In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a
graded commutative algebra finitely generated over a
field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra.
These notions have been extended to
filtered algebras, and graded or filtered
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over these algebras, as well as to
coherent sheaves over
projective schemes.
The typical situations where these notions are used are the following:
* The quotient by a homogeneous
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 considered ...
of a
multivariate polynomial ring, graded by the total degree.
* The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree.
* The filtration of a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...
by the powers of its
maximal ideal. In this case the Hilbert polynomial is called the
Hilbert–Samuel polynomial.
The
Hilbert series of an algebra or a module is a special case of the
Hilbert–Poincaré series of a
graded vector space.
The Hilbert polynomial and Hilbert series are important in computational
algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations. In addition, they provide useful invariants for families of algebraic varieties because a flat family
has the same Hilbert polynomial over any closed point
. This is used in the construction of the
Hilbert scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a ...
and
Quot scheme.
Definitions and main properties
Consider a finitely generated
graded commutative algebra over a
field , which is finitely generated by elements of positive degree. This means that
:
and that
.
The Hilbert function
:
maps the integer to the dimension of the -vector space . The Hilbert series, which is called
Hilbert–Poincaré series in the more general setting of graded vector spaces, is the
formal series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
:
If is generated by homogeneous elements of positive degrees
, then the sum of the Hilbert series is a rational fraction
:
where is a polynomial with integer coefficients.
If is generated by elements of degree 1 then the sum of the Hilbert series may be rewritten as
:
where is a polynomial with integer coefficients, and
is the
Krull dimension of .
In this case the series expansion of this rational fraction is
:
where
:
is the
binomial coefficient for
and is 0 otherwise.
If
:
the coefficient of
in
is thus
:
For
the term of index in this sum is a polynomial in of degree
with leading coefficient
This shows that there exists a unique polynomial
with rational coefficients which is equal to
for large enough. This polynomial is the Hilbert polynomial, and has the form
:
The least such that
for is called the Hilbert regularity. It may be lower than
.
The Hilbert polynomial is a
numerical polynomial, since the dimensions are integers, but the polynomial almost never has integer coefficients .
All these definitions may be extended to finitely generated
graded modules over , with the only difference that a factor appears in the Hilbert series, where is the minimal degree of the generators of the module, which may be negative.
The Hilbert function, the Hilbert series and the Hilbert polynomial of a
filtered algebra are those of the associated graded algebra.
The Hilbert polynomial of a
projective variety in is defined as the Hilbert polynomial of the
homogeneous coordinate ring of .
Graded algebra and polynomial rings
Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if is a graded algebra generated over the field by homogeneous elements of degree 1, then the map which sends onto defines an homomorphism of graded rings from