Hilbert–Samuel function

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

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

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

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

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

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Examples

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

χ ( 1 ) = 6 , χ ( 2 ) = 18 , χ ( 3 ) = 36 , χ ( 4 ) = 60 , and in general χ ( n ) = 3 n ( n + 1 ) for n ≥ 0. {\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.} ³

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 P I , M {\displaystyle P_{I,M}} the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.

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

0 → M ′ → M → M ″ → 0 {\displaystyle 0\to M'\to M\to M''\to 0}

is an exact sequence of finitely generated R-modules and if M / I M {\displaystyle M/IM} has finite length,⁴ then we have:⁵

P I , M = P I , M ′ + P I , M ″ − F {\displaystyle P_{I,M}=P_{I,M'}+P_{I,M''}-F}

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

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

0 → ( I n M ∩ M ′ ) / I n M ′ → M ′ / I n M ′ → M / I n M → M ″ / I n M ″ → 0 , {\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:

χ M I ( n − 1 ) = χ M ′ I ( n − 1 ) + χ M ″ I ( n − 1 ) − ℓ ( ( I n M ∩ M ′ ) / I n M ′ ) {\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,

I n M ∩ M ′ = I n − k ( ( I k M ) ∩ M ′ ) ⊂ I n − k M ′ . {\displaystyle I^{n}M\cap M'=I^{n-k}((I^{k}M)\cap M')\subset I^{n-k}M'.}

Thus,

ℓ ( ( I n M ∩ M ′ ) / I n M ′ ) ≤ χ M ′ I ( n − 1 ) − χ M ′ I ( n − k − 1 ) {\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 A {\displaystyle A} is a local ring of Krull dimension d {\displaystyle d} , with m {\displaystyle m} -primary ideal I {\displaystyle I} , its Hilbert polynomial has leading term of the form e d ! ⋅ n d {\displaystyle {\frac {e}{d!}}\cdot n^{d}} for some integer e {\displaystyle e} . This integer e {\displaystyle e} is called the multiplicity of the ideal I {\displaystyle I} . When I = m {\displaystyle I=m} is the maximal ideal of A {\displaystyle A} , one also says e {\displaystyle e} is the multiplicity of the local ring A {\displaystyle A} .

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

References

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. ↩
Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969. ↩
Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969. ↩
This implies that M ′ / I M ′ {\displaystyle M'/IM'} and M ″ / I M ″ {\displaystyle M''/IM''} also have finite length. ↩
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. /wiki/David_Eisenbud ↩

Examples

Degree bounds

Multiplicity

See also

References