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 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