On the Formula j(M)+α(M)=2n Over the Rings D n ( D n ) and P*

Abstract

In the study of the holonomic modules over D n ( D n ) and ()p, it is claimed and used that gr ( D n )(gr( D n ) and gr (()p) are regular Noetherian rings with pure dimension 2n, where D n is the stalk of the sheaf of differential operators withholomorphic coefficients, and ()p is the stalk of the sheaf () of microlocal differential operators. This property is used to prove j(M)+d(M)=2n for any finitely generated modules over D n ( D n ) and ()p by using the generalized Roos Theorem. In [1], it was proved that gr( D n )(gr( D n )) and gr(()p) do not have pure dimension, so we cannot apply the generalized Roos Theorem directly. In this paper, we reestablish the formula j ( M )+ d ( M )=2 n for any finitely generated modules over D n ( D n ) and()p.