In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module M over a commutative Noetherian local ring A and a primary ideal I of A is the map χ M I : N → N such that, for all n ∈ N ,
χ M I ( n ) = ℓ ( M / I n M ) where ℓ denotes the length over A . It is related to the Hilbert function of the associated graded module gr I ( M ) by the identity
χ M I ( n ) = ∑ i = 0 n H ( gr I ( M ) , i ) . For sufficiently large n , it coincides with a polynomial function of degree equal to dim ( gr I ( M ) ) .
For the ring of formal power series in two variables k [ [ x , y ] ] taken as a module over itself and graded by the order and the ideal generated by the monomials x2 and y3 we have
χ ( 1 ) = 1 , χ ( 2 ) = 3 , χ ( 3 ) = 5 , χ ( 4 ) = 6 and χ ( k ) = 6 for k > 4. 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 the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.
Proof: Tensoring the given exact sequence with 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 , 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 ′ ) .
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 ′ . Thus,
ℓ ( ( I n M ∩ M ′ ) / I n M ′ ) ≤ χ M ′ I ( n − 1 ) − χ M ′ I ( n − k − 1 ) .
This gives the desired degree bound.
