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Grade Filtration of Linear Functional Systems

Grade Filtration of Linear Functional Systems 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Grade Filtration of Linear Functional Systems

Acta Applicandae Mathematicae , Volume 127 (1) – Dec 28, 2012

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References (69)

Publisher
Springer Journals
Copyright
Copyright © 2012 by Springer Science+Business Media Dordrecht
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Statistical Physics, Dynamical Systems and Complexity; Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-012-9791-2
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Dec 28, 2012

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