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 \cdots \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