Thu 2 Feb 2006
Rob Pawloski and I became intersted in Groebner Bases when the ground ring is no longer assumed to be a field. It seemed that there are many results that can should be proven using Groebner bases, but the existing theory was limited to a ground field.
For example, a simple GB arguement can yield a -basis for
where . A slightly “harder” example is the Poincare-Birkhoff-Witt Theorem, which gives a -module isomorphism between the universal enveloping algebra and symmetric algebra of a Lie algebra. However, neither of these results should be dependent on the ground ring being a field (they are not) and it would be nice if one had a Grobner basis argument available to apply to this in the arbitrary (commutative) ground ring case. This is what we have done in
Unital Grobner Bases Over Arbitrary Commutative Ground Rings
The only condition that one needs is that the leading coefficients of your proposed Grobner basis are in fact units of the ring. If you know non-commutative Grobners Bases, then put simply, everything still works GB-wise over an arbitrary (commutative) ground ring.What is interesting, however, is that one needs to introduce the “-Polynomials” at the beginning. Let us explain this a bit. Over a field, one can still look at the leading monomial ideal, with out reference to a GB, and obtain a basis of the quotient ring. Then, one can look at GB as having control over the relations in your ideal. In the non-field case, one does not have a priori a basis for the quotient. It is exactly the introduction of the GB that lets you say anything about the quotient.
One quick application: If is a Lie algebra with Lie bracket over a ring , then one look at the Lie Algebra over given by . Then the existence of the PBW basis (which is immediate from the GB theory) over and the fact that evaluating gives the symmetric algebra on , while at is the universal enveloping algebra of gives a Deformation Quanitzation of in terms of the Kostant-Kirilov Poisson bracket.
Though this is a well known result, it shows how Grobner Bases over arbitrary ground rings is an imporant basic tool.



