Homogeneous coordinate ring
Encyclopedia
In algebraic geometry
, the homogeneous coordinate ring R of an algebraic variety
V given as a subvariety
of projective space
of a given dimension N is by definition the quotient ring
where I is the homogeneous ideal defining V, K is the algebraically closed field
over which V is defined, and
is the polynomial ring
in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates
, for a given choice of basis (in the vector space
underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra
.
, and so R is an integral domain. The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero. From the point of view of scheme theory these cases may be dealt with on the same footing by means of the Proj construction
.
The correspondence between homogeneous ideals I and varieties is bijective for ideals not containing the ideal J generated by all the Xi, which corresponds to the empty set because not all homogeneous coordinates can vanish at a point of projective space.
techniques to algebraic geometry, it has been traditional since David Hilbert
(though modern terminology is different) to apply free resolutions of R, considered as a graded module over the polynomial ring. This yields information about syzygies, namely relations between generators of the ideal I. In a classical perspective, such generators are simply the equations one writes down to define V. If V is a hypersurface
there need only be one equation, and for complete intersection
s the number of equations can be taken as the codimension; but the general projective variety has no defining set of equations that is so transparent. Detailed studies, for example of canonical curves and the equations defining abelian varieties
, show the geometric interest of systematic techniques to handle these cases. The subject also grew out of elimination theory
in its classical form, in which reduction modulo I is supposed to become an algorithmic process (now handled by Gröbner bases in practice).
There are for general reasons free resolutions of R as graded module over K[X0, X1, X2, ..., XN]. A resolution is defined as minimal if the image in each module morphism of free module
s
in the resolution lies in JFi − 1. As a consequence of Nakayama's lemma φ then takes a given basis in Fi to a minimal set of generators in Fi − 1. The concept of minimal free resolution is well-defined in a strong sense, in that such a resolution is unique (up to
isomorphism of chain complex
es) and occurs as a direct summand in any free resolution. This property of being intrinsic to R allows the definition of the graded Betti numbers, namely the βi, j which are the number of grade-j images coming from Fi (more precisely, by thinking of φ as a matrix of homogeneous polynomials, the count of entries of that homogeneous degree incremented by the gradings acquired inductively from the right). In other words weights in all the free modules may be inferred from the resolution, and the graded Betti numbers count the number of generators of a given weight in a given module of the resolution. The discussion of these invariants of V in a given projective embedding is a research area, even in the case of curves.
There are examples where the minimal free resolution is known explicitly. For a rational normal curve it is an Eagon–Northcott complex. For elliptic curve
s in projective space the resolution may be constructed as a mapping cone of Eagon–Northcott complexes.
may be read off the minimum resolution of the ideal I defining the projective variety. In terms of the imputed "shifts" ai, j in the i-th module Fi, it is the maximum over i of the ai, j − i; it is therefore small when the shifts increase only by increments of 1 as we move to the left in the resolution (linear syzygies only).
. This condition implies that V is a normal variety, but not conversely: the property of projective normality is not independent of the projective embedding, as is shown by the example of a quartic curve in three dimensions. Another equivalent condition is in terms of the linear system of divisors
on V cut out by the tautological line bundle L on projective space, and its d-th powers for d = 1, 2, 3, ... ; when V is non-singular, it is projectively normal if and only if each such linear system is a complete linear system. In a more geometric way one can think of L as the Serre twist sheaf O(1) on projective space, and use it to twist the structure sheaf OV k times, for any k. Then V is called k-normal if the global sections of O(k) map surjectively to those of OV(k), for a given k; if V is 1-normal it is called linearly normal, and projective normality is the condition that V is k-normal for all k ≥ 1. Linear normality may be said geometrically: V as projective variety cannot be obtained by an isomorphic linear projection from a projective space of higher dimension, except in the trivial way of lying in a proper linear subspace. Projective normality may similarly be translated, by using enough Veronese mappings to reduce it to conditions of linear normality.
Looking at the issue from the point of view of a given very ample line bundle giving rise to the projective embedding of V, such a line bundle (invertible sheaf
) is said to be normally generated if V as embedded is projectively normal. Projective normality is the first condition N0 of a sequence of conditions defined by Green and Lazarsfeld. For this
is considered as graded module over the homogeneous coordinate ring of the projective space, and a minimal free resolution taken. Condition Np applied to the first p graded Betti numbers, requiring they vanish when j > i + 1. For curves Green showed that condition Np is satisfied when deg(L) ≥ 2g + 1 + p, which for p = 0 was a classical result of Guido Castelnuovo
.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, the homogeneous coordinate ring R of an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V given as a subvariety
Subvariety
In botanical nomenclature, a subvariety is a taxonomic rank below that of variety but above that of form : it is an infraspecific taxon. Its name consists of three parts: a genus name, a specific epithet and an infraspecific epithet. To indicate the rank, the abbreviation "subvar." should be put...
of projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
of a given dimension N is by definition the quotient ring
- R = K[X0, X1, X2, ..., XN]/I
where I is the homogeneous ideal defining V, K is the algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
over which V is defined, and
- K[X0, X1, X2, ..., XN]
is the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
, for a given choice of basis (in the vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra
Symmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
.
