Proj construction

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.
Let formula_1 be a graded ring, whereformula_2is the direct sum decomposition associated with the gradation. The irrelevant ideal of formula_1 is the ideal of elements of positive degreeformula_4We say an ideal is homogeneous if it is generated by homogeneous elements. Then, as a set,formula_5For brevity we will sometimes write formula_6 for formula_7.
Proj as a topological space.
We may define a topology, called the Zariski topology, on formula_7 by defining the closed sets to be those of the form
where formula_10 is a homogeneous ideal of formula_1. As in the case of affine schemes it is quickly verified that the formula_12 form the closed sets of a topology on formula_6.
Indeed, if formula_14 are a family of ideals, then we have formula_15 and if the indexing set "I" is finite, then formula_16.
Equivalently, we may take the open sets as a starting point and define
A common shorthand is to denote "D"("Sf") by "D"("f"), where "Sf" is the ideal generated by "f". For any ideal "a", the sets "D"("a") and "V"("a") are complementary, and hence the same proof as before shows that the sets "D"("a") form a topology on formula_7. The advantage of this approach is that the sets "D"("f"), where "f" ranges over all homogeneous elements of the ring "S", form a base for this topology, which is an indispensable tool for the analysis of formula_7, just as the analogous fact for the spectrum of a ring is likewise indispensable.
Proj as a scheme.
We also construct a sheaf on formula_7, 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 formula_21 of formula_7 (which is by definition a set of homogeneous prime ideals of "formula_1" not containing formula_24) we define the ring formula_25 to be the set of all functions
(where formula_27 denotes the subring of the ring of fractions formula_28 consisting of fractions of homogeneous elements of the same degree) such that for each prime ideal formula_29 of formula_21:
It follows immediately from the definition that the formula_25 form a sheaf of rings formula_43 on formula_7, and it may be shown that the pair (formula_7, formula_43) is in fact a scheme (this is accomplished by showing that each of the open subsets formula_47 is in fact an affine scheme).
The sheaf associated to a graded module.
The essential property of "formula_1" for the above construction was the ability to form localizations formula_27 for each prime ideal formula_29 of formula_1. This property is also possessed by any graded module formula_52 over "formula_1", and therefore with the appropriate minor modifications the preceding section constructs for any such formula_52 a sheaf, denoted formula_55, of formula_43-modules on formula_7. This sheaf is quasicoherent by construction. If "formula_1" is generated by finitely many elements of degree formula_59 (e.g. a polynomial ring or a homogenous quotient of it), all quasicoherent sheaves on formula_7 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 "formula_52" to be "formula_1" itself with a different grading: namely, we let the degree formula_63 elements of formula_52 be the degree formula_65 elements of "formula_1", soformula_67and denote formula_68. We then obtain formula_55 as a quasicoherent sheaf on formula_7, denoted formula_71 or simply formula_72, called the twisting sheaf of Serre. It can be checked that formula_72 is in fact an invertible sheaf.
One reason for the utility of formula_72 is that it recovers the algebraic information of "formula_1" that was lost when, in the construction of formula_43, 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 formula_77 here form only the degree-zero elements of "formula_1". If we define
then each formula_80 contains the degree-formula_81 information about formula_1, denoted formula_83, and taken together they contain all the grading information that was lost. Likewise, for any sheaf of graded formula_77-modules formula_85 we define
and expect this “twisted” sheaf to contain grading information about "formula_85". In particular, if formula_85 is the sheaf associated to a graded formula_1-module formula_52 we likewise expect it to contain lost grading information about "formula_52". This suggests, though erroneously, that "formula_1" can in fact be reconstructed from these sheaves; asformula_93however, this is true in the case that "formula_1" 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 "formula_95" is a ring, we define projective "n"-space over formula_95 to be the scheme
The grading on the polynomial ring formula_98 is defined by letting each formula_99 have degree one and every element of "formula_95", degree zero. Comparing this to the definition of formula_72, above, we see that the sections of formula_72 are in fact linear homogeneous polynomials, generated by the formula_103 themselves. This suggests another interpretation of formula_72, namely as the sheaf of “coordinates” for formula_7, since the formula_103 are literally the coordinates for projective formula_81-space.
Examples of Proj.
Proj over the affine line.
