# Projective spectrum of a ring

A scheme $ X = \mathop{\rm Proj} ( R) $
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 géometrie algebrique" 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 |

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Projective spectrum of a ring.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Projective_spectrum_of_a_ring&oldid=54855