##
Topologies and the Artin-Rees lemma
*August 19, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: Artin-Rees lemma, filtered modules, filtered rings, filtrations, I-adic filtration

2 comments

Tags: Artin-Rees lemma, filtered modules, filtered rings, filtrations, I-adic filtration

2 comments

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. (more…)

##
Gradings, filtrations, and gr
*August 18, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: filtered modules, filtered rings, gr, graded modules, graded rings, Noetherian rings

5 comments

Tags: filtered modules, filtered rings, gr, graded modules, graded rings, Noetherian rings

5 comments

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 1Agraded ringis ring together with a decomposition$latex \displaystyle A = \bigoplus_{n=-\infty}^\infty A_n \ \mathrm{as \ abelian \ groups},&fg=000000$such that $latex {A_i \cdot A_j \subset A_{i+j}}&fg=000000$. The set $latex {A_i}&fg=000000$ is said to consist of

homogeneous elementsof degree $latex {i}&fg=000000$. (more…)