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EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS

Francisco Jesús Castro Jiménez, Michel Granger

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Abstract

We use the notion of a standard basis to study algebras of linear differential \noperators and finite type modules over these algebras. We consider the \npolynomial and the holomorphic cases as well as the formal case. Our aim is to demonstrate how to calculate classical invariants of germs of coherent \n(left) modules over the sheaf D of linear differential operators over Cn. The main invariants we deal with are: the characteristic variety, its dimension and the multiplicity of this variety at a point of the cotangent space. In the final chapter we shall study more refined invariants of D-modules linked to the \nquestion of irregularity: The slopes of a D-module along a smooth hypersurface of the base space.

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What this paper is about

We use the notion of a standard basis to study algebras of linear differential \noperators and finite type modules over these algebras. We consider the \npolynomial and the holomorphic cases as well as the formal case. Our aim is to demonstrate how to calculate classical invariants of germs of coherent \n(left) modules over the sheaf D of linear differential operators over Cn. The main invariants we deal with are: the characteristic variety, its dimension and the multiplicity of this variety at a point of the cotangent space. In the final chapter we shall study more refined invariants of D-modules linked to the \nquestion of irregularity: The slopes of a D-module along a smooth hypersurface of the base space.

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Available abstract

We use the notion of a standard basis to study algebras of linear differential \noperators and finite type modules over these algebras. We consider the \npolynomial and the holomorphic cases as well as the formal case. Our aim is to demonstrate how to calculate classical invariants of germs of coherent \n(left) modules over the sheaf D of linear differential operators over Cn. The main invariants we deal with are: the characteristic variety, its dimension and the multiplicity of this variety at a point of the cotangent space. In the final chapter we shall study more refined invariants of D-modules linked to the \nquestion of irregularity: The slopes of a D-module along a smooth hypersurface of the base space.

Key concepts: Mathematics, Variety (cybernetics), Hypersurface, Holomorphic function, Pure mathematics, Dimension (graph theory), Functor, Differential operator

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