Hilbert–Samuel function

In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel,^[1] of a nonzero finitely generated module ${\displaystyle M}$ over a commutative Noetherian local ring ${\displaystyle A}$ and a primary ideal ${\displaystyle I}$ of ${\displaystyle A}$ is the map ${\displaystyle \chi _{M}^{I}:\mathbb {N} \rightarrow \mathbb {N} }$ such that, for all ${\displaystyle n\in \mathbb {N} }$,

${\displaystyle \chi _{M}^{I}(n)=\ell (M/I^{n}M)}$

where ${\displaystyle \ell }$ denotes the length over ${\displaystyle A}$. It is related to the Hilbert function of the associated graded module ${\displaystyle \operatorname {gr} _{I}(M)}$ by the identity

${\displaystyle \chi _{M}^{I}(n)=\sum _{i=0}^{n}H(\operatorname {gr} _{I}(M),i).}$

For sufficiently large ${\displaystyle n}$, it coincides with a polynomial function of degree equal to ${\displaystyle \dim(\operatorname {gr} _{I}(M))}$, often called the Hilbert-Samuel polynomial (or Hilbert polynomial).^[2]

Examples

For the ring of formal power series in two variables ${\displaystyle k[[x,y]]}$ taken as a module over itself and the ideal ${\displaystyle I}$ generated by the monomials x² and y³ we have

${\displaystyle \chi (1)=6,\quad \chi (2)=18,\quad \chi (3)=36,\quad \chi (4)=60,{\text{ and in general }}\chi (n)=3n(n+1){\text{ for }}n\geq 0.}$^[2]

Degree bounds

Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence. However, it is still reasonably close to being additive, as a consequence of the Artin–Rees lemma. We denote by ${\displaystyle P_{I,M}}$ the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.

Theorem — Let ${\displaystyle (R,m)}$ be a Noetherian local ring and I an m-primary ideal. If

${\displaystyle 0\to M'\to M\to M''\to 0}$

is an exact sequence of finitely generated R-modules and if ${\displaystyle M/IM}$ has finite length,^[3] then we have:^[4]

${\displaystyle P_{I,M}=P_{I,M'}+P_{I,M''}-F}$

where F is a polynomial of degree strictly less than that of ${\displaystyle P_{I,M'}}$ and having positive leading coefficient. In particular, if ${\displaystyle M'\simeq M}$, then the degree of ${\displaystyle P_{I,M''}}$ is strictly less than that of ${\displaystyle P_{I,M}=P_{I,M'}}$.

Proof: Tensoring the given exact sequence with ${\displaystyle R/I^{n}}$ and computing the kernel we get the exact sequence:

${\displaystyle 0\to (I^{n}M\cap M')/I^{n}M'\to M'/I^{n}M'\to M/I^{n}M\to M''/I^{n}M''\to 0,}$

which gives us:

${\displaystyle \chi _{M}^{I}(n-1)=\chi _{M'}^{I}(n-1)+\chi _{M''}^{I}(n-1)-\ell ((I^{n}M\cap M')/I^{n}M')}$.

The third term on the right can be estimated by Artin-Rees. Indeed, by the lemma, for large n and some k,

${\displaystyle I^{n}M\cap M'=I^{n-k}((I^{k}M)\cap M')\subset I^{n-k}M'.}$

Thus,

${\displaystyle \ell ((I^{n}M\cap M')/I^{n}M')\leq \chi _{M'}^{I}(n-1)-\chi _{M'}^{I}(n-k-1)}$.

This gives the desired degree bound.

Multiplicity

If ${\displaystyle A}$ is a local ring of Krull dimension ${\displaystyle d}$, with ${\displaystyle m}$-primary ideal ${\displaystyle I}$, its Hilbert polynomial has leading term of the form ${\displaystyle {\frac {e}{d!}}\cdot n^{d}}$ for some integer ${\displaystyle e}$. This integer ${\displaystyle e}$ is called the multiplicity of the ideal ${\displaystyle I}$. When ${\displaystyle I=m}$ is the maximal ideal of ${\displaystyle A}$, one also says ${\displaystyle e}$ is the multiplicity of the local ring ${\displaystyle A}$.

The multiplicity of a point ${\displaystyle x}$ of a scheme ${\displaystyle X}$ is defined to be the multiplicity of the corresponding local ring ${\displaystyle {\mathcal {O}}_{X,x}}$.

See also

• j-multiplicity

References

1. H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I. Ann. of Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203.
2. Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969.
3. This implies that ${\displaystyle M'/IM'}$ and ${\displaystyle M''/IM''}$ also have finite length.
4. Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8. Lemma 12.3.