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
\pi:X\toS
s\inS
Consider a finitely generated graded commutative algebra over a field, which is finitely generated by elements of positive degree. This means that
S=oplusiSi
S0=K
The Hilbert function
HFS:n\longmapsto\dimKSn
HSS(t)=\sum
infty | |
n=0 |
n. | |
HF | |
S(n)t |
If is generated by homogeneous elements of positive degrees
d1,\ldots,dh
HS | |||||||||||||||||
|
,
If is generated by elements of degree 1 then the sum of the Hilbert series may be rewritten as
HS | ||||
|
,
\delta
In this case the series expansion of this rational fraction is
HSS(t)=P(t)\left(1+\deltat+ … +\binom{n+\delta-1}{\delta-1}tn+ … \right)
\binom{n+\delta-1}{\delta-1}=
(n+\delta-1)(n+\delta-2) … (n+1) | |
(\delta-1)! |
n>-\delta,
If
d | |
P(t)=\sum | |
i=0 |
i, | |
a | |
it |
tn
HSS(t)
HFS(n)=
d | |
\sum | |
i=0 |
ai\binom{n-i+\delta-1}{\delta-1}.
For
n\gei-\delta+1,
\delta-1
ai/(\delta-1)!.
HPS(n)
HFS(n)
HPS(n)=
P(1) | |
(\delta-1)! |
n\delta-1+termsoflowerdegreeinn.
The least such that
HPS(n)=HFS(n)
\degP-\delta+1
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 .
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
Rn=K[X1,\ldots,Xn]
Rn/I
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.
Hilbert series and Hilbert polynomial are additive relatively to exact sequences. More precisely, if
0 → A → B → C → 0
HSB=HSA+HSC
HPB=HPA+HPC.
Let be a graded algebra and a homogeneous element of degree in which is not a zero divisor. Then we have
HSA/(f)
d)HS | |
(t)=(1-t | |
A(t). |
0 → A[d] \xrightarrow{f} A → A/f → 0,
A[d]
HS | |
A[d] |
dHS | |
(t)=t | |
A(t). |
The Hilbert series of the polynomial ring
Rn=K[x1,\ldots,xn]
n
HS | |
Rn |
(t)=
1 | |
(1-t)n |
.
HP | |
Rn |
(k)={{k+n-1}\choose{n-1}}=
(k+1) … (k+n-1) | |
(n-1)! |
.
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
xn
HSK(t)=1.
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
HSA/(f)(t)=(1-t)HSA(t)
HS | ||||
|
This formula for the Hilbert series implies that the degree of the Hilbert polynomial is, and that its leading coefficient is
P(1) | |
d! |
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.
I\subsetk[x0,x1,\ldots,xn]
R=k[x0,\ldots,xn]/I
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
d
h0,\ldots,hd
0\longrightarrow\left(R/\langleh0,\ldots,hk-1\rangle\right)[1]\stackrel{hk}{\longrightarrow}R/\langleh1,\ldots,hk-1\rangle\longrightarrowR/\langleh1,\ldots,hk\rangle\longrightarrow0,
k=0,\ldots,d.
HS | |
R/\langleh0,\ldots,hd-1\rangle |
(t)=
dHS | ||||
(1-t) | ||||
|
,
P(t)
The ring
R1=R/\langleh0,\ldots,hd-1\rangle
V0
hd
hd=0.
V0.
V0
R0=R1/\langlehd-1\rangle
hd-1
R1,
0\longrightarrowR1\stackrel{hd-1}{\longrightarrow}R1\longrightarrowR0\longrightarrow0,
HS | |
R0 |
(t)=
(1-t)HS | |
R1 |
(t)=P(t).
Thus
R0
For proving Bézout's theorem, one may proceed similarly. If
f
\delta
0\longrightarrowR[\delta]\stackrel{f}{\longrightarrow}R\longrightarrowR/\langlef\rangle\longrightarrow0,
HSR/\langle(t)=\left(1-t\delta\right)HSR(t).
Looking on the numerators this proves the following generalization of Bézout's theorem:
Theorem - If is a homogeneous polynomial of degree
\delta
f
\delta.
In a more geometrical form, this may restated as:
Theorem - If a projective hypersurface of degree does not contain any irreducible component of an algebraic set of degree, then the degree of their intersection is .
The usual Bézout's theorem is easily deduced by starting from a hypersurface, and intersecting it with other hypersurfaces, one after the other.
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
f1,\ldots,fk
R=K[x1,\ldots,xn]
\delta1,\ldots,\deltak.
Ri=R/\langlef1,\ldots,fi\rangle,
0 →
[\deltai] | |
R | |
i-1 |
\xrightarrow{fi} Ri-1 → Ri → 0.
The additivity of Hilbert series implies thus
HS | |
Ri |
\deltai | |
(t)=(1-t |
)HS | |
Ri-1 |
(t).
HS | (t)= | |
Rk |
| |||||||||||
(1-t)n |
=
| |||||||||||
(1-t)n-k |
.
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.
Every graded module over a graded regular ring has a graded free resolution because of the Hilbert syzygy theorem, meaning there exists an exact sequence
0\toLk\to … \toL1\toM\to0,
Li
The additivity of Hilbert series implies that
HSM(t)
k | |
=\sum | |
i=1 |
(-1)i-1
HS | |
Li |
(t).
If
R=k[x1,\ldots,xn]
Li,
HSM(t)
HSR(t)=1/(1-t)n.
\delta1,\ldots,\deltah,
HSL(t)=
| |||||||||||
(1-t)n |
.
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.
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,
R=K[x1,\ldots,xn]
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.
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
l{F}
l{F}
pl{F
l{F}(m)
This function is indeed a polynomial.[1] For large m it agrees with dim
H0(X,l{F}(m))
\tilde{M}
Since the category of coherent sheaves on a projective variety
X
X
(d1,d2)
0\to
l{O} | |
Pn |
(-d1-d2)\xrightarrow{\begin{bmatrix}f2\ -f1\end{bmatrix}}
l{O} | |
Pn |
(-d1) ⊕ l{O}
Pn |
(-d2)\xrightarrow{\begin{bmatrix}f1&f2\end{bmatrix}}
l{O} | |
Pn |
\tol{O}X\to0