Hilbert-Samuel polynomial

The Hilbert-Samuel polynomial is a term from the mathematical sub-areas of commutative algebra and algebraic geometry . It is used there in dimension theory and in the calculation of the points of intersection. While the degree is important for dimensional theory, the coefficients play a role for the intersection theory of algebraic geometry. It was named after David Hilbert and Pierre Samuel .

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

Definitions

Be it

${\displaystyle R=\bigoplus _{n\in \mathbb {N} }R_{n}}$

a graduated ring with the following properties:

1. ${\displaystyle R_{0}}$is a module of finite length ${\displaystyle R_{0}}$
2. ${\displaystyle R}$is created as a ring of and finitely many elements . ${\displaystyle R_{0}}$ ${\displaystyle a_{1},\dots ,a_{r}}$
3. ${\displaystyle M}$be a graduated finite module. ${\displaystyle R}$

Then the function

${\displaystyle f\colon \mathbb {N} \to \mathbb {N} }$

${\displaystyle f\colon n\mapsto l_{R_{0}}\left(\bigoplus _{i=0}^{n-1}M_{i}\right)}$

Called the Hilbert-Samuel function

Under the conditions of the definition (and with these designations) the following sentence applies:

• For large , the Hilbert-Samuel function is a polynomial off . It is positive and the highest coefficient of is. ${\displaystyle n}$ ${\displaystyle P[X]}$ ${\displaystyle \mathbb {Q} [X]}$ ${\displaystyle grad(P[X])\leq r}$ ${\displaystyle P[X]}$

This means that there is a: and a , so that applies to all : ${\displaystyle P[X]\in \mathbb {Q} [X]}$ ${\displaystyle :k\in \mathbb {N} }$ ${\displaystyle n>k}$

${\displaystyle f(n)=P(n)}$

This polynomial is called the Hilbert-Samuel polynomial

Dimensional theory

Is a local ring with maximum ideal , and ${\displaystyle A}$ ${\displaystyle m}$

${\displaystyle \mathrm {gr} _{m}(A):=\bigoplus _{n\in \mathbb {N} }m^{n}/m^{n+1}}$

the graduated ring to this ideal. Then the following applies to the degree of the Hilbert polynomial of this ring (considered as a module over itself): ${\displaystyle \mathrm {P} _{m}(\mathrm {gr} _{m}(A))}$

${\displaystyle \mathrm {grad} (\mathrm {P} _{m}(\mathrm {gr} _{m}(A)))=\mathrm {dim} (A)}$

( is the Krull dimension of the ring) ${\displaystyle \mathrm {dim} (A)}$

literature

• Atiyah , Macdonald : Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9
• Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
• Robin Hartshorne : Algebraic Geometry , Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9