Multiplicity of a module

with respect to an ideal

Let $A$ be a commutative ring with unit. A module $M$ over $A$ is said to be of finite length $n$ if there is a sequence of submodules (a Jordan–Hölder sequence) $M _ {0} \subset \dots \subset M _ {n}$ such that each of the quotients $M _ {i} / M _ {i+} 1$, $i = 0 \dots n - 1$, is a simple $A$- module. (The number $n$ does not depend on the sequence chosen, by the Jordan–Hölder theorem.) Now let $M$ be an $A$- module of finite type and $\mathfrak a$ an ideal contained in the radical of $A$ and such that $M / \mathfrak a M$ is of finite length, and let $M \neq 0$ be of Krull dimension $d$. (The Krull dimension of a module $M$ is equal to the dimension of the ring $A / \mathfrak q ( M)$ where $\mathfrak q ( M)$ is the annihilator of $M$, i.e. $\mathfrak q ( M) = \{ {a \in A } : {a M = 0 } \}$.) Then there exists a unique integer $e _ {A} ( \mathfrak a ; M )$ such that

$$\textrm{ length } _ {A} ( M / \mathfrak a ^ {n+} 1 M ) = \ e _ {A} ( \mathfrak a ; M ) \frac{n ^ {d} }{d!} + ( \textrm{ lower degree terms } )$$

for $n$ large enough. The number $e _ {A} ( \mathfrak a ; M )$ is called the multiplicity of $M$ with respect to $\mathfrak a$. The multiplicity of an ideal $\mathfrak a$ is $e ( \mathfrak a ) = e _ {A} ( \mathfrak a ; A )$. Thus, the multiplicity of the maximal ideal $\mathfrak m$ of a local ring $A$ of dimension $d$ is equal to $( d - 1 ) !$ times the leading coefficient of the Hilbert–Samuel polynomial of $A$, cf. Local ring.

There are some mild terminological discrepancies in the literature with respect to the Hilbert–Samuel polynomial. Let $\psi ( n) = \textrm{ length } _ {A} ( M / \mathfrak a ^ {n+} 1 M )$ and $\chi ( n) = \textrm{ length } _ {A} ( \mathfrak a ^ {n} M / \mathfrak a ^ {n+} 1 M )$. Then both $\psi ( n)$ and $\chi ( n)$ are sometimes called Hilbert–Samuel functions. For both $\psi ( n)$ and $\chi ( n)$ there are polynomials in $n$( of degree $d$ and $d - 1$, respectively) such that $\psi ( n)$ and $\chi ( n)$ coincide with these polynomials for large $n$. Both these polynomials occur in the literature under the name Hilbert–Samuel polynomial.

For a more general set-up cf. [a1].

The multiplicity of a local ring $A$ is the multiplicity of its maximal ideal $\mathfrak m$, $e _ {A} ( \mathfrak m ; A )$.

References

[a1]	N. Bourbaki, "Algèbre commutative" , Masson (1983) pp. Chapt. 8, §4: Dimension MR2333539 MR2284892 MR0260715 MR0194450 MR0217051 MR0171800 Zbl 0579.13001
[a2]	M. Nagata, "Local rings" , Interscience (1962) pp. Chapt. III, §23 MR0155856 Zbl 0123.03402
[a3]	D. Mumford, "Algebraic geometry" , 1. Complex projective varieties , Springer (1976) pp. Appendix to Chapt. 6 MR0453732 Zbl 0356.14002
[a4]	O. Zariski, P. Samuel, "Commutative algebra" , 2 , v. Nostrand (1960) pp. Chapt. VIII, §10 MR0120249 Zbl 0121.27801