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Concept
Proj construction
Summary
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.
Let be a graded ring, whereis the direct sum decomposition associated with the gradation. The irrelevant ideal of is the ideal of elements of positive degreeWe say an ideal is homogeneous if it is generated by homogeneous elements. Then, as a set, For brevity we will sometimes write for .
We may define a topology, called the Zariski topology, on by defining the closed sets to be those of the form
where is a homogeneous ideal of . As in the case of affine schemes it is quickly verified that the form the closed sets of a topology on .
Indeed, if are a family of ideals, then we have and if the indexing set I is finite, then .
Equivalently, we may take the open sets as a starting point and define
A common shorthand is to denote by , where is the ideal generated by . For any ideal , the sets and are complementary, and hence the same proof as before shows that the sets form a topology on . The advantage of this approach is that the sets , where ranges over all homogeneous elements of the ring , 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.
We also construct a sheaf on , 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.
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