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Difference between revisions of "Projective spectrum of a ring"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford,   "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck,   "Eléments de géometrie algebrique" ''Publ. Math. IHES'' , '''1–4''' (1960–1967)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géometrie algebrique" ''Publ. Math. IHES'' , '''1–4''' (1960–1967) {{MR|0238860}} {{MR|0217086}} {{MR|0199181}} {{MR|0173675}} {{MR|0163911}} {{MR|0217085}} {{MR|0217084}} {{MR|0163910}} {{MR|0163909}} {{MR|0217083}} {{MR|0163908}} {{ZBL|0203.23301}} {{ZBL|0144.19904}} {{ZBL|0135.39701}} {{ZBL|0136.15901}} {{ZBL|0118.36206}} </TD></TR></table>
  
  
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. 91</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Revision as of 21:55, 30 March 2012

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) 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
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=23937
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article