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.
Equivalently, we may take the open sets as a starting point and define A common shorthand is to denote
, form a base for this topology, which is an indispensable tool for the analysis of
, just as the analogous fact for the spectrum of a ring is likewise indispensable.
, 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.
(which is by definition a set of homogeneous prime ideals of
consisting of fractions of homogeneous elements of the same degree) such that for each prime ideal
) is in fact a scheme (this is accomplished by showing that each of the open subsets
(e.g. a polynomial ring or a homogenous quotient of it), all quasicoherent sheaves on
A special case of the sheaf associated to a graded module is when we take
we define and expect this “twisted” sheaf to contain grading information about
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.
has a canonical projective morphism to the affine line
which is also a smooth morphism of schemes (which can be checked using the Jacobian criterion).
-variables can be converted into a projective scheme using the proj construction for the graded algebra
Weighted projective spaces can be constructed using a polynomial ring whose variables have non-standard degrees.
The proj construction extends to bigraded and multigraded rings.
Geometrically, this corresponds to taking products of projective schemes.
There is an embedding of such schemes into projective space by taking the total graded algebra
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.
-modules on a locally ringed space): that is, a sheaf with a direct sum decomposition where each
-algebra and the resulting direct sum decomposition is a grading of this algebra as a ring.
and a “projection” map p onto X such that for every open affine U of X, This definition suggests that we construct
, 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
(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
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
and construct global proj of this quotient sheaf of algebras