ac.commutative algebra - How much theory works out for "almost commutative" rings?

Don't get too excited about the theory of algebraic geometry for almost commutative algebras. A ring can be almost commutative and still have some very weird behavior. The Weyl algebras (the differential operators on affine n-space) are a great example, since they are almost commutative, and yet:

• They are simple rings. Therefore, either they have no 'closed subschemes', or the notion of closed subscheme must correspond to something different than a quotient.


• The have global dimension n, even though their associated graded algebra has global dimension 2n. So, global dimension can jump up, even along flat deformations.


• There exist non-free projective modules of the nth Weyl algebra (in fact, stably-free modules!). Thus, intuitively, Spec(D) should have non-trivial line bundles, even though it is 'almost' affine 2n-space.

Just having a ring of quotients isn't actually that strong a condition on a ring. For instance, Goldie's theorem says that any right Noetherian domain has a ring of quotients, and that is a pretty broad class of rings.

Also, what sheaf are you thinking of D as giving you? You have all these Ore localizations, and so you can try to build something like a scheme out of this. However, you start to run into some problems, because closed subspaces will no longer correspond to quotient rings. In commutative algebraic geometry, we take advantage of the miracle that the kernel of a quotient map is the intersection of a finite number of primary ideals, each of which correspond to a prime ideal and hence a localization. In noncommutative rings, there is no such connection between two-sided ideals and Ore sets.

Here's something that might work better (or maybe this is what you are talking about in the first place). If you have a positively filtered algebra A whose associated graded algebra is commutative, then A_0 is commutative, and so you can try to think of A as a sheaf of algebras on Spec(A_0). The almost commutativity requirement here assures us that any multiplicative set in A_0 is Ore in A, and so we do get a genuine sheaf of algebras on Spec(A_0). For D_X, this gives the sheaf of differential operators on X. Other algebras that work very similarly are the enveloping algebras of Lie algebroids, and also rings of twisted differential operators.