Hilbert–Samuel function

In <a href="/facts/Commutative_algebra/mKtUdPCT">commutative algebra</a> the Hilbert–Samuel function, named after <a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a> and <a href="/facts/Pierre_Samuel/r63UjoTx">Pierre Samuel</a>, of a nonzero <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> 
 
 
 
 M
 
 
 {\displaystyle M}
 
 over a commutative <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian</a> <a href="/facts/Local_ring/gRTdg5Qx">local ring</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and a <a href="/facts/Primary_ideal/kvY9HXVL">primary ideal</a> 
 
 
 
 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 <a href="/facts/Length_of_a_module/UpWodT4k">length</a> over 
 
 
 
 A
 
 
 {\displaystyle A}
 
. It is related to the <a href="/facts/Hilbert_function/FtFu54gb">Hilbert function</a> of the <a href="/facts/Associated_graded_module/mxdU4Dfm">associated graded module</a> 
 
 
 
 
 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 <a href="/facts/Hilbert_polynomial/FtFu54gb">Hilbert polynomial</a>).

Hilbert–Samuel function open-in-new

Hilbert–Samuel function