Formulation
Since V is assumed to be a variety, and so an irreducible algebraic set, the ideal I can be chosen to be a prime idealPrime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
, and so R is an integral domain. The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero. From the point of view of scheme theory these cases may be dealt with on the same footing by means of the Proj construction
Proj construction
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...
.
The correspondence between homogeneous ideals I and varieties is bijective for ideals not containing the ideal J generated by all the Xi, which corresponds to the empty set because not all homogeneous coordinates can vanish at a point of projective space.
Resolutions and syzygies
In application of homological algebraHomological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
techniques to algebraic geometry, it has been traditional since David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
(though modern terminology is different) to apply free resolutions of R, considered as a graded module over the polynomial ring. This yields information about syzygies, namely relations between generators of the ideal I. In a classical perspective, such generators are simply the equations one writes down to define V. If V is a hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
there need only be one equation, and for complete intersection
Complete intersection
In mathematics, an algebraic variety V in projective space is a complete intersection if it can be defined by the vanishing of the number of homogeneous polynomials indicated by its codimension...
s the number of equations can be taken as the codimension; but the general projective variety has no defining set of equations that is so transparent. Detailed studies, for example of canonical curves and the equations defining abelian varieties
Equations defining abelian varieties
In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety...
, show the geometric interest of systematic techniques to handle these cases. The subject also grew out of elimination theory
Elimination theory
In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables....
in its classical form, in which reduction modulo I is supposed to become an algorithmic process (now handled by Gröbner bases in practice).
There are for general reasons free resolutions of R as graded module over K[X0, X1, X2, ..., XN]. A resolution is defined as minimal if the image in each module morphism of free module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
s
- φ:Fi → Fi − 1
in the resolution lies in JFi − 1. As a consequence of Nakayama's lemma φ then takes a given basis in Fi to a minimal set of generators in Fi − 1. The concept of minimal free resolution is well-defined in a strong sense, in that such a resolution is unique (up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
isomorphism of chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
es) and occurs as a direct summand in any free resolution. This property of being intrinsic to R allows the definition of the graded Betti numbers, namely the βi, j which are the number of grade-j images coming from Fi (more precisely, by thinking of φ as a matrix of homogeneous polynomials, the count of entries of that homogeneous degree incremented by the gradings acquired inductively from the right). In other words weights in all the free modules may be inferred from the resolution, and the graded Betti numbers count the number of generators of a given weight in a given module of the resolution. The discussion of these invariants of V in a given projective embedding is a research area, even in the case of curves.
There are examples where the minimal free resolution is known explicitly. For a rational normal curve it is an Eagon–Northcott complex. For elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s in projective space the resolution may be constructed as a mapping cone of Eagon–Northcott complexes.
Regularity
The Castelnuovo–Mumford regularityCastelnuovo–Mumford regularity
In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf F over projective space Pn is the smallest integer r such that it is r-regular, meaning thatH^i=0 \,...
may be read off the minimum resolution of the ideal I defining the projective variety. In terms of the imputed "shifts" ai, j in the i-th module Fi, it is the maximum over i of the ai, j − i; it is therefore small when the shifts increase only by increments of 1 as we move to the left in the resolution (linear syzygies only).
Projective normality
The variety V in its projective embedding is projectively normal if R is integrally closedIntegrally closed
In mathematics, more specifically in abstract algebra, the concept of integrally closed has two meanings, one for groups and one for rings. -Commutative rings:...
. This condition implies that V is a normal variety, but not conversely: the property of projective normality is not independent of the projective embedding, as is shown by the example of a quartic curve in three dimensions. Another equivalent condition is in terms of the linear system of divisors
Linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family....
on V cut out by the tautological line bundle L on projective space, and its d-th powers for d = 1, 2, 3, ... ; when V is non-singular, it is projectively normal if and only if each such linear system is a complete linear system. In a more geometric way one can think of L as the Serre twist sheaf O(1) on projective space, and use it to twist the structure sheaf OV k times, for any k. Then V is called k-normal if the global sections of O(k) map surjectively to those of OV(k), for a given k; if V is 1-normal it is called linearly normal, and projective normality is the condition that V is k-normal for all k ≥ 1. Linear normality may be said geometrically: V as projective variety cannot be obtained by an isomorphic linear projection from a projective space of higher dimension, except in the trivial way of lying in a proper linear subspace. Projective normality may similarly be translated, by using enough Veronese mappings to reduce it to conditions of linear normality.
Looking at the issue from the point of view of a given very ample line bundle giving rise to the projective embedding of V, such a line bundle (invertible sheaf
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 OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
) is said to be normally generated if V as embedded is projectively normal. Projective normality is the first condition N0 of a sequence of conditions defined by Green and Lazarsfeld. For this
is considered as graded module over the homogeneous coordinate ring of the projective space, and a minimal free resolution taken. Condition Np applied to the first p graded Betti numbers, requiring they vanish when j > i + 1. For curves Green showed that condition Np is satisfied when deg(L) ≥ 2g + 1 + p, which for p = 0 was a classical result of Guido Castelnuovo
Guido Castelnuovo
Guido Castelnuovo was an Italian mathematician. His father, Enrico Castelnuovo, was a novelist and campaigner for the unification of Italy...
.