# Projective spectrum of a ring

A scheme associated with a graded ring (cf. also Graded module). As a set of points, is a set of homogeneous prime ideals such that does not contain . The topology on is defined by the following basis of open sets: for , . The structure sheaf of the locally ringed space is defined on the basis open sets as follows: , that is, the subring of the elements of degree of the ring of fractions with respect to the multiplicative system .

The most important example of a projective spectrum is . The set of its -valued points for any field is in natural correspondence with the set of points of the -dimensional projective space over the field .

If all the rings as -modules are spanned by ( terms), then an additional structure is defined on . Namely, the covering and the units determine a Čech -cocycle on to which an invertible sheaf, denoted by , corresponds. The symbol usually denotes the -th tensor power of . There exists a canonical homomorphism , indicating the geometric meaning of the grading of the ring (see [1]). If, for example, , then corresponds to a sheaf of hyperplane sections in .

#### References

[1] | D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) |

[2] | A. Grothendieck, "Eléments de géometrie algebrique" Publ. Math. IHES , 1–4 (1960–1967) |

#### Comments

See also Projective scheme.

#### References

[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 |

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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=18513