## Gradings, filtrations, and grAugust 18, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
Tags: , , , , ,

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.

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…)