In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, that produces objects with the typical properties of projective spaces and projective varieties. It is a fundamental tool in scheme theory. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In mathematics, a projective space is a fundamental construction from any vector space. ... This article is about algebraic varieties. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

In this article, all rings will be assumed to be commutative and with identity. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...

Proj of a graded ring

Proj as a set

Let S be a graded ring where In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ...

We define the set Proj S to be the set of homogeneous prime ideals that do not contain In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...

.

For brevity we will sometimes use X for Proj S.

Proj as a topological space

We may define a topology, called the Zariski topology, on Proj S by defining the closed sets to be those of the form Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

where a is a homogeneous ideal of S. As in the case of affine schemes it is quickly verified that the V(a) form the closed sets of a topology on X. Equivalently, we may take the open sets as a starting point and define In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a grading. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...

A common shorthand is to denote D(Sf) by D(f), where Sf is the ideal generated by f. For any a, D(a) and V(a) are obviously complementary and hence the same proof as before shows that the D(a) are a topology on Proj S. The advantage of this approach is that the D(f), where f ranges over all homogeneous elements of S, form a base for this topology, which is an indispensable tool for the analysis of Proj S just as the analogous fact for the spectrum of a ring is likewise indispensable. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...

Proj as a scheme

We also construct a sheaf on Proj S, called the “structure sheaf” as in the affine case, which makes it into a scheme. As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following. For any open set U of Proj S (which is by definition a set of homogeneous prime ideals of S not containing S ₊ ) we define the ring O_X(U) to be the set of all functions In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

(where S_(p) denotes the subring of the ring of fractions S_p consisting of fractions of homogeneous elements of the same degree) such that for each prime ideal p of U:

1. f(p) is an element of S_(p);
2. There exists an open subset V of U containing p and homogeneous elements s, t of S of the same degree such that for each prime ideal q of V:
   • t is not in q;
   • f(q) = s/t.

It follows immediately from the definition that the O_X(U) form a sheaf of rings O_X on Proj S, and it may be shown that the pair (Proj S, O_X) is in fact a scheme (this is accomplished by showing that each of the open subsets D(f) is in fact an affine scheme).

The sheaf associated to a graded module

The essential property of S for the above construction was the ability to form localizations S_(p) for each prime ideal p of S. This property is also possessed by any graded module M over S, and therefore with the appropriate minor modifications the preceding section constructs for any such M a sheaf, denoted , of graded O_X-modules on Proj S.

The twisting sheaf of Serre

A special case of the sheaf associated to a graded module is when we take M to be S itself with a different grading: namely, we let the degree-d elements of M be the degree-(d + 1) elements of S, and denote M = S(1). We then obtain as a sheaf of graded O_X-modules on Proj S, denoted O_X(1) or simply O(1), called the twisting sheaf of Serre. It can be checked that O(1) is in fact an invertible sheaf. In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. ...

One reason for the utility of O(1) is that it recovers the algebraic information of S that was lost when, in the construction of O_X, we passed to fractions of degree zero. In the case Spec A for a ring A, the global sections of the structure sheaf form A itself, whereas the global sections of O_X here form only the degree-zero elements of S. If we define

then each O(n) contains the degree-n information about S, and taken together they contain all the grading information that was lost. Likewise, for any sheaf of graded O_X-modules N we define

and expect this “twisted” sheaf to contain grading information about N. In particular, if N is the sheaf associated to a graded S-module M we likewise expect it to contain lost grading information about M. This suggests, though erroneously, that S can in fact be reconstructed from these sheaves; however, this is true in the case that S is a polynomial ring, below.

Projective n-space

If A is a ring, we define projective n-space over A to be the scheme Scheme can refer to: The Scheme programming language. ...

The grading on the polynomial ring is defined by letting each x_i have degree one and every element of A, degree zero. Comparing this to the definition of O(1), above, we see that the sections of O(1) are in fact linear homogeneous polynomials, generated by the x_i themselves. This suggests another interpretation of O(1), namely as the sheaf of “coordinates” for Proj S, since the x_i are literally the coordinates for projective n-space.

Global Proj

A generalization of the Proj construction replaces the ring S with a sheaf of algebras and produces, as the end result, a scheme which might be thought of as a fibration of Proj's of rings. This construction is often used, for example, to construct projective space bundles over a base scheme. In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. ... Grothendiecks relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of objects explicitly depending on parameters, as the basic field of study, rather than a single such object. ...

Assumptions

Formally, let X be any scheme and S be a sheaf of graded O_X-algebras (the definition of which is similar to the definition of O_X-modules on a locally ringed space): that is, a sheaf with a direct sum decomposition In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...

where each S_i is an O_X-module such that for every open subset U of X, S(U) is an O_X(U)-algebra and the resulting direct sum decomposition

is a grading of this algebra as a ring. Here we assume that S₀ = O_X. We make the additional assumption that S is a quasi-coherent sheaf; this is a “consistency” assumption on the sections over different open sets that is necessary for the construction to proceed. In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX-modules OXm → OXn. ...

Construction

In this setup we may construct a scheme Proj S and a “projection” map p onto X such that for every open affine U of X, In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

This definition suggests that we construct Proj S by first defining schemes Y_U for each open affine U, by setting

and maps , and then showing that these data can be glued together “over” each intersection of two open affines U and V to form a scheme Y which we define to be Proj S. It is not hard to show that defining each p_U to be the map corresponding to the inclusion of O_X(U) into S(U) as the elements of degree zero yields the necessary consistency of the p_U, while the consistency of the Y_U themselves follows from the quasi-coherence assumption on S.

The twisting sheaf

If S has the additional property that S₁ is a coherent sheaf and locally generates S over S₀ (that is, when we pass to the stalk of the sheaf S at a point x of X, which is a graded algebra whose degree-zero elements form the ring O_X,x then the degree-one elements form a finitely-generated module over O_X,x and also generate the stalk as an algebra over it) then we may make a further construction. Over each open affine U, Proj S(U) bears an invertible sheaf O(1), and the assumption we have just made ensures that these sheaves may be glued just like the Y_U above; the resulting sheaf on Proj S is also denoted O(1) and serves much the same purpose for Proj S as the twisting sheaf on the Proj of a ring does. In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX-modules OXm → OXn. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. ...

Projective space bundles

As mentioned above, we obtain projective space bundles as a special case of this construction. To do this, we take S to be locally free as an O_X-algebra, which means that there exists an open cover of X by open affines Spec A such that restricted to each of these, S is the sheaf associated to a polynomial ring over A. This is stronger than being simply quasi-coherent and implies, in particular, that the number of variables in each such ring is constant on connected components of X. By the construction above, we now have on a cover of X consisting of schemes U = Spec A In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X...

and hence is a projective space bundle.

See also

• scheme
• graded ring
• projective space