In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental tool in scheme theory.

In this article, all rings will be assumed to be commutative and with identity.

Proj of a graded ring

Proj as a set

be a commutative graded ringis the direct sum decomposition associated with the gradation. The irrelevant ideal is the idealWe say an ideal is homogeneous.

Proj as a topological space

We may define a topology, called the Zariski topology by defining the closed sets to be those of the form

is a homogeneous ideal form the closed sets of a topology.

and if the indexing set I

Equivalently, we may take the open sets as a starting point and define

is the ideal, form a base, just as the analogous fact for the spectrum of a ring is likewise indispensable.

Proj as a scheme

We also construct a sheaf, 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 (which is by definition a set of homogeneous prime ideals of to be the set of all functions

denotes the subring of the ring of fractions:

1. ;

of :

• ;

is in fact an affine scheme).

The sheaf associated to a graded module

The essential property of . This property is also possessed by any graded module over . This sheaf is quasicoherent by construction. If (e.g. a polynomial ring arise from graded modules by this construction. The corresponding graded module is not unique.

The twisting sheaf of Serre

A special case of the sheaf associated to a graded module is when we take to be elements of , called the twisting sheaf of Serre is in fact an invertible sheaf.

is that it recovers the algebraic information of , 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 here form only the degree-zero elements of . If we define

we define

and expect this “twisted” sheaf to contain grading information about we likewise expect it to contain lost grading information about . This suggests, though erroneously, that However, this is true in the case that is a polynomial ring, below. This situation is to be contrasted with the fact that the Spec functor is adjoint to the global sections functor in the category of locally ringed spaces.

Projective n-space

If is a ring, we define projective n to be the scheme

have degree one and every element of -space.

Examples of Proj

Proj over the affine line

whose fibers are elliptic curveswhich is also a smooth morphism of schemes (which can be checked using the Jacobian criterion).

Projective hypersurfaces and varieties

The projective hypersurface is an example of a Fermat quintic threefold which is also a Calabi–Yau manifoldgiving an embedding of projective varieties into projective schemes.

Weighted projective space

Weighted projective spaces has weight 2.

Bigraded rings

. Then, the tensor productare the canonical projections coming from the injections of these algebras from the tensor product diagram of commutative algebras.

Global Proj

A generalization of the Proj construction replaces the ring S with a sheaf of algebras and produces, as the 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.

Assumptions

Formally, let X be any scheme and S-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

-module such that for every open subset U of X, S(U-algebra and the resulting direct sum decomposition

. 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.

Construction

and a “projection” map p onto X such that for every open affine U of X,

for each open affine U, by setting

, 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 into S(U themselves follows from the quasi-coherence assumption on S.

The twisting sheaf

If S is a coherent sheaf and locally generates S (that is, when we pass to the stalk of the sheaf S at a point x of 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) is also denoted O as the twisting sheaf on the Proj of a ring does.

Proj of a quasi-coherent sheaf

. The sheaf of symmetric algebras is a projective morphism.

.

is of this form.

Projective space bundles

, we get a projective bundle. Indeed, if we take an open cover of X is free over A, then

is a projective space bundle. Many families of varieties can be constructed as subschemes of these projective bundles, such as the Weierstrass family of elliptic curves. For more details, see the main article.

Example of Global Proj

Global Proj can be used to construct Lefschetz pencils of degree k. We can consider the ideal sheaf.

Another application is the blow-up of a scheme with respect to a coherent sheaf of ideals.

See also

• Projective space

• Algebraic geometry of projective spaces

• Projectivization

References