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For a nonzero finite module and an ideal of definition, Hilbert-Samuel multiplicity is a positive integer

Statement

Assume the Axiom of Choice.

Let $(R, m)$ be a Noetherian local ring, let $M \neq 0$ be a finite $R$ -module, and let $I$ be an ideal of definition for $M$ . Then the Hilbert-Samuel multiplicity $e_{I} (M)$ is a positive integer.

Facts & Assumptions

Given: The Axiom of Choice, a Noetherian local ring $(R, m)$ , a nonzero finite $R$ -module $M$ , and an ideal of definition $I$ for $M$ .

[L1]

The Hilbert-Samuel function agrees for large $n$ with a polynomial written in binomial form $χ_{I, M} (n) = \sum_{j = 0}^{d} a_{j} (\binom{n + j}{j})$ for integers $a_{j}$ (The Hilbert-Samuel function agrees eventually with a rational polynomial in binomial form).

[L2]

If a finite module over a local ring satisfies $M / I^{n + 1} M = 0$ , then $M = 0$ , because the empty generating family lifts across the Jacobson-radical ideal $I^{n + 1} \subseteq m$ (Assuming the Axiom of Choice, generators modulo an ideal in the Jacobson radical lift to generators).

[L3]

Hilbert-Samuel multiplicity is the factorial-scaled leading coefficient of the eventual Hilbert-Samuel polynomial (Hilbert-Samuel multiplicity as the factorial-scaled leading coefficient).

Proof

technique · direct

1.1

By [L1], there are integers $a_{0}, \dots, a_{d}$ and a polynomial $P_{I, M} (n) = \sum_{j = 0}^{d} a_{j} (\binom{n + j}{j})$ such that $χ_{I, M} (n) = P_{I, M} (n)$ for all sufficiently large $n$ .

L1given

1.2

For every $n \geq 0$ , the quotient $M / I^{n + 1} M$ is nonzero. Indeed, if $M / I^{n + 1} M = 0$ , then $M = I^{n + 1} M$ ; since $I^{n + 1} \subseteq m = J (R)$ , [L2] would force $M = 0$ , contradicting the hypothesis. Thus $χ_{I, M} (n) = ℓ_{R} (M / I^{n + 1} M) > 0$ for every $n$ .

L2givenalgebra

2.1

The polynomial $P_{I, M}$ therefore takes positive values for all sufficiently large integers, so its leading coefficient is positive. In the binomial expansion of step 1.1 the leading coefficient is $a_{d} / d!$ , hence $a_{d} > 0$ .

step 1.1step 1.2algebra

3.1

By [L3], the Hilbert-Samuel multiplicity is $e_{I} (M) = d! \cdot (leading coefficient of P_{I, M}) = a_{d}$ . Since $a_{d} \in Z_{> 0}$ , the multiplicity is a positive integer.

L3step 2.1algebra∎

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Sources

Stacks Project, Section 10.59: Noetherian local rings (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, §21 (standard reference, not scraped)