0}

-graded

\mathcal{O}_X

-algebras that is generated in degree 1. The relative scheme

X(\mathcal{A}) = \Proj_X(\mathcal{A})

comes equipped with a structure morphism

p: X(\mathcal{A}) \to X

. The relative twisting sheaves on

X(\mathcal{A})

are defined as the quasi-coherent sheaves

\mathcal{O}_{X(\mathcal{A})}(n) = \widetilde{\mathcal{A}(n)}

OX(A)(n)=A(n)

for $n \in \mathbb{Z}$ , where $\mathcal{A}(n)$ denotes the graded shift of $\mathcal{A}$ satisfying $\mathcal{A}(n)_k = \mathcal{A}_{k+n}$ .^[25]^[26] These twisting sheaves generalize the classical Serre twisting sheaves on projective space to the relative setting over an arbitrary base $X$ . Locally on affine opens $U \subset X$ , where $\mathcal{A}|_U$ corresponds to a graded algebra $A$ , the restriction $\mathcal{O}_{X(\mathcal{A})}(n)|_{U(\mathcal{A})}$ coincides with the usual twist $\widetilde{A(n)}$

on the absolute Proj $U(A)$ . On a basic open subscheme $D_+(\overline{f}) \subset X(\mathcal{A})$ , where $\overline{f}$ is the image of a homogeneous section $f \in \Gamma(U, \mathcal{A}_d)$ for some affine $U \subset X$ and $d > 0$ , the sections are given by $\Gamma(D_+(\overline{f}), \mathcal{O}_{X(\mathcal{A})}(n)) = (A_{(f)})_n$ , the degree- $n$ homogeneous component of the localization of the corresponding graded algebra at $f$ .^[5] For sheaves incorporating twists from the base $X$ , the mixed twisting sheaves are formed as $p^* \mathcal{O}_X(m) \otimes_{\mathcal{O}_{X(\mathcal{A})}} \mathcal{O}_{X(\mathcal{A})}(n)$ for $m, n \in \mathbb{Z}$ , though the pure relative twists $\mathcal{O}_{X(\mathcal{A})}(n)$ capture the grading intrinsic to $\mathcal{A}$ . However, the focus here remains on the pure $\mathcal{A}$ -twists, which encode the projective structure over $X$ .^[25] Under projectivity assumptions on $X(\mathcal{A})/X$ —such as when $\mathcal{A}$ is generated by a finite-type quasi-coherent $\mathcal{O}_X$ -module in degree 1—the higher relative cohomology of these twisting sheaves vanishes: $R^i p_* \mathcal{O}_{X(\mathcal{A})}(n) = 0$ for all $i > 0$ and $n \gg 0$ . This relative vanishing theorem facilitates computations of cohomology on $X(\mathcal{A})$ via the Leray spectral sequence, reducing them to base cohomology on $X$ .^[24]

Global Proj of quasi-coherent sheaves

To extend the global Proj construction to incorporate a quasi-coherent sheaf

\mathcal{F}

on the base scheme

X

, consider a graded

\mathcal{O}_X

-algebra

\mathcal{A} = \bigoplus_{n \geq 0} \mathcal{A}_n

. The twisted algebra

\mathcal{A}_\mathcal{F}

is defined as the bigraded

\mathcal{O}_X

-algebra

\mathcal{A}_\mathcal{F} = \bigoplus_{n \geq 0} \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{A}_n

, where

\mathcal{F}

is regarded as being in bidegree

(1,0)

.^[27] This construction generalizes the base global Proj

\underline{\Proj}_X(\mathcal{A})

by embedding

\mathcal{F}

into the grading structure, allowing for relative projective schemes twisted by arbitrary quasi-coherent sheaves on

X

.^[25] The scheme

\underline{\Proj}_X(\mathcal{A}_\mathcal{F})

is then formed as the relative Proj over

X

of this bigraded algebra, resulting in a morphism

\underline{\Proj}_X(\mathcal{A}_\mathcal{F}) \to \underline{\Proj}_X(\mathcal{A})

that is a fibration with fibers given by

\Proj( \mathcal{A}_x \otimes_{\mathcal{O}_{X,x}} \mathcal{F}_x )

at each point

x \in X

.^[25] Here, the fiber over

x

reflects the local Proj construction twisted by the stalk

\mathcal{F}_x

, preserving the projective nature while incorporating the local structure of

\mathcal{F}

. This fibration ensures that the relative scheme captures the twisting effect globally over

