Operations on D-modules

Summary

In this chapter we study bounded complexes of D _X -modules and perform various operations. We prove that the homological dimension of the abelian category of left D _X -modules is equal to 2 · dim(X) + 1. for every complex manifold X.

We introduce the derived category D ^b (D _X ) whose objects are bounded complexes of left D _X -modules. Various operations from Chapter I are extended to derived categories in section 1 and 2.

The construction of direct and inverse images of complexes of D _X -modules is carried out in section 3. Temperate localisations along analytic sets give rise to functors on D ^b (D _X ) which are studied in section 5. The remaining sections are devoted to special situations. If Y ⊂ X is a closed analytic submanifold we establish an equivalence of categories between coherent D _Y -modules and the category of coherent D _X -modules supported by Y. Preservation of coherence and the behaviour of characteristic varieties under direct images is studied in section 7, where Spencer’s resolution applied to coherent D-modules with globally defined good filtrations plays an essential role.

Non-characteristic inverse images are studied in section 8. Here coherence is preserved and the characteristic variety of a non-characteristic inverse image determined. There is also a formula for the solution complex of the non-characteristic inverse image which is derived from the Cauchy-Kowalevski Theorem for a single differential operator with analytic coefficients. Some special constructions which lead to direct images in a more naive set-up as compared with the direct image functor expressed by derived functors occur in section 9.

Fuchsian filtrations are studied in section 10. They will be used later on to study regular holonomic modules. A duality functor on the derived category of coherent complexes of D-modules is constructed in section 11. We prove that this functor commutes with direct images of coherent D-modules equipped with globally defined good filtrations.