Topologies and the Artin-Rees lemma

Today I’ll continue the series on graded rings and filtrations by discussing the resulting topologies and the Artin-Rees lemma.

All filtrations henceforth are descending.

 Topologies

 Recall that a topological group is a topological space with a group structure in which the group operations of composition and inversion are continuous—in other words, a group object in the category of topological spaces.

 

Gradings, filtrations, and gr

Bourbaki has a whole chapter in Commutative Algebra devoted to “graduations, filtrations, and topologies,” which indicates the importance of these concepts. That’s the theme for the next few posts I’ll do here, although I will (of course) be more concise.

In general, all rings will be commutative.

 Gradings

 The idea of a graded ring is necessary to define projective space. 

Definition 1    A graded ring is ring together with a decomposition

 

such that . The set is said to consist of homogeneous elements of degree .