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Hilbert-Samuel leading coefficients are additive at the top polynomial degree

Statement

Assume the Axiom of Choice (The Axiom of Choice).

Let $(R, m)$ be a Noetherian local ring, let $0 \to M^{'} \to M \to M^{''} \to 0$ be a short exact sequence of finite $R$ -modules, and let $I \subseteq m$ be an ideal of definition for $M$ . If $M = 0$ , put $d = 0$ ; otherwise put $d = \max {\deg P_{I, M^{'}}, \deg P_{I, M}, \deg P_{I, M^{''}}},$ ignoring a zero module when taking the maximum, and let $e_{I}^{[d]} (N) : = d! [n^{d}] P_{I, N} (n),$ with value $0$ when $N = 0$ or $\deg P_{I, N} < d$ . Then $e_{I}^{[d]} (M) = e_{I}^{[d]} (M^{'}) + e_{I}^{[d]} (M^{''}) .$ In particular, if all three nonzero modules have Hilbert-Samuel polynomial of degree $d$ , then $e_{I} (M) = e_{I} (M^{'}) + e_{I} (M^{''}) .$

Facts & Assumptions

Given: The Axiom of Choice, a Noetherian local ring $(R, m)$ , an ideal of definition $I \subseteq m$ for $M$ , and a short exact sequence $0 \to M^{'} \to M \to M^{''} \to 0$ of finite modules.

[L1]

Artin-Rees gives an exact eventual formula for the filtration induced on the submodule $M^{'}$ (Artin-Rees controls intersections of submodules with high ideal powers).

[L2]

Length is additive on short exact sequences (Module length is additive in short exact sequences).

[L3]

Under the Axiom of Choice, every finite module with an ideal of definition has an eventual Hilbert-Samuel polynomial (The Hilbert-Samuel function agrees eventually with a rational polynomial in binomial form).

Proof

technique · direct

1.1L1L2L3givenalgebra

The finite modules $M^{'}$ , $M^{''}$ , and every finite submodule $U \subseteq M^{'}$ also have $I$ as an ideal of definition. Indeed, finite length of $M / I M$ gives $m^{s} M \subseteq I M$ for some $s$ , hence $m^{s t} M \subseteq I^{t} M$ . By Artin-Rees [L1], for sufficiently large $t$ one has $U \cap I^{t} M \subseteq I U$ , so $m^{s t} U \subseteq I U$ . The finite module $U / I U$ is then killed by a power of the finitely generated maximal ideal and has finite length; $M^{''} / I M^{''}$ , as a quotient of $M / I M$ , also has finite length. Thus [L3] supplies the eventual polynomials of $M^{'}$ , $M^{''}$ , $N$ below, and $M$ . Artin-Rees as recorded in [L1] gives $c \geq 0$ and $N : = M^{'} \cap I^{c} M$ such that $I^{c} M^{'} \subseteq N \subseteq M^{'}$ and $M^{'} \cap I^{n + 1} M = I^{n + 1 - c} N$ for all large $n$ . Since $M^{'} / I^{c} M^{'}$ has finite length, so does $M^{'} / N$ . The exact sequence $0 \to M^{'} / (M^{'} \cap I^{n + 1} M) \to M / I^{n + 1} M \to M^{''} / I^{n + 1} M^{''} \to 0$ and [L2] therefore give $χ_{I, M} (n) = χ_{I, M^{''}} (n) + χ_{I, N} (n - c) + ℓ_{R} (M^{'} / N)$ for all large $n$ .

2.1L2step 1.1algebra

Put $C : = ℓ_{R} (M^{'} / N)$ . The inclusions $I^{c} M^{'} \subseteq N \subseteq M^{'}$ give $I^{n + c + 1} M^{'} \subseteq I^{n + 1} N \subseteq I^{n + 1} M^{'},$ while [L2] gives $ℓ_{R} (M^{'} / I^{n + 1} N) = χ_{I, N} (n) + C .$ Consequently $χ_{I, M^{'}} (n) \leq χ_{I, N} (n) + C \leq χ_{I, M^{'}} (n + c)$ for all large $n$ . If $M^{'}$ has positive Hilbert-Samuel degree, this squeeze shows that $P_{I, N}$ and $P_{I, M^{'}}$ have the same degree and leading coefficient. If $P_{I, M^{'}}$ has degree zero, the two outer terms in the squeeze are the same constant polynomial, so $P_{I, N} + C = P_{I, M^{'}}$ . The same conclusion is immediate when $M^{'} = 0$ .

3.1step 1.1step 2.1algebra

The eventual identity in step 1.1 is the polynomial identity $P_{I, M} (n) = P_{I, M^{''}} (n) + P_{I, N} (n - c) + C .$ By step 2.1, the polynomial $P_{I, N} (n - c) + C$ has the same degree- $d$ coefficient as $P_{I, M^{'}}$ : for positive degree this is invariance of the leading coefficient under a shift, for degree zero it is the constant-polynomial equality, and below degree $d$ both coefficients vanish. Comparing degree- $d$ coefficients therefore gives $e_{I}^{[d]} (M) = e_{I}^{[d]} (M^{'}) + e_{I}^{[d]} (M^{''}) .$ This also covers the all-zero sequence by the convention $d = 0$ . When all three modules are nonzero of degree $d$ , the displayed quantities are their ordinary Hilbert-Samuel multiplicities.

4.1step 3.1∎

Therefore Hilbert-Samuel leading coefficients are additive in the stated top-degree sense.

Depends on

Used by

The degree of the Hilbert-Samuel polynomial equals the dimension of the support Theorem

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Stacks Project, Lemma 10.59.10 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, Proposition (20.20) (standard reference, not scraped)