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Grade filtration of linear functional systems. (English) Zbl 1327.16037
Summary: The grade (purity) filtration of a finitely generated left module \(M\) over an Auslander regular ring \(D\) is a built-in classification of the elements of \(M\) in terms of their grades (or their (co)dimensions if \(D\) is also a Cohen-Macaulay ring). In this paper, we show how grade filtration can be explicitly characterized by means of elementary methods of homological algebra. Our approach avoids using sophisticated methods such as bidualizing complexes, spectral sequences, associated cohomology, or Spencer cohomology used in the literature of algebraic analysis. Efficient implementations dedicated to the computation of grade filtration can then be easily developed in the standard computer algebra systems. Moreover, this characterization of grade filtration is shown to induce a new presentation of the left \(D\)-module \(M\) which is defined by a block-triangular matrix formed by equidimensional diagonal blocks. The linear functional system associated with the left \(D\)-module \(M\) can then be integrated in cascade by successively solving inhomogeneous linear functional systems defined by equidimensional homogeneous linear systems of increasing dimension. This equivalent linear system generally simplifies the computation of closed-form solutions of the original linear system. In particular, many classes of underdetermined/overdetermined linear systems of partial differential equations can be explicitly integrated by the Maple package PurityFiltration and the GAP package homalg, but not by the standard PDE solvers of computer algebra systems such as Maple.

MSC:
16W70 Filtered associative rings; filtrational and graded techniques
16S32 Rings of differential operators (associative algebraic aspects)
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32C38 Sheaves of differential operators and their modules, \(D\)-modules
16Z05 Computational aspects of associative rings (general theory)
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
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