Hilbert series and Hilbert polynomial

In commutative algebra, 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 over these algebras, as well as to coherent sheaves over projective schemes.
The typical situations where these notions are used are the following:
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 formula_1 has the same Hilbert polynomial over any closed point formula_2. This is used in the construction of the Hilbert scheme 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 formula_4.
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
If is generated by homogeneous elements of positive degrees formula_7, 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 formula_10 is the Krull dimension of .
In this case the series expansion of this rational fraction is
where
is the binomial coefficient for formula_13 and is 0 otherwise.
If 
the coefficient of formula_15 in formula_16 is thus
For formula_18 the term of index in this sum is a polynomial in of degree formula_19 with leading coefficient formula_20 This shows that there exists a unique polynomial formula_21 with rational coefficients which is equal to formula_22 for large enough. This polynomial is the Hilbert polynomial, and has the form
The least such that formula_24 for is called the Hilbert regularity. It may be lower than formula_25.
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 formula_26 onto . Its kernel is a homogeneous ideal and this defines an isomorphism of graded algebra between formula_27 and .
Thus, the graded algebras generated by elements of degree 1 are exactly, up to an isomorphism, the quotients of polynomial rings by homogeneous ideals. Therefore, the remainder of this article will be restricted to the quotients of polynomial rings by ideals.
Properties of Hilbert series.
Additivity.
Hilbert series and Hilbert polynomial are additive relatively to exact sequences. More precisely, if 
is an exact sequence of graded or filtered modules, then we have 
and 
This follows immediately from the same property for the dimension of vector spaces.
Quotient by a non-zero divisor.
Let be a graded algebra and a homogeneous element of degree in which is not a zero divisor. Then we have 
It follows from the additivity on the exact sequence 
where the arrow labeled is the multiplication by , and formula_33 is the graded module which is obtained from by shifting the degrees by , in order that the multiplication by has degree 0. This implies that formula_34
Hilbert series and Hilbert polynomial of a polynomial ring.
The Hilbert series of the polynomial ring formula_35 in formula_36 indeterminates is 
It follows that the Hilbert polynomial is 
The proof that the Hilbert series has this simple form is obtained by applying recursively the previous formula for the quotient by a non zero divisor (here formula_39) and remarking that formula_40
Shape of the Hilbert series and dimension.
A graded algebra generated by homogeneous elements of degree 1 has Krull dimension zero if the maximal homogeneous ideal, that is the ideal generated by the homogeneous elements of degree 1, is nilpotent. This implies that the dimension of as a -vector space is finite and the Hilbert series of is a polynomial such that is equal to the dimension of as a -vector space.
If the Krull dimension of is positive, there is a homogeneous element of degree one which is not a zero divisor (in fact almost all elements of degree one have this property). The Krull dimension of is the Krull dimension of minus one.
The additivity of Hilbert series shows that formula_41. Iterating this a number of times equal to the Krull dimension of , we get eventually an algebra of dimension 0 whose Hilbert series is a polynomial . This show that the Hilbert series of is
where the polynomial is such that and is the Krull dimension of .
This formula for the Hilbert series implies that the degree of the Hilbert polynomial is , and that its leading coefficient is formula_43.
Degree of a projective variety and Bézout's theorem.
The Hilbert series allows us to compute the degree of an algebraic variety as the value at 1 of the numerator of the Hilbert series. This provides also a rather simple proof of Bézout's theorem.
For showing the relationship between the degree of a projective algebraic set and the Hilbert series, consider an projective algebraic set , defined as the set of the zeros of a homogeneous ideal formula_44, where is a field, and let formula_45 be the ring of the regular functions on the algebraic set.
In this section, one does not need irreducibility of algebraic sets nor primality of ideals. Also, as Hilbert series are not changed by extending the field of coefficients, the field is supposed, without loss of generality, to be algebraically closed.
The dimension of is equal to the Krull dimension minus one of , and the degree of is the number of points of intersection, counted with multiplicities, of with the intersection of formula_46 hyperplanes in general position. This implies the existence, in , of a regular sequence formula_47 of homogeneous polynomials of degree one. The definition of a regular sequence implies the existence of exact sequences
for formula_49 This implies that 
where formula_51 is the numerator of the Hilbert series of .
The ring formula_52 has Krull dimension one, and is the ring of regular functions of a projective algebraic set formula_53 of dimension 0 consisting of a finite number of points, which may be multiple points. As formula_54 belongs to a regular sequence, none of these points belong to the hyperplane of equation formula_55 The complement of this hyperplane is an affine space that contains formula_56 This makes formula_53 an affine algebraic set, which has formula_58 as its ring of regular functions. The linear polynomial formula_59 is not a zero divisor in formula_60 and one has thus an exact sequence
which implies that 
Here we are using Hilbert series of filtered algebras, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra.
