Chapter 7 - Graded and filtered modules

Summary

Simple rings are very hard to study because most techniques in ring theory depend on the existence of two-sided ideals. In the case of the Weyl algebra, however, we have a way out. As we saw in Ch. 2, one may define a degree for the elements of the Weyl algebra. Using this degree, we construct a commutative ring, k[x] works as a shadow of A_n. We may then draw an outline of what A_n really looks like. This is the best method we have for understanding the structure of A_n and of its modules.

GRADED RINGS

An important feature of a polynomial ring is that it admits a degree function. We want to generalize and formalize what it means for an algebra to have a degree. This leads to the definition of graded rings. These rings find their justification in algebraic geometry, more precisely in projective algebraic geometry; for details see [Hartshorne, Ch. 1, §2], For the sake of completeness, we define graded rings without assuming commutativity.

Let R be a K-algebra. We say that R is graded if there are K-vector subspaces R_i, i ∈ ℕ, such that

1. (1) R = ⊕_i∈ℕR_i,

2. (2) R_i · R_i ⊆ R_i+.

The R_i are called the homogeneous components of R. The elements of R_i are the homogeneous elements of degree i. If R_i = 0 when i < 0 then we say that the grading is positive. From now on all graded algebras will have a positive grading unless explicitly stated otherwise.