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Topologies and the Artin-Rees lemma August 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 gr August 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…)

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