the product of a unit and a “Weierstrass polynomial” in $x_n$.

[@cokernel p. 903]\[Mreg\] Let $M$ be a ${\mathcal{D}}$-module. We say that $M$ is *$x_n$-regular* if for every $m \in M$, there exists an $x_n$-regular $f \in R$ such that $E_{f \partial}(m)$ is a finitely generated $R$-module. (Any $m \in M$ for which this holds is said to be an *$x_n$-regular element*.)

\[holMreg\] In what follows, we will frequently consider ${\mathcal{D}}$-modules $M$ which are holonomic, hence cyclic, together with a choice of $m$ such that $M = {\mathcal{D}}\cdot m$. It is not hard to check that if $E_{\tau}(m)$ is a finitely generated $R$-module for some derivation $\tau \in {\mathcal{D}}$, then $E_{\tau}(\delta(m))$ is also finitely generated for every $\delta \in {\mathcal{D}}$. It follows that if $M = {\mathcal{D}}\cdot m$ is cyclic, then $M$ is $x_n$-regular as long as $E_{\tau}(m)$ is finitely generated over $R$, and this does not depend on the choice of ${\mathcal{D}}$-module generator $m$ for $M$.

Having now stated all necessary definitions, we begin working toward the crucial technical result (Lemma \[coklemma\]) on the cokernel of $\partial$ acting on a holonomic, $x_n$-regular ${\mathcal{D}}$-module. The following proposition can be viewed as a generalization of the Weierstrass preparation theorem (which we 