DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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The Hilbert function and formal Hilbert series of a graded module with finite-length pieces

Definition

Let $S = ⨁_{n \geq 0} S_{n}$ be a graded ring and $M = ⨁_{n \in Z} M_{n}$ a graded $S$ -module. Assume each homogeneous piece $M_{n}$ has finite length as an $S_{0}$ -module and that $M_{n} = 0$ for all sufficiently negative $n$ .

The Hilbert function of $M$ is

H_{M} (n) : = ℓ_{S_{0}} (M_{n}) (n \in Z) .

Its formal Hilbert series is the formal Laurent series

{HS}_{M} (t) : = \sum_{n \in Z} H_{M} (n) t^{n} .

Thus this is an ordinary formal power series after a shift. For a twist, one has

{HS}_{M (a)} (t) = t^{- a} {HS}_{M} (t) .

Depends on

Used by

A finite graded module over a standard graded algebra has rational Hilbert series and eventual polynomial growth Theorem

Dependency tree · two levels

5 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

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, §20 (standard reference, not scraped)
Stacks Project, Section 10.58: Noetherian graded rings (standard reference, not scraped)