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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Nonnegatively graded rings and modules, homogeneous elements, and twists

Definition

A nonnegatively graded ring is a commutative ring $S = ⨁_{n \geq 0} S_{n}$ such that $S_{n} S_{m} \subseteq S_{n + m}$ for all $m, n \geq 0$ . An element of $S_{n}$ is called homogeneous of degree $n$ .

If $S$ is graded, a graded $S$ -module is an $S$ -module $M = ⨁_{n \in Z} M_{n}$ with $S_{i} M_{j} \subseteq M_{i + j}$ for all $i \geq 0$ and $j \in Z$ . An element of $M_{j}$ is homogeneous of degree $j$ .

For an integer $a$ , the twist $M (a)$ is the graded module with $M (a)_{n} = M_{n + a} .$ Thus a homogeneous element of degree $n + a$ in $M$ is viewed as degree $n$ in $M (a)$ .

The graded ring is standard graded over $S_{0}$ when $S$ is generated as an $S_{0}$ -algebra by finitely many degree-one elements.

Used by

…and 3 more results.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Stacks Project, Section 10.56: Graded rings (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, §20 (standard reference, not scraped)