If we let the base ring be formula_108, thenformula_109has a canonical projective morphism to the affine line formula_110 whose fibers are elliptic curves except at the points formula_111 where the curves degenerate into nodal curves. So there is a fibrationformula_112which is also a smooth morphism of schemes (which can be checked using the Jacobian criterion).
Projective hypersurfaces and varieties.
The projective hypersurface formula_113 is an example of a Fermat quintic threefold which is also a Calabi–Yau manifold. In addition to projective hypersurfaces, any projective variety cut out by a system of homogeneous polynomialsformula_114in formula_115-variables can be converted into a projective scheme using the proj construction for the graded algebraformula_116giving an embedding of projective varieties into projective schemes.
Weighted projective space.
Weighted projective spaces can be constructed using a polynomial ring whose variables have non-standard degrees. For example, the weighted projective space formula_117 corresponds to taking formula_118 of the ring formula_119 where formula_120 have weight formula_59 while formula_122 has weight 2.
Bigraded rings.
The proj construction extends to bigraded and multigraded rings. Geometrically, this corresponds to taking products of projective schemes. For example, given the graded ringsformula_123with the degree of each generator formula_59. Then, the tensor product of these algebras over formula_125 gives the bigraded algebraformula_126where the formula_127 have weight formula_128 and the formula_129 have weight formula_130. Then the proj construction givesformula_131which is a product of projective schemes. There is an embedding of such schemes into projective space by taking the total graded algebraformula_132where a degree formula_133 element is considered as a degree formula_134 element. This means the formula_135-th graded piece of formula_136 is the moduleformula_137In addition, the scheme formula_138 now comes with bigraded sheaves formula_139 which are the tensor product of the sheaves formula_140 whereformula_141and formula_142are 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 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.
Assumptions.
Formally, let "X" be any scheme and "S" be a sheaf of graded formula_43-algebras (the definition of which is similar to the definition of formula_43-modules on a locally ringed space): that is, a sheaf with a direct sum decomposition
where each formula_146 is an formula_43-module such that for every open subset "U" of "X", "S"("U") is an formula_25-algebra and the resulting direct sum decomposition
is a grading of this algebra as a ring. Here we assume that formula_150. 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.
In this setup we may construct a scheme formula_151 and a “projection” map "p" onto "X" such that for every open affine "U" of "X",
This definition suggests that we construct formula_151 by first defining schemes formula_154 for each open affine "U", by setting
and maps formula_156, 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 formula_151. It is not hard to show that defining each formula_158 to be the map corresponding to the inclusion of formula_25 into "S"("U") as the elements of degree zero yields the necessary consistency of the formula_158, while the consistency of the formula_154 themselves follows from the quasi-coherence assumption on "S".
The twisting sheaf.
If "S" has the additional property that formula_162 is a coherent sheaf and locally generates "S" over formula_163 (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 formula_164 then the degree-one elements form a finitely-generated module over formula_164 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 formula_154 above; the resulting sheaf on formula_151 is also denoted "O"(1) and serves much the same purpose for formula_151 as the twisting sheaf on the Proj of a ring does.
Proj of a quasi-coherent sheaf.
Let formula_169 be a quasi-coherent sheaf on a scheme formula_6. The sheaf of symmetric algebras formula_171 is naturally a quasi-coherent sheaf of graded formula_43-modules, generated by elements of degree 1. The resulting scheme is denoted by formula_173. If formula_169 is of finite type, then its canonical morphism formula_175 is a projective morphism.
For any formula_176, the fiber of the above morphism over formula_177 is the projective space formula_178 associated to the dual of the vector space formula_179 over formula_180.
If formula_181 is a quasi-coherent sheaf of graded formula_43-modules, generated by formula_183 and such that formula_183 is of finite type, then formula_185 is a closed subscheme of formula_186 and is then projective over formula_6. In fact, every closed subscheme of a projective formula_173 is of this form.
Projective space bundles.
As a special case, when formula_169 is locally free of rank formula_190, we get a projective bundle formula_173 over formula_6 of relative dimension formula_81. Indeed, if we take an open cover of "X" by open affines formula_194 such that when restricted to each of these, formula_169 is free over "A", then
and hence formula_173 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. For example, let formula_198 and take homogeneous polynomials formula_199 of degree k. We can consider the ideal sheaf formula_200 of formula_201 and construct global proj of this quotient sheaf of algebras formula_202. This can be described explicitly as the projective morphism formula_203.