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Projective spectrum of a ring

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A scheme associated with a graded ring R = \sum _ {n=0} ^ \infty R _ {n} ( cf. also Graded module). As a set of points, X is a set of homogeneous prime ideals \mathfrak p \subset R such that \mathfrak p does not contain \sum _ {n=1} ^ \infty R _ {n} . The topology on X is defined by the following basis of open sets: X _ {f} = \{ {\mathfrak p } : {f \notin \mathfrak p } \} for f \in R _ {n} , n > 0 . The structure sheaf {\mathcal O} _ {X} of the locally ringed space X is defined on the basis open sets as follows: \Gamma ( X _ {f} , {\mathcal O} _ {X} ) = [ R _ {(} f) ] _ {0} , that is, the subring of the elements of degree 0 of the ring R _ {(} f) of fractions with respect to the multiplicative system \{ f ^ { n } \} _ {n \geq 0 } .

The most important example of a projective spectrum is P ^ {n} = \mathop{\rm Proj} \mathbf Z [ T _ {0} \dots T _ {n} ] . The set of its k - valued points P _ {k} ^ {n} for any field k is in natural correspondence with the set of points of the n - dimensional projective space over the field k .

If all the rings R _ {m} as R _ {0} - modules are spanned by R _ {1} \otimes \dots \otimes R _ {1} ( m terms), then an additional structure is defined on \mathop{\rm Proj} ( R) . Namely, the covering \{ {X _ {f} } : {f \in R _ {1} } \} and the units f / g determine a Čech 1 - cocycle on \mathop{\rm Proj} ( R) to which an invertible sheaf, denoted by {\mathcal O} ( 1) , corresponds. The symbol {\mathcal O} ( n) usually denotes the n - th tensor power {\mathcal O} ( 1) ^ {\otimes n } of {\mathcal O} ( 1) . There exists a canonical homomorphism \phi _ {n} : R _ {n} \rightarrow \Gamma ( X , {\mathcal O} ( n) ) , indicating the geometric meaning of the grading of the ring R ( see [1]). If, for example, R = k [ T _ {0} \dots T _ {n} ] , then {\mathcal O} ( 1) corresponds to a sheaf of hyperplane sections in P _ {k} ^ {n} .

References

[1] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701
[2] A. Grothendieck, "Eléments de géométrie algébrique" Publ. Math. IHES , 1–4 (1960–1967) MR0238860 MR0217086 MR0199181 MR0173675 MR0163911 MR0217085 MR0217084 MR0163910 MR0163909 MR0217083 MR0163908 Zbl 0203.23301 Zbl 0144.19904 Zbl 0135.39701 Zbl 0136.15901 Zbl 0118.36206

Comments

See also Projective scheme.

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Projective spectrum of a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_spectrum_of_a_ring&oldid=55852
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article