nLab filtered ring

Definition

A filtered ring (resp. filtered algebra) is a monoid object in the category of filtered abelian groups (resp. filtered vector spaces).

One considers positive and negative filtrations, as well as $\mathbb{Z}$-filtrations.

To-do list: complete filtrations, associated graded ring, symbol map, Poisson structure on the associated graded algebra if the latter is commutative.

Examples

A major example is the universal enveloping algebra of any Lie algebra.

Related concepts

See also Lazard's criterion and microlocalization.

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

References

• wikipedia filtration (mathematics), completion (ring_theory)