SÉRIES E POLINÔMIOS DE HILBERT NO SISTEMA PROGRESSIMAL INFINITESIMAL DE GRACELI.

P = PROGRESSÃO.

S = VARIÁVEL COMPLEXA.

         ELEMENTOS DO SISTEMA PROGRESSIMAL INFINITESIMAL DE GRACELI 

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Hilbert series and Hilbert polynomial NO SISTEMA PROGRESSIMAL INFINTESIMAL DE GRACELI.  SISTEMA DE EXPOENTE GRACELI].

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 quotient by a homogeneous ideal 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 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 ${\displaystyle \pi :X\to S}$ has the same Hilbert polynomial over any closed point ${\displaystyle s\in S}$. This is used in the construction of the Hilbert scheme and Quot scheme.

• 1Definitions and main properties[edit]

Consider a finitely generated graded commutative algebra S over a field K, which is finitely generated by elements of positive degree. This means that

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${\displaystyle S=\bigoplus _{i\geq 0}S_{i}}$

and that ${\displaystyle S_{0}=K}$.

The Hilbert function

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${\displaystyle HF_{S}:n\longmapsto \dim _{K}S_{n}}$

maps the integer n to the dimension of the K-vector space S_n. The Hilbert series, which is called Hilbert–Poincaré series in the more general setting of graded vector spaces, is the formal series

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${\displaystyle HS_{S}(t)=\sum _{n=0}^{\infty }HF_{S}(n)t^{n}.}$

If S is generated by h homogeneous elements of positive degrees ${\displaystyle d_{1},\ldots ,d_{h}}$, then the sum of the Hilbert series is a rational fraction

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${\displaystyle HS_{S}(t)={\frac {Q(t)}{\prod _{i=1}^{h}\left(1-t^{d_{i}}\right)}},}$

where Q is a polynomial with integer coefficients.

If S is generated by elements of degree 1 then the sum of the Hilbert series may be rewritten as

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${\displaystyle HS_{S}(t)={\frac {P(t)}{(1-t)^{\delta }}},}$

where P is a polynomial with integer coefficients, and ${\displaystyle \delta }$ is the Krull dimension of S.

In this case the series expansion of this rational fraction is

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${\displaystyle HS_{S}(t)=P(t)\left(1+\delta t+\cdots +{\binom {n+\delta -1}{\delta -1}}t^{n}+\cdots \right)}$

where

${\displaystyle {\binom {n+\delta -1}{\delta -1}}={\frac {(n+\delta -1)(n+\delta -2)\cdots (n+1)}{(\delta -1)!}}}$

is the binomial coefficient for ${\displaystyle n>-\delta ,}$ and is 0 otherwise.

If

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the coefficient of ${\displaystyle t^{n}}$ in ${\displaystyle HS_{S}(t)}$ is thus

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${\displaystyle HF_{S}(n)=\sum _{i=0}^{d}a_{i}{\binom {n-i+\delta -1}{\delta -1}}.}$

For ${\displaystyle n\geq i-\delta +1,}$ the term of index i in this sum is a polynomial in n of degree ${\displaystyle \delta -1}$ with leading coefficient ${\displaystyle a_{i}/(\delta -1)!.}$ This shows that there exists a unique polynomial ${\displaystyle HP_{S}(n)}$ with rational coefficients which is equal to ${\displaystyle HF_{S}(n)}$ for n large enough. This polynomial is the Hilbert polynomial, and has the form

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${\displaystyle HP_{S}(n)={\frac {P(1)}{(\delta -1)!}}n^{\delta -1}+{\text{ terms of lower degree in }}n.}$

The least n₀ such that ${\displaystyle HP_{S}(n)=HF_{S}(n)}$ for n ≥ n₀ is called the Hilbert regularity. It may be lower than ${\displaystyle \deg P-\delta +1}$.

The Hilbert polynomial is a numerical polynomial, since the dimensions are integers, but the polynomial almost never has integer coefficients (Schenck 2003, pp. 41).

All these definitions may be extended to finitely generated graded modules over S, with the only difference that a factor t^m appears in the Hilbert series, where m 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 V in Pⁿ is defined as the Hilbert polynomial of the homogeneous coordinate ring of V.

Graded algebra and polynomial rings[edit]

Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if S is a graded algebra generated over the field K by n homogeneous elements g₁, ..., g_n of degree 1, then the map which sends X_i onto g_i defines an homomorphism of graded rings from ${\displaystyle R_{n}=K[X_{1},\ldots ,X_{n}]}$ onto S. Its kernel is a homogeneous ideal I and this defines an isomorphism of graded algebra between ${\displaystyle R_{n}/I}$ and S.

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[edit]

Additivity[edit]

Hilbert series and Hilbert polynomial are additive relatively to exact sequences. More precisely, if

${\displaystyle 0\;\rightarrow \;A\;\rightarrow \;B\;\rightarrow \;C\;\rightarrow \;0}$

is an exact sequence of graded or filtered modules, then we have

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${\displaystyle HS_{B}=HS_{A}+HS_{C}}$

and

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${\displaystyle HP_{B}=HP_{A}+HP_{C}.}$

This follows immediately from the same property for the dimension of vector spaces.

where the arrow labeled f is the multiplication by f, and ${\displaystyle A^{[d]}}$ is the graded module which is obtained from A by shifting the degrees by d, in order that the multiplication by f has degree 0. This implies that 

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${\displaystyle HS_{A^{[d]}}(t)=t^{d}\,HS_{A}(t)\,.}$

Hilbert series and Hilbert polynomial of a polynomial ring[edit]

The Hilbert series of the polynomial ring ${\displaystyle R_{n}=K[x_{1},\ldots ,x_{n}]}$ in ${\displaystyle n}$ indeterminates is

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${\displaystyle HS_{R_{n}}(t)={\frac {1}{(1-t)^{n}}}\,.}$

It follows that the Hilbert polynomial is

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${\displaystyle HP_{R_{n}}(k)={{k+n-1} \choose {n-1}}={\frac {(k+1)\cdots (k+n-1)}{(n-1)!}}\,.}$

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 ${\displaystyle x_{n}}$) and remarking that ${\displaystyle HS_{K}(t)=1\,.}$

Shape of the Hilbert series and dimension[edit]

A graded algebra A 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 A as a K-vector space is finite and the Hilbert series of A is a polynomial P(t) such that P(1) is equal to the dimension of A as a K-vector space.

If the Krull dimension of A is positive, there is a homogeneous element f of degree one which is not a zero divisor (in fact almost all elements of degree one have this property). The Krull dimension of A/(f) is the Krull dimension of A minus one.

The additivity of Hilbert series shows that ${\displaystyle HS_{A/(f)}(t)=(1-t)\,HS_{A}(t)}$. Iterating this a number of times equal to the Krull dimension of A, we get eventually an algebra of dimension 0 whose Hilbert series is a polynomial P(t). This show that the Hilbert series of A is

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${\displaystyle HS_{A}(t)={\frac {P(t)}{(1-t)^{d}}}}$

where the polynomial P(t) is such that P(1) ≠ 0 and d is the Krull dimension of A.

This formula for the Hilbert series implies that the degree of the Hilbert polynomial is d, and that its leading coefficient is ${\displaystyle {\frac {P(1)}{d!}}}$.