TITLE: Groebner bases for differential operators with field coefficients (reference request)
QUESTION [5 upvotes]: Let $K$ be a field, $\partial_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial_1\ldots \partial_n]$ (it is implicitly assumed that the derivations do not commute with elements of $K$ and the commutation relation is the obvious $\partial_i k - k\partial_i=\partial_i(k)$).
For example, $R$ can be the ring of differential operators with rational function coefficients, etc.
It appears that much of the theory of Groebner bases for polynomial rings $S=K[x_1\ldots x_n]$ can be trivially restated for $R=K[\partial_1\ldots \partial_n]$, since when we work with a monomial order, we usually track only what happens to the leading monomial, and it is the same for $R$ and $S$. I am pretty sure that this is also true for modules over $R$. (If I am wrong and this is not true, an explanation of this will be a perfect answer to this question:) )
Is there a reference which restates the standard results over $S$ for modules over $R$?
Most of the references I have found deal with the more(?) complicated case of polynomial coefficients. Other than that, I am aware of two references, both by Nobuki Takayama,
"Gröbner basis and the problem of contiguous relations" and a chapter in "Gröbner Bases: Statistics and Software Systems". The first basically introduces a version of Buchberger's algorithm for modules over $R$ and the second briefly goes over the very basic facts for ideals in $R$.

REPLY [2 votes]: Elaborating on Michael's comments and your concerns about Saito, Sturmfels, Takayama, "Grobner Deformations of Hypergeometric Differential Equations", the case this book addresses is the case of the Weyl algebra over a characteristic 0 field $k$. I.e., the noncommutative polynomial ring $$D=k[x_1,\ldots, x_n, \partial_1,\ldots, \partial_n]$$, subject to the commutator relations $[\partial_i, x_j]=\delta_{ij}$. There's also a few points at which they address the larger ring $$k(x_1,\ldots, x_n)\otimes_{k[x_1,\ldots, x_n]} D.$$ Is this what you're looking for?
As for online things, some of the above is also covered in Oaku and Takayama's "Algorithms for D-modules —restriction, tensor product, localization, and local cohomology groups".