Hilbert–Poincaré Series
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and in particular in the field of
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, a Hilbert–Poincaré series (also known under the name
Hilbert series 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 homoge ...
), named after
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
and
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, is an adaptation of the notion of
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
to the context of graded algebraic structures (where the dimension of the entire structure is often infinite). It is a
formal power 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 sum ...
in one indeterminate, say t, where the coefficient of t^n gives the dimension (or rank) of the sub-structure of elements homogeneous of degree n. It is closely related to the Hilbert polynomial in cases when the latter exists; however, the Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the Hilbert polynomial even if the latter exists. In good cases, the Hilbert–Poincaré series can be expressed as a
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
of its argument t.


Definition

Let ''K'' be a field, and let V=\textstyle\bigoplus_V_i be an \mathbb-
graded vector space In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces. Integer gradation Let \mathbb be th ...
over ''K'', where each subspace V_i of vectors of degree ''i'' is finite-dimensional. Then the Hilbert–Poincaré series of ''V'' is the
formal power 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 sum ...
:\sum_\dim_K(V_i)t^i. A similar definition can be given for an \N-graded ''R''-module over any
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
''R'' in which each submodule of elements homogeneous of a fixed degree ''n'' is free of finite rank; it suffices to replace the dimension by the rank. Often the graded vector space or module of which the Hilbert–Poincaré series is considered has additional structure, for instance, that of a ring, but the Hilbert–Poincaré series is independent of the multiplicative or other structure. Example: Since there are \binom monomials of degree ''k'' in variables X_0, \dots, X_n (by induction, say), one can deduce that the sum of the Hilbert–Poincaré series of K _0, \dots, X_n/math> is the
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
1/(1-t)^.


Hilbert–Serre theorem

Suppose ''M'' is a finitely generated graded module over A _1, \dots, x_n \deg x_i = d_i with an
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are na ...
(e.g., a field) ''A''. Then the Poincaré series of ''M'' is a polynomial with integral coefficients divided by \prod (1-t^). The standard proof today is an induction on ''n''. Hilbert's original proof made a use of
Hilbert's syzygy theorem In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at ...
(a
projective resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to def ...
of ''M''), which gives more homological information. Here is a proof by induction on the number ''n'' of indeterminates. If n = 0, then, since ''M'' has finite length, M_k = 0 if ''k'' is large enough. Next, suppose the theorem is true for n - 1 and consider the exact sequence of
graded module In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
s (exact degree-wise), with the notation N(l)_k = N_, :0 \to K(-d_n) \to M(-d_n) \overset \to M \to C \to 0. Since the length is additive, Poincaré series are also additive. Hence, we have: :P(M, t) = -P(K(-d_n), t) + P(M(-d_n), t) - P(C, t). We can write P(M(-d_n), t) = t^ P(M, t). Since ''K'' is killed by x_n, we can regard it as a graded module over A _0, \dots, x_/math>; the same is true for ''C''. The theorem thus now follows from the inductive hypothesis.


Chain complex

An example of graded vector space is associated to a
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
, or cochain complex ''C'' of vector spaces; the latter takes the form : 0\to C^0 \stackrel C^1\stackrel C^2 \stackrel \cdots \stackrel C^n \longrightarrow 0. The Hilbert–Poincaré series (here often called the Poincaré polynomial) of the graded vector space \bigoplus_iC^i for this complex is :P_C(t) = \sum_^n \dim(C^j)t^j. The Hilbert–Poincaré polynomial of the
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
, with cohomology spaces ''H''''j'' = ''H''''j''(''C''), is :P_H(t) = \sum_^n \dim(H^j)t^j. A famous relation between the two is that there is a polynomial Q(t) with non-negative coefficients, such that P_C(t) - P_H(t) = (1+t)Q(t).


References

* {{DEFAULTSORT:Hilbert-Poincare series Homological algebra Linear algebra Commutative algebra Mathematical series Henri Poincaré Poincare series