these papers, with simpler and better proofs superseding more complicated ones. The purpose of this expository paper is to assemble these preliminary results and proofs in one place, giving only the shortest argument in each case.

Ideally, this paper would be entirely self-contained except for basic commutative algebra, but the amount of necessary background material on ${\mathcal{D}}$-modules is too large for this ideal to be reasonable. Our compromise is the following. Björk’s book [@bjork] is our basic reference for the theory of ${\mathcal{D}}$-modules (it is also the basic reference cited in van den Essen’s papers), and we freely quote without proof results appearing in this book. We will also appeal to Gabber’s deep result [@gabber Thm. I] on the involutivity of characteristic ideals without providing a proof. We will, however, give full proofs for all preliminary results taken from van den Essen’s papers. We stress that nothing in this paper is original, neither the results nor the proofs; our goal in writing it is merely to make available a complete proof of Theorem \[mainthm\] in one document.

In this paper, we only state and prove precisely what we need for Theorem \[mainthm\]. The papers of van den Essen cited here 