DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Associated graded algebra of a filtered algebra

Definition

Let $A$ be a unital algebra with an increasing multiplicative filtration

$0 = F_{- 1} A \subseteq F_{0} A \subseteq F_{1} A \subseteq \dots,$

where $1 \in F_{0} A$ and $(F_{m} A) (F_{n} A) \subseteq F_{m + n} A$ . Its associated graded algebra is

${gr}_{F} A = ⨁_{n \geq 0} F_{n} A / F_{n - 1} A,$

with multiplication on homogeneous classes

$(a + F_{m - 1} A) (b + F_{n - 1} A) = a b + F_{m + n - 1} A .$

This product is well-defined. Indeed, if $a$ changes by $u \in F_{m - 1} A$ and $b$ by $v \in F_{n - 1} A$ , the product changes by $u b + a v + u v \in F_{m + n - 1} A$ . Associativity and the unit class descend from $A$ .

Depends on

Dependency tree · 2 levels

Sources

Etingof, MIT 18.745 notes, §12.2, printed p. 70