Algebraic Analysis of residue currents and an algorithm for computing Noether operators

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We have investigated Noether operators attached to a zero-dimensional primary ideal, the associated algebraic local cohomology and Grothendieck duality in the context of algebraic analysis. We have studies the structure of holonomic D-modules attached to a quasi-Homogeneous isolated singularity.1.A concept of Noether operators attaced to a zero-dimensional primary ideal is introduced. Their fundamental properties are clarified. An algorithm that compute Noether operator basis. are derived.2.An algorithm for computing holonomic D-module that leads zero-dimensional algebraic local cohomology class is constructed.3.Hermite-Jacobi reproducing kernel is investigated. A method for computing dual basis w.r.t. Grothendieck duality is constructed.4.A new method that compute Grothendieck local residue is derived.5.Semi quasi-homogeneous isolated singularities with inner modarity (at most) four are considered. Holonomic D-modules attached to these singularities are investigated. The factthat he multiplicity of such holonomic system is equal to the difference of Milnor number and Tjurina namber has proved by case by case computation.Besides, we have investigated Noether operator attaced to higher dimensional primary ideal and residue currents.