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## Topologies and the Artin-Rees lemmaAugust 19, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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 grAugust 18, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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 1    A graded ring is ring ${A}$ together with a decomposition

$\displaystyle A = \bigoplus_{n=-\infty}^\infty A_n \ \mathrm{as \ abelian \ groups},$

such that ${A_i \cdot A_j \subset A_{i+j}}$. The set ${A_i}$ is said to consist of homogeneous elements of degree ${i}$. (more…)