Hilbert–Samuel function
Encyclopedia
In commutative algebra
the Hilbert–Samuel function, named after David Hilbert
and Pierre Samuel
, of a nonzero finitely generated module
over a commutative Noetherian
local ring
and a primary ideal
of 
is the map 
such that, for all 
,
where
denotes the length
over
. It is related to the Hilbert function of the associated graded module 
by the identity
For sufficiently large
, it coincides with a polynomial function of degree equal to 
.
of formal power series
in two variables
taken as a module over itself and graded by the order and the ideal generated by the monomials x2 and y3 we have
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
the Hilbert–Samuel function, named after David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
and Pierre Samuel
Pierre Samuel
Pierre Samuel was a French mathematician, known for his work in commutative algebra and its applications to algebraic geometry. The two-volume work Commutative Algebra that he wrote with Oscar Zariski is a classic. Other books of his covered projective geometry and algebraic number theory...
, of a nonzero finitely generated module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over a commutative NoetherianNoetherian
In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...
local ring
Local ring
In abstract algebra, more particularly in 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 algebraic number fields examined at a particular place, or...
and a primary idealPrimary ideal
In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n...
of is the map such that, for all ,where
denotes the lengthLength of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
over
. It is related to the Hilbert function of the associated graded module by the identity
For sufficiently large
, it coincides with a polynomial function of degree equal to .Examples
For the ringRing (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
in two variables
taken as a module over itself and graded by the order and the ideal generated by the monomials x2 and y3 we have