MacRae invariant for \({\mathcal D}\)-modules (Q1891712)

A module \(M\) over the ring \(R\) is called `elementary' when it has a presentation of type \(0 \to R^p \to R^p \to M \to 0\); in this case its MacRae invariant \(G(M)\) is just the ideal generated by the determinant of the previous map. More generally, if \(M\) has a resolution \(0 \to L_i \to M \to 0\), with each \(L_i\) elementary, then one defines its MacRae invariant as the submodule generated by \(\prod [G(L_i)]^{(-1)^i}\) in the total quotient ring of \(R\). This definition can be extended to sheaves of modules. In this paper, the author shows that, when \(X\) is an analytic variety and \({\mathcal M}\) is a module over the sheaf \({\mathcal D}_X\) of differential operators on \(X\), then the MacRae invariant of the associated graded modules \(gr_\Phi {\mathcal M}\) does not depend on the choice of the good filtration \(\Phi\), but only on \({\mathcal M}\) itself. The author points out the connection between the MacRae invariant of \({\mathcal M}\) and its holonomy deficiency and studies the behaviour of this invariant under direct and inverse images of sheaves.