X

.^[27] Quasi-coherent sheaves on

\underline{\Proj}_X(\mathcal{A}_\mathcal{F})

are constructed from relative graded modules over

\mathcal{A}_\mathcal{F}

. Specifically, for a quasi-coherent graded

\mathcal{A}_\mathcal{F}

-module

\mathcal{M}

, the associated sheaf

\widetilde{\mathcal{M}}

is defined on basic opens by taking the degree-zero part of the localized module, yielding a quasi-coherent $\mathcal{O}$ -module on the relative Proj. In particular, the sheaf $\widetilde{\mathcal{F}}$

associated to $\mathcal{F}$ (viewed as the degree-zero relative module) provides the canonical twisting structure on $\underline{\Proj}_X(\mathcal{A}_\mathcal{F})$ .^[25] A representative example arises when $\mathcal{A}$ is the trivial graded algebra $\mathcal{O}_X$ (polynomials in one variable) and $\mathcal{F} = \mathcal{L}$ is an invertible sheaf (line bundle) on $X$ . Then $\mathcal{A}_\mathcal{L} = \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n}$ , and $\underline{\Proj}_X(\mathcal{A}_\mathcal{L})$ is isomorphic to the projective bundle $\mathbf{P}(\mathcal{E})$ for the rank-one sheaf $\mathcal{E} = \mathcal{O}_X \oplus \mathcal{L}$ , with the tautological line bundle corresponding to the twist by $\mathcal{L}$ .^[28] This illustrates how the construction recovers standard projective bundles in the case of line bundles, highlighting its role in generalizing twisting sheaves from the base global Proj.^[27]

Projective space bundles

Projective space bundles arise as a special case of the Global Proj construction applied to the symmetric algebra of the dual of a locally free sheaf on a scheme

X

. Specifically, given a locally free sheaf

\mathcal{E}

X

of rank

r+1

, the projective bundle

\mathbb{P}(\mathcal{E})

is defined as

\mathbb{P}(\mathcal{E}) = \GlobalProj_X(\Sym(\mathcal{E}^\vee))

, where the grading is given by the components

\Sym^n(\mathcal{E}^\vee)

in degree

n

.^[29] The natural projection

p: \mathbb{P}(\mathcal{E}) \to X

makes

\mathbb{P}(\mathcal{E})

a scheme over

X

whose fiber over each point

x \in X

is the projective space

\mathbb{P}^r

parametrizing one-dimensional subspaces (lines) of the fiber

\mathcal{E}_x

.^[29] A key feature of this construction is the tautological line bundle

\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)

\mathbb{P}(\mathcal{E})

, which fits into the Euler exact sequence

0 \to \mathcal{O}_{\mathbb{P}(\mathcal{E})} \to p^*\mathcal{E} \otimes \mathcal{O}_{\mathbb{P}(\mathcal{E})}(1) \to T_{\mathbb{P}(\mathcal{E})/X} \to 0,

where

T_{\mathbb{P}(\mathcal{E})/X}

denotes the relative tangent bundle along the fibers. This sequence captures the geometry of lines in the fibers of

\mathcal{E}

, with

\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)

being the subbundle whose restriction to each fiber

\mathbb{P}^r

is the standard tautological line bundle on projective space. The dual

\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)

then provides the twisting sheaf for the Proj construction. The global sections of powers of the twisting sheaf are given by

H^0(\mathbb{P}(\mathcal{E}), \mathcal{O}_{\mathbb{P}(\mathcal{E})}(n)) = \Sym^n(\mathcal{E}^\vee)

, reflecting the grading of the symmetric algebra used in the definition.^[30] This isomorphism holds because

\mathcal{E}

is locally free, ensuring that the higher cohomology vanishes appropriately on the projective bundle.^[30] A representative example is the case where

