Algebraic local cohomology with parameters and parametric standard bases for zero-dimensional ideals

A computation method of algebraic local cohomology classes, associated with zero-dimensional ideals with parameters, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the structure of algebraic local cohomology classes. This decomposition informs us on several properties of input ideals and the output of the proposed algorithm completely describes the multiplicity structure of input ideals. An algorithm for computing a parametric standard basis of a given zero-dimensional ideal, with respect to an arbitrary local term order, is also described as an application of the computation method. The algorithm can always output “reduced” standard basis of a given zero-dimensional ideal, even if the zero-dimensional ideal has parameters.