filtered ring in nLab

Contents

Contents

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 ℤ\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.

See also Lazard's criterion and microlocalization.

References

• wikipedia filtration (mathematics), completion (ring_theory)