Thus formula_63 is an Artinian ring, which is a -vector space of dimension , and Jordan–Hölder theorem may be used for proving that is the degree of the algebraic set . In fact, the multiplicity of a point is the number of occurrences of the corresponding maximal ideal in a composition series.
For proving Bézout's theorem, one may proceed similarly. If formula_64 is a homogeneous polynomial of degree formula_10, which is not a zero divisor in , the exact sequence 
shows that
Looking on the numerators this proves the following generalization of Bézout's theorem:
In a more geometrical form, this may restated as:
The usual Bézout's theorem is easily deduced by starting from a hypersurface, and intersecting it with other hypersurfaces, one after the other.
Complete intersection.
A projective algebraic set is a complete intersection if its defining ideal is generated by a regular sequence. In this case, there is a simple explicit formula for the Hilbert series.
Let formula_71 be homogeneous polynomials in formula_72, of respective degrees formula_73 Setting formula_74 one has the following exact sequences 
The additivity of Hilbert series implies thus 
A simple recursion gives 
This shows that the complete intersection defined by a regular sequence of polynomials has a codimension of , and that its degree is the product of the degrees of the polynomials in the sequence.
Relation with free resolutions.
Every graded module over a graded regular ring has a graded free resolution, meaning there exists an exact sequence
where the formula_79 are graded free modules, and the arrows are graded linear maps of degree zero.
The additivity of Hilbert series implies that
If formula_81 is a polynomial ring, and if one knows the degrees of the basis elements of the formula_82 then the formulas of the preceding sections allow deducing formula_83 from formula_84 In fact, these formulas imply that, if a graded free module has a basis of homogeneous elements of degrees formula_85 then its Hilbert series is 
These formulas may be viewed as a way for computing Hilbert series. This is rarely the case, as, with the known algorithms, the computation of the Hilbert series and the computation of a free resolution start from the same Gröbner basis, from which the Hilbert series may be directly computed with a computational complexity which is not higher than that the complexity of the computation of the free resolution.
Computation of Hilbert series and Hilbert polynomial.
The Hilbert polynomial is easily deducible from the Hilbert series (see above). This section describes how the Hilbert series may be computed in the case of a quotient of a polynomial ring, filtered or graded by the total degree.
Thus let "K" a field, formula_87 be a polynomial ring and "I" be an ideal in "R". Let "H" be the homogeneous ideal generated by the homogeneous parts of highest degree of the elements of "I". If "I" is homogeneous, then "H"="I". Finally let "B" be a Gröbner basis of "I" for a monomial ordering refining the total degree partial ordering and "G" the (homogeneous) ideal generated by the leading monomials of the elements of "B".
The computation of the Hilbert series is based on the fact that "the filtered algebra R/I and the graded algebras R/H and R/G have the same Hilbert series".
Thus the computation of the Hilbert series is reduced, through the computation of a Gröbner basis, to the same problem for an ideal generated by monomials, which is usually much easier than the computation of the Gröbner basis. The computational complexity of the whole computation depends mainly on the regularity, which is the degree of the numerator of the Hilbert series. In fact the Gröbner basis may be computed by linear algebra over the polynomials of degree bounded by the regularity.
The computation of Hilbert series and Hilbert polynomials are available in most computer algebra systems. For example in both Maple and Magma these functions are named "HilbertSeries" and "HilbertPolynomial".
Generalization to coherent sheaves.
In algebraic geometry, graded rings generated by elements of degree 1 produce projective schemes by Proj construction while finitely generated graded modules correspond to coherent sheaves. If formula_88 is a coherent sheaf over a projective scheme "X", we define the Hilbert polynomial of formula_88 as a function formula_90, where "χ" is the Euler characteristic of coherent sheaf, and formula_91 a Serre twist. The Euler characteristic in this case is a well-defined number by Grothendieck's finiteness theorem.
This function is indeed a polynomial. For large "m" it agrees with dim formula_92 by Serre's vanishing theorem. If "M" is a finitely generated graded module and formula_93 the associated coherent sheaf the two definitions of Hilbert polynomial agree.
Graded free resolutions.
Since the category of coherent sheaves on a projective variety formula_94 is equivalent to the category of graded-modules modulo a finite number of graded-pieces, we can use the results in the previous section to construct Hilbert polynomials of coherent sheaves. For example, a complete intersection formula_94 of multi-degree formula_96 has the resolution