Hilbert–Samuel function

<h2 id="examples">Examples</h2>
For the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> of <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a> 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.}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
<h2 id="degree-bounds">Degree bounds</h2>
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 <a href="/facts/Artin%25E2%2580%2593Rees_lemma/cdskVx25">Artin–Rees lemma</a>. 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-<a href="/facts/Primary_ideal/kvY9HXVL">primary ideal</a>. 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,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> then we have:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

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.

<h2 id="multiplicity">Multiplicity</h2>
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}}
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/J-multiplicity/epC4d3XF">j-multiplicity</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">This implies that 
 
 
 
 
 M
 ′
 
 
 /
 
 I
 
 M
 ′
 
 
 
 {\displaystyle M'/IM'}
 
 and 
 
 
 
 
 M
 ″
 
 
 /
 
 I
 
 M
 ″
 
 
 
 {\displaystyle M''/IM''}
 
 also have finite length. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">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. <a href="/wiki/David_Eisenbud" target="_blank">/wiki/David_Eisenbud</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

Hilbert–Samuel function open-in-new

Hilbert–Samuel function