nLab associated graded ring

Definition

A filtered ring is a filtered object in the category Ring of rings.

The associated graded ring to a filtered ring is the corresponding associated graded object.

Properties

A version of PBW theorem states that if a Lie algebra $g$ over a field $k$ is flat as a $k$-module over a commutative ground ring $k\supset \mathbb{Q}$ containing rationals, then the associated graded ring $Gr U(g)$ is isomorphic to the symmetric algebra $Sym(g)$ of the underlying $k$-module of $g$.

Related concepts

filtered objects

• filtered topological space

  • spectral sequence of a tower of fibrations

• filtered vector space

• filtered chain complex

  • spectral sequence of a filtered complex

  • Hodge filtration

• filtered ring

• filtered object in an (∞,1)-category

• filtered stable homotopy type

  • spectral sequence of a filtered stable homotopy type

• filtered probability space

associated graded objects

• associated graded vector space

• associated graded ring