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 Template:ZBL
[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 Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL

Comments

See also Projective scheme.

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 MR0463157 Template:ZBL