Minimal filtered free resolutions for analytic \(D\)-modules.
This paper discusses the definition of a minimal filtered free resolution for a filtered module over the ring \({\mathcal D}^{(h)}\), a homogenization of the ring \(\mathcal D\) of differential operators of analytic coefficients on \(\mathbb{C}^n\). Free resolutions are a basic tool for studying \(\mathcal D\)-modules. For example, in order to compute the restriction or the local cohomology of a \(\mathcal D\)-module we need a free resolution adapted to the \(V\)-filtration. However, such resolutions are not unique and it is difficult to define the notion of minimal filtered resolution of a \(\mathcal D\)-module directly. For this purpose, the homogenized ring of differential operators \({\mathcal D}^{(h)}\) is introduced in the first section. In the next section the authors consider filtered graded \({\mathcal D}^{(h)}\)-modules with respect to a general weight vector \((u,v)\in\mathbb{Z}^{2n}\) for the variables \(x\) and the corresponding derivations \(\partial\). In section 3, they show how to come back from \({\mathcal D}^{(h)}\)-modules to \(\mathcal D\)-modules and in the last section, they prove a generalization to submodules of \(({\mathcal D}^{(h)})^r\) of the division theorem for ideals in \({\mathcal D}^{(h)}\).
scientific article; zbMATH DE number 2093444