In
algebraic geometry, Proj is a construction analogous to the
spectrum-of-a-ring construction of
affine schemes, which produces objects with the typical properties of
projective spaces and
projective varieties
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
. The construction, while not
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
ial, is a fundamental tool in
scheme theory
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...
.
In this article, all
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
will be assumed to be commutative and with identity.
Proj of a graded ring
Proj as a set
Let
be a
graded ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
, where
is the
direct sum decomposition associated with the gradation. The
irrelevant ideal In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the ir ...
of
is the ideal of elements of positive degree
We say an ideal is
homogeneous if it is generated by homogeneous elements. Then, as a set,
For brevity we will sometimes write
for
.
Proj as a topological space
We may define a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, called the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
, on
by defining the closed sets to be those of the form
:
where
is a
homogeneous ideal
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
of
. As in the case of affine schemes it is quickly verified that the
form the closed sets of a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on
.
Indeed, if
are a family of ideals, then we have
and if the indexing set ''I'' is finite, then
.
Equivalently, we may take the open sets as a starting point and define
:
A common shorthand is to denote
by
, where
is the
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
generated by
. For any ideal
, the sets
and
are complementary, and hence the same proof as before shows that the sets
form a topology on
. The advantage of this approach is that the sets
, where
ranges over all homogeneous elements of the ring
, form a
base for this topology, which is an indispensable tool for the analysis of
, just as the analogous fact for the spectrum of a ring is likewise indispensable.
Proj as a scheme
We also construct a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
on
, called the “structure sheaf” as in the affine case, which makes it into a
scheme. As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following. For any open set
of
(which is by definition a set of homogeneous prime ideals of ''
'' not containing
) we define the ring
to be the set of all functions
:
(where
denotes the subring of the ring of fractions
consisting of fractions of homogeneous elements of the same degree) such that for each prime ideal
of
:
#
is an element of
;
# There exists an open subset
containing
and homogeneous elements
of ''
'' of the same degree such that for each prime ideal
of
:
#*
is not in
;
#*
It follows immediately from the definition that the
form a sheaf of rings
on
, and it may be shown that the pair (
,
) is in fact a scheme (this is accomplished by showing that each of the open subsets
is in fact an affine scheme).
The sheaf associated to a graded module
The essential property of ''
'' for the above construction was the ability to form localizations
for each prime ideal
of
. This property is also possessed by any
graded module
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
over ''
'', and therefore with the appropriate minor modifications the preceding section constructs for any such
a sheaf, denoted
, of
-modules on
. This sheaf is
quasicoherent by construction. If ''
'' is generated by finitely many elements of degree
(e.g. a polynomial ring or a homogenous quotient of it), all quasicoherent sheaves on
arise from graded modules by this construction. The corresponding graded module is not unique.
The twisting sheaf of Serre
A special case of the sheaf associated to a graded module is when we take ''
'' to be ''
'' itself with a different grading: namely, we let the degree
elements of
be the degree
elements of ''
'', so
and denote
. We then obtain
as a quasicoherent sheaf on
, denoted
or simply
, called the
twisting sheaf of
Serre. It can be checked that
is in fact an
invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
.
One reason for the utility of
is that it recovers the algebraic information of ''
'' that was lost when, in the construction of
, we passed to fractions of degree zero. In the case Spec ''A'' for a ring ''A'', the global sections of the structure sheaf form ''A'' itself, whereas the global sections of
here form only the degree-zero elements of ''
''. If we define
:
then each
contains the degree-
information about
, denoted
, and taken together they contain all the grading information that was lost. Likewise, for any sheaf of graded
-modules
we define
:
and expect this “twisted” sheaf to contain grading information about ''
''. In particular, if
is the sheaf associated to a graded
-module
we likewise expect it to contain lost grading information about ''
''. This suggests, though erroneously, that ''
'' can in fact be reconstructed from these sheaves; as
however, this is true in the case that ''
'' is a polynomial ring, below. This situation is to be contrasted with the fact that the
spec functor is adjoint to the
global sections functor
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
in the category of
locally ringed spaces.
Projective ''n''-space
If ''
'' is a ring, we define projective ''n''-space over
to be the
scheme
:
The grading on the polynomial ring
is defined by letting each
have degree one and every element of ''
'', degree zero. Comparing this to the definition of
, above, we see that the sections of
are in fact linear homogeneous polynomials, generated by the
themselves. This suggests another interpretation of
, namely as the sheaf of “coordinates” for
, since the
are literally the coordinates for projective
-space.
Examples of Proj
Proj over the affine line
If we let the base ring be