TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-23 (gpt-6-sol)

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Multiplicity under parameter reduction with annihilator correction

Statement

Assume the Axiom of Choice.

Let $(R, m)$ be a Noetherian local ring, let $M \neq 0$ be a finite $R$ -module, and put $d = \dim Supp (M)$ . Assume $d \geq 1$ and let $x, x_{2}, \dots, x_{d}$ be a system of parameters for $M$ . Put $Q = (x, x_{2}, \dots, x_{d})$ and $Q^{'} = (x_{2}, \dots, x_{d})$ . For a finite module $N$ on which $J$ is an ideal of definition, and for an integer $r \geq 0$ with either $N = 0$ or $r \geq \deg P_{J, N}$ , write $e_{J}^{[r]} (N) : = r! [n^{r}] P_{J, N} (n),$ with value $0$ when $N = 0$ or $\deg P_{J, N} < r$ . Then $e_{Q}^{[d]} (M) = e_{Q^{'}}^{[d - 1]} (M / x M) - e_{Q^{'}}^{[d - 1]} (0 :_{M} x) .$ In particular, if $x$ is $M$ -regular, then $e_{Q} (M) = e_{Q^{'}} (M / x M) .$

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), a Noetherian local ring $(R, m)$ , a nonzero finite $R$ -module $M$ of support dimension $d \geq 1$ , and a system of parameters $x, x_{2}, \dots, x_{d}$ as above.

[L1]

The degree of a nonzero finite module's Hilbert-Samuel polynomial equals its support dimension under AC (The degree of the Hilbert-Samuel polynomial equals the dimension of the support).

[L2]

For a finite module and an ideal of definition generated by $r$ elements, its degree- $r$ coefficient is the Euler characteristic of the Koszul complex; in particular its polynomial has degree at most $r$ (degree-r Hilbert–Samuel coefficient as Koszul Euler characteristic).

[L3]

When $M / (x, Q^{'}) M$ has finite length, the quotient $C = M / x M$ and annihilator $K = 0 :_{M} x$ each have $Q^{'}$ -quotients of finite length, and their Koszul Euler characteristics satisfy the first-element subtraction formula (koszul euler characteristic first element reduction).

[L4]

Degree-indexed multiplicity and Koszul Euler characteristic use the displayed polynomial and homological sign conventions (koszul euler characteristic and degree indexed multiplicity).

Proof

technique · direct

1.1L2L3given

Put $K = (0 :_{M} x)$ and $C = M / x M$ . The parameter hypothesis gives finite length of $M / Q M$ , so [L3] makes $Q^{'}$ a module-relative ideal of definition for both $C$ and $K$ (including the zero-module case). The $(d - 1)$ -generator degree bound in [L2] gives $\deg P_{Q^{'}, C}, \deg P_{Q^{'}, K} \leq d - 1$ whenever the polynomials are nonzero.

2.1L2L3L4step 1.1

By [L2], $e_{Q}^{[d]} (M) = χ (K (x, x_{2}, \dots, x_{d}; M))$ . The exact first-element reduction [L3] makes this $χ (K (x_{2}, \dots, x_{d}; C)) - χ (K (x_{2}, \dots, x_{d}; K))$ . Apply [L2] to each $Q^{'}$ -complex, with the degree-indexed convention [L4], to obtain $e_{Q}^{[d]} (M) = e_{Q^{'}}^{[d - 1]} (C) - e_{Q^{'}}^{[d - 1]} (K) .$

3.1L1step 1.1step 2.1algebra

If $x$ is $M$ -regular, then $K = 0$ . By [L1], $\deg P_{Q, M} = d$ , so $e_{Q}^{[d]} (M) = e_{Q} (M) \neq 0$ . Step 2.1 therefore makes $e_{Q^{'}}^{[d - 1]} (C)$ nonzero. The degree bound in step 1.1 forces $\deg P_{Q^{'}, C} = d - 1$ , and hence $e_{Q^{'}}^{[d - 1]} (C) = e_{Q^{'}} (C)$ . This proves $e_{Q} (M) = e_{Q^{'}} (M / x M) .$

4.1step 2.1step 3.1∎

Therefore the parameter-reduction formula holds.

Depends on

Used by

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Sources

Stacks Project, Lemmas 10.60.13 and 10.60.14 (standard reference, not scraped)
Stacks Project, Theorem 43.15.5: multiplicity as Koszul Euler characteristic (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, §21 (standard reference, not scraped)