I've been reading about D-modules, and have seen a proof that D_X, the ring of differential operators on a variety, is "almost commutative", that is, that its associated graded ring is commutative. Now, I know that with the Ore conditions, we can localize almost commutative rings, and so we get a legitimate sheaf D to do geometry with. But how far does the analogy go? What theorems are true for commutative rings but can't be modified reasonably to be true for (nice, say, left and right noetherian or somesuch) 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.
Check out the paper differential operator on noncommutative ring, Rosenberg-Lunts define the noncommutative version of Grothendieck differential operator. This framework works perfect for arbitrary noncommutative ring. As an application. They developed quantum D-module theory as a special case of Noncommutative D-module theory. They treated the much more general case in Differential Calculus in noncommutative algebraic geometry
As Greg mentioned above, considering example of Weyl algebra. In commutative geometry, there is not good notion for closed subscheme(at least for diagonal). So,the crucial step is to find what is the correct notion for closed subscheme for noncommutative ring which led to the definition of noncommutative subscheme(naturally goes to the content of noncommutative algebraic geometry). In the paper I mentioned above, they gave the correct notion for noncommutative subscheme and correct definition for diagonal. Then they defined noncommutative differential operator as certain kind of differential bimodule and developed localization theory for differential operators. Then one can localize differential operators.(Which means (noncommutative)D-modules automatically becomes noncommutative schemes)
Notice These framework works for any noncommutative ring, either noncommutative D-module(very noncommutative, such as ring of quantized differential operators) or usual commutative D-module(itself noncommutative but almost commutative)