On the formula \(j(M)+d(M)=2n\) over the rings \(D_ n\) \((\widehat {D}_ n)\) and \({\mathcal E}_ P\) (Q1318866)

A filtered ring \(R\) is said to be Zariski if the associated graded ring \(\text{gr} (R)\) is Noetherian and if good filtrations on \(R\) modules induce good filtrations on submodules and are separated. Suppose further that \(\text{gr} (R)\) is commutative regular. Let \(M\) be a finitely generated \(R\) module with a good filtration, and consider the \(\text{gr} (R)\) module \(\text{gr} (M)\); both can be localized at a prime ideal \(P\). The author lets \(d_ P(M)\) (respectively \(e_ P(M)\)) denote the dimension (multiplicity) of \(\text{gr} (M)_ P\) over \(\text{gr} (R)_ P\). Then he defines \(d(M)\) to be the supremum of \(\{d_ P(M) | P\) a maximal ideal\} and \(e(M)\) to be the supremum over the \(e_ P(M)\) such that \(P\) is maximal and \(d(M) = d_ P(M)\). He also defines \(d'(M)\) and \(e'(M)\) by similar formulas, except restricting the ideals \(P\) to be maximal homogeneous prime ideals. Finally, the grade number \(j_ R(M)\) is the inf such that \(\text{Ext}^ J_ R (M,R) \neq 0\). Under the assumptions above, the author shows that if all the maximal homogeneous prime ideals of \(\text{gr} (R)\) have identical height \(\omega'\) then \(j(M) + d'(M) = \omega'\). He applies this to the case where \(R=D_ n\) (the ring of differential operators on convergent complex power series in \(n\) variables), where \(R=\widehat{D}_ n\) (operators on formal power series) and \(R = {\mathcal E}_ p\) (the ring of germs of microlocal differential operators). In all three cases he shows that \(j(M) + d(M) = 2n\) and in the first two that \(d'(M) = d(M)\), from which it follows in the third case that \(d(M) = d'(M)+1\).