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A [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753601.png" /> associated with a graded ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753602.png" /> (cf. also [[Graded module|Graded module]]). As a set of points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753603.png" /> is a set of homogeneous prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753604.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753605.png" /> does not contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753606.png" />. The topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753607.png" /> is defined by the following basis of open sets: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753608.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p0753609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536010.png" />. The structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536011.png" /> of the locally ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536012.png" /> is defined on the basis open sets as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536013.png" />, that is, the subring of the elements of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536014.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536015.png" /> of fractions with respect to the multiplicative system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536016.png" />.
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The most important example of a projective spectrum is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536017.png" />. The set of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536018.png" />-valued points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536019.png" /> for any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536020.png" /> is in natural correspondence with the set of points of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536021.png" />-dimensional projective space over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536022.png" />.
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If all the rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536023.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536024.png" />-modules are spanned by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536026.png" /> terms), then an additional structure is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536027.png" />. Namely, the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536028.png" /> and the units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536029.png" /> determine a Čech <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536030.png" />-cocycle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536031.png" /> to which an [[Invertible sheaf|invertible sheaf]], denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536032.png" />, corresponds. The symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536033.png" /> usually denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536034.png" />-th tensor power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536036.png" />. There exists a canonical homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536037.png" />, indicating the geometric meaning of the grading of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536038.png" /> (see [[#References|[1]]]). If, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536040.png" /> corresponds to a sheaf of hyperplane sections in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075360/p07536041.png" />.
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A [[Scheme|scheme]]  $  X = \mathop{\rm Proj} ( R) $
 +
associated with a graded ring  $  R = \sum _ {n=0}  ^  \infty  R _ {n} $(
 +
cf. also [[Graded module|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 $
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is defined by the following basis of open sets:  $  X _ {f} = \{ {\mathfrak p } : {f \notin \mathfrak p } \} $
 +
for  $  f \in R _ {n} $,
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$  n > 0 $.  
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The structure sheaf  $  {\mathcal O} _ {X} $
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of the locally ringed space  $  X $
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is defined on the basis open sets as follows: $  \Gamma ( X _ {f} , {\mathcal O} _ {X} ) = [ R _ {(} f) ] _ {0} $,
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that is, the subring of the elements of degree  $  0 $
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of the ring $  R _ {(} f) $
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of fractions with respect to the multiplicative system  $  \{ f ^ { n } \} _ {n \geq  0 }  $.
  
====References====
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The most important example of a projective spectrum is  $  P  ^ {n} =  \mathop{\rm Proj} \mathbf Z [ T _ {0} \dots T _ {n} ] $.  
<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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The set of its  $  k $-
 +
valued points  $  P _ {k} ^ {n} $
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for any field  $  k $
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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} $
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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 $
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determine a Čech  $  1 $-
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cocycle on  $  \mathop{\rm Proj} ( R) $
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to which an [[Invertible sheaf|invertible sheaf]], denoted by  $  {\mathcal O} ( 1) $,
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corresponds. The symbol  $  {\mathcal O} ( n) $
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usually denotes the  $  n $-
 +
th tensor power  $  {\mathcal O} ( 1) ^ {\otimes n } $
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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 [[#References|[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====
 +
<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 géométrie algébrique" ''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>
  
 
====Comments====
 
====Comments====

Latest revision as of 05:57, 16 July 2024


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 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=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