X

is a smooth projective curve and

\mathcal{E}

is a rank-2 locally free sheaf, yielding a

\mathbb{P}^1

-bundle over the curve, known as a ruled surface.^[31] Such surfaces, like the Hirzebruch surfaces when

X = \mathbb{P}^1

, illustrate how projective bundles encode the geometry of vector bundles over lower-dimensional bases.^[31]

Properties and morphisms

Basic properties of Proj

If the graded ring

S

is Noetherian, then

\operatorname{Proj} S

is a Noetherian scheme.^[32] This follows from the fact that

\operatorname{Proj} S

admits a finite covering by affine schemes

\operatorname{Spec} S_{(f_i)}

, where each

S_{(f_i)}

is Noetherian as a localization of the Noetherian ring

S

.^[1] For a Noetherian graded ring

S

generated over

S_0

by finitely many elements of degree 1 (standard grading), the dimension of

\operatorname{Proj} S

\dim S - 1

.^[1] This holds because the standard open sets

D_+(f)

covering

\operatorname{Proj} S

have dimension

\dim S_{(f)} - 1

, and

\dim S_{(f)} = \dim S

for homogeneous

f

of positive degree, with the relative dimension over

\operatorname{Spec} S_0

accounting for the grading.^[33] The scheme

\operatorname{Proj} S

is integral if

S

is an integral domain (assuming

\operatorname{Proj} S

is nonempty, i.e.,

S_+

contains a non-zerodivisor).^[34] In this case, each affine open

D_+(f) \cong \operatorname{Spec} S_{(f)}

has ring of sections

S_{(f)}

, a localization of the domain

S

and hence itself a domain, making the scheme reduced and irreducible.^[1] The morphism

\operatorname{Proj} S \to \operatorname{Spec} S_0

is proper if

S

is finitely generated as an

S_0

-algebra.^[1] Under this assumption, the morphism is projective (hence proper, being of finite type, separated, and universally closed), as

\operatorname{Proj} S

is locally projective over

\operatorname{Spec} S_0

.^[35] Likewise, for Global Proj, if

\mathcal{A}

is a quasi-coherent sheaf of graded

\mathcal{O}_X

-algebras finitely generated in degree 1, then

\operatorname{Proj}_X \mathcal{A} \to X

is proper.^[24]

Morphisms between Proj schemes

A graded homomorphism

\phi: S \to T

between graded rings

S

and

T

induces a morphism of schemes

\Proj T \to \Proj S

, provided that the preimage under

\phi

of every relevant homogeneous prime ideal in

T

is relevant in

S

. Specifically, the map on points sends a homogeneous prime ideal

\mathfrak{p} \subset T

(not containing

T_+

) to

\phi^{-1}(\mathfrak{p}) \subset S

(which does not contain

S_+

), and this extends to a morphism of schemes on the distinguished open sets

D_+(\mathfrak{f})

for homogeneous elements

\mathfrak{f} \in T_d

via the identification with

\Spec T_{(\mathfrak{f})}

.^[36] This construction is functorial: composition of graded homomorphisms corresponds to composition of the induced morphisms, and identity maps induce identities. The morphism is well-defined when

\phi

maps

S_+

into

T_+

and ensures the preimages are proper for the Proj construction. If

\phi

is surjective on components of sufficiently high degree, the induced morphism is a closed immersion.^[36] If

\phi: S \to T

is a degree-preserving (i.e.,

\phi(S_i) \subseteq T_i

) flat ring homomorphism, then the induced morphism

\Proj T \to \Proj S

is flat. This follows from the local structure on distinguished opens, where flatness of the ring map implies flatness of the corresponding affine morphisms, combined with the gluing of the Proj scheme. A key example is the Veronese embedding, induced by the degree-

d

graded homomorphism

\phi: S \to S^{(d)}

, where

S^{(d)} = \bigoplus_{n \geq 0} S_{dn}

is the

d

-th Veronese subring of

S

. This map sends

\Proj S \to \Proj S^{(d)}

, embedding the scheme into a higher-dimensional projective space via monomials of degree

d

, and it is a closed immersion. For instance, when

S = k[x_0, \dots, x_n]

