Topologies and the Artin-Rees lemma August 19, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: Artin-Rees lemma, filtered modules, filtered rings, filtrations, I-adic filtration
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.
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. (more…)
Gradings, filtrations, and gr August 18, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: filtered modules, filtered rings, gr, graded modules, graded rings, Noetherian rings
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.
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 . (more…)