Purity filtration of multidimensional linear systems

In this paper, we show how the purity filtration of a finitely presented module, associated with a multidimensional linear system, can be explicitly characterized by means of classical concepts of module theory and homological algebra. Our approach avoids the use of sophisticated homological algebra methods such as spectral sequences used in [3], [4], [5], associated cohomology used in [9], and Spencer cohomology used in [12], [13]. It allows us to develop efficient implementations in the PURITYFILTRATION and AbelianSystems packages. The purity filtration gives an intrinsic classification of the torsion elements of the module by means of their grades, and thus a classification of the autonomous elements of the multidimensional linear system by means of their codimensions. The results developed here are used in [16] to determine an equivalent block-triangular linear system of the multidimensional linear system formed by equidimensional diagonal blocks. This equivalent linear system highly simplifies the computation of a Monge parametrization of the original linear system.