, it realizes the

d

-uple embedding of

\mathbb{P}^n_k

into

\mathbb{P}^N_k

with

N = \binom{n+d}{d} - 1

.^[18] For the Global Proj construction over a base scheme

Y = \operatorname{Spec} R

with a graded

R

-algebra

\mathcal{A}

, base change along a morphism

X \to Y

pulls back the sheaf of algebras

\mathcal{A}

to a graded

\mathcal{O}_X

-algebra, and the resulting Global Proj over

X

has fibers that are the base changes of the original fibers over

Y

. This compatibility ensures that the relative Proj functor preserves fiber structures under base change.^[24]

Relation to affine schemes

The scheme

\operatorname{Proj} S

for a graded ring

S

is covered by the standard open subschemes

D_+(f)

for homogeneous elements

f \in S_d

with

d \geq 1

, each of which is affine and isomorphic to

\operatorname{Spec}(S_{(f)}^0)

, where

S_{(f)}^0

denotes the degree-zero elements of the localized graded ring

S_{(f)}

.^[1] This provides a basis of affine open sets for the scheme structure on

\operatorname{Proj} S

.^[4] Despite this affine covering,

\operatorname{Proj} S

is generally not an affine scheme. For instance, in the case of projective space

\mathbb{P}^n_R = \operatorname{Proj} R[x_0, \dots, x_n]

over a ring

R

, the global sections

\Gamma(\mathbb{P}^n_R, \mathcal{O}_{\mathbb{P}^n_R})

are precisely

R

, the constants, whereas if

\mathbb{P}^n_R

were affine, say

\operatorname{Spec} A

, then

A = \Gamma(\mathbb{P}^n_R, \mathcal{O}_{\mathbb{P}^n_R}) = R

, implying

\mathbb{P}^n_R \cong \operatorname{Spec} R

, which fails for

n \geq 1

due to differing dimensions and topology.^[9] More generally, the global sections of the structure sheaf on

\operatorname{Proj} S

coincide with

S_0

when

S

is generated by its degree-1 part, but the compatibility with the affine opens

D_+(f_i)

covering

\operatorname{Proj} S

forces the ring of global functions to be too restrictive for

\operatorname{Proj} S

to be affine unless trivial.^[5] The construction of

\operatorname{Proj} S

relates closely to

\operatorname{Spec} S

, the spectrum of the underlying graded ring viewed as ungraded. The space

\operatorname{Spec} S

includes all prime ideals, whereas

\operatorname{Proj} S

consists only of homogeneous primes not containing the irrelevant ideal

S_+

, effectively excluding the "vertex" of the affine cone

\operatorname{Spec} S

corresponding to ideals containing

S_+

.^[1] This exclusion removes the point at infinity in the cone interpretation, making

\operatorname{Proj} S

a non-affine quotient-like object despite its affine open cover.^[37] In the relative setting, the global

\operatorname{Proj}_X(\mathcal{A})

of a quasi-coherent graded

\mathcal{O}_X

-algebra

\mathcal{A}

on a scheme

X

comes equipped with a morphism

\pi: \operatorname{Proj}_X(\mathcal{A}) \to X

. This relative scheme is affine over

X

only in the trivial case where

\mathcal{A} = \mathcal{O}_X

concentrated in degree zero, in which case

\operatorname{Proj}_X(\mathcal{A})

is empty (and hence affine over

X

).^[38] For non-trivial

\mathcal{A}

, such as when

\mathcal{A}

is generated in positive degrees, the fibers are non-affine (e.g., projective spaces), preventing relative affineness.^[9] A key result characterizing affineness is that

\operatorname{Proj} S

is an affine scheme if and only if

S = S_0

for some

t \in S_1

, in which case

\operatorname{Proj} S \cong \operatorname{Spec} S_0

. However, this structure contradicts the typical projectivity of

\operatorname{Proj} S

constructions, as projective schemes over a field are proper but non-finite unless points, rendering non-trivial projective examples non-affine.^[1]

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