The sheaf DX and its modules
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This chapter is devoted to the construction and basic properties of rings of differential operators. The ring D n of differential operators on the local ring of convergent power series is constructed and its ring-theoretic properties are analyzed in the first section. The sheaf D X on a complex analytic manifold is studied in the subsequent sections. We establish its coherence and construct good filtrations on every coherent D X -module M. This enables us to associate a conic analytic set in the holomorphic cotangent bundle, called the characteristic variety and denoted by SS(M). Counting multiplicities there exists the characteristic cycle of M.
An important result is that SS(M) is involutive in the symplectic manifold T*(X). We prove this together with certain homological results with the aid of material in the appendix. Special topics include Spencer’s resolution of coherent D X -modules with good filtrations and some non-trivial facts about the left D X -module O X . The notable fact is that every left D X -module whose underlying O X -module is coherent is locally isomorphic to a finite direct sum of O X in the category of left D X -modules. The proof relies on the existence of solutions to Pfaffian equations. A section about twisted rings of differential operators is included at the end of this chapter.
The material in this chapter is of course basic and will be used throughout the book.
KeywordsExact Sequence Irreducible Component Left Ideal Noetherian Ring Principal Symbol
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