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 |
Projective spectrum of a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_spectrum_of_a_ring&oldid=23937