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In `F33f`_`[commutative algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Commutative_algebra]`_`f the `!Hilbert–Samuel function`!, named after `F33f`_`[David Hilbert`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=David_Hilbert]`_`f and `F33f`_`[Pierre Samuel`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pierre_Samuel]`_`f,`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f] of a nonzero `F33f`_`[finitely generated`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Finitely_generated_module]`_`f `F33f`_`[module`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Module_(mathematics)]`_`f M {\\displaystyle M} over a commutative `F33f`_`[Noetherian`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Noetherian_ring]`_`f `F33f`_`[local ring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Local_ring]`_`f A {\\displaystyle A} and a `F33f`_`[primary ideal`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Primary_ideal]`_`f I {\\displaystyle I} of A {\\displaystyle A} is the map χ 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 χ M I ( n ) = ℓ ℓ ( M / I n M ) {\\displaystyle \\chi _{M}^{I}(n)=\\ell (M/I^{n}M)}

where ℓ ℓ {\\displaystyle \\ell } denotes the `F33f`_`[length`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Length_of_a_module]`_`f over A {\\displaystyle A} . It is related to the `F33f`_`[Hilbert function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hilbert_function]`_`f of the `F33f`_`[associated graded module`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Associated_graded_module]`_`f gr I ⁡ ⁡ ( M ) {\\displaystyle \\operatorname {gr} _{I}(M)} by the identity

χ 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 `F33f`_`[Hilbert polynomial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hilbert_polynomial]`_`f).`:cite-ref-ica-2-0[`F5bf`_`[2`#cite-note-ica-2]`_`f]

>>Contents

• `F0af`_`[Examples`#examples]`_`f

• `F0af`_`[Degree bounds`#degree-bounds]`_`f

• `F0af`_`[Multiplicity`#multiplicity]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Examples

For the `F33f`_`[ring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ring_(mathematics)]`_`f of `F33f`_`[formal power series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Formal_power_series]`_`f 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 `*x`*2 and `*y`*3 we have

χ χ ( 1 ) = 6 , χ χ ( 2 ) = 18 , χ χ ( 3 ) = 36 , χ χ ( 4 ) = 60 , and in general χ 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.} `:cite-ref-ica-2-1[`F5bf`_`[2`#cite-note-ica-2]`_`f]

>>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 `F33f`_`[Artin–Rees lemma`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Artin–Rees_lemma]`_`f. 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-`F33f`_`[primary ideal`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Primary_ideal]`_`f. 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,`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f] then we have:`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]

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 χ M I ( n − − 1 ) = χ χ M ′ I ( n − − 1 ) + χ χ M ″ I ( n − − 1 ) − − ℓ 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 ℓ ( ( I n M ∩ ∩ M ′ ) / I n M ′ ) ≤ ≤ χ χ M ′ I ( n − − 1 ) − − χ 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}} .

>>See also

• `F33f`_`[j-multiplicity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=J-multiplicity]`_`f

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f 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.

`:cite-note-ica-2`!2.`! `F0af`_`[↑`#cite-ref-ica-2-0]`_`f Atiyah, M. F. and MacDonald, I. G. `*Introduction to Commutative Algebra`*. Reading, MA: Addison–Wesley, 1969.

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f This implies that M ′ / I M ′ {\\displaystyle M'/IM'} and M ″ / I M ″ {\\displaystyle M''/IM''} also have finite length.

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `F33f`_`[Eisenbud, David`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=David_Eisenbud]`_`f, `*Commutative Algebra with a View Toward Algebraic Geometry`*, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-387-94268-8. Lemma 12.3.

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