Abstract
Due to the theoretical work and computer implementations of, for instance, Barakat, Quadrat and Robertz and their coauthors the theory of finitely generated (f.g.) modules over non-commutative regular noetherian rings of partial differential operators with variable coefficients like the Weyl algebras and over other similar rings has become constructive in recent years. In particular these authors compute the purity or grade filtration of a f.g. module by homological means and discuss its significance for the associated behavior. Pommaret and Quadrat noted this significance already in 1999. In this note it is shown that over an arbitrary commutative noetherian ring of operators the purity filtration of a finitely generated module can be easily computed by means of the the primary decomposition of its zero submodule and indeed, with smaller complexity, by inductively computing equidimensional parts. In most books on Constructive Commutative Algebra the connection between this primary decomposition, the equidimensional parts and the purity filtration is implicitly stated for cyclic modules. For many commutative rings of operators the standard signal modules are injective cogenerators. In this case the purity filtration of the module gives rise to a corresponding filtration of the dual behavior, and the primary decomposition induces additional sum representations of the pure dimensional factors of this filtration. For non-commutative rings of operators the standard signal modules are in general neither injective nor cogenerators, and for such signal modules the usefulness of the purity filtration of the module for the determination of the behavior and its structural properties is not obvious. It is also shown by a counter-example that dimensional purity of the module or behavior does not imply dimensional purity of the initial conditions according to Riquier of the associated homogeneous Cauchy problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barakat, M. (2009). Spectral filtrations via generalized morphisms. arXiv/org/abs/:0904.0240 (April 30, 2009).
Barakat, M. (2010). Purity filtration and the fine structure of autonomy. In Proceedings of the 19th international symposium on the mathematical theory of networks and systems (MTNS 2010) (pp. 1657–1661), Budapest.
Björk, J.-E. (1979). Rings of differential operators. Amsterdam: North-Holland.
Bourbaki, N. (1972). Commutative algebra. Paris: Hermann.
Cartan, H., & Eilenberg, S. (1956). Homological algebra. Princeton: Princeton University Press.
Eisenbud, D., Huneke, C., & Vasconcelos, W. (1992). Direct methods for primary decompostion. Inventiones Mathematicae, 110, 207–235.
Greuel, G.-M., & Pfister, G. (2002). A singular introduction to commutative algebra. Berlin: Springer.
Janet, M. (1924). Les modules des formes algébriques et la théorie générale des systèmes différentiels. Ann. Sci. École Norm. Sup., 41, 27–65.
Kreuzer, M., & Robbiano, L. (2005). Computational commutative algebra 2. Berlin: Springer.
Matsumura, H. (1986). Commutative ring theory. Cambridge: Cambridge University Press.
Oberst, U. (2010). The significance of Gabriel localization for stability and stabilization of multidimensional input/output behaviors. In Proceedings of the 19th international symposium on the mathematical theory of networks and systems (MTNS 2010) (pp. 1651–1655), Budapest.
Oberst, U., & Pauer, F. (2001). The constructive solution of linear systems of partial difference and differential equations with constant coefficients. Multidimensional Systems and Signal Processing, 12, 253–308.
Oberst, U., & Scheicher, M. (2012). Time-autonomy and time-controllability of discrete multidimensional behaviours. International Journal of Control, 85, 990–1009.
Plesken, W., & Robertz, D. (2005). Janet’s approach to presentations and resolutions for polynomials and linear pdes. Archiv der Mathematik, 84, 22–37.
Pommaret, J. F., & Quadrat, A. (1999). Algebraic analysis of linear multidimensional control systems. IMA Journal of Mathematical Control and Information, 16, 275–297.
Quadrat, A. (2013). Grade filtration of linear functional systems. Acta Applicandae Mathematicae, 1–62. doi:10.1007/s10440-012-9791-2.
Quadrat, A., Robertz, D., Baer’s extension problem for multidimensional linears systems. In Proceedings of the 18th international symposium on the mathematical theory of networks and systems (MTNS 2008) (pp. 1–12), Blacksburg (VA).
Riquier, C. (1910). Les systèmes d’Équations aux dérivées partielles. Paris: Gautiers-Villars.
Robertz, D. (2012). Formal algorithmic elimination for PDEs. RWTH Aachen: Habilitationsschrift.
Roos, J.-E. (1962). Bidualité et structure des foncteurs dérivés de \(\mathop {\text{ lim }}\limits _{\longleftarrow }\) dans la catégorie des modules sur un anneau régulier. C.R. Acad. Sci. Paris, 254, 1556–1558.
Scheicher, M., & Oberst, U. (2013). Multidimensional stability by Serre categories and the construction and parametrization of observers via Gabriel localizations. SIAM Journal on Control and Optimization, 51, 1873–1908.
Vasconcelos, W. (1998). Computational methods in commutative algebra and algebraic geometry. Berlin: Springer.
Wood, J., Oberst, U., Rogers, E., & Owens, D. H. (2000). A behavioral approach to the pole structure of one-dimensional and multidimensional linear systems. SIAM Journal on Control and Optimization, 38, 627–661.
Wood, J., Rogers, E., & Owens, D. H. (1998). A formal theory of matrix primeness. Mathematics of Control, Signals, and Systems, 11, 40–78.
Author information
Authors and Affiliations
Corresponding author
Additional information
A paper in FWF-project P 22535-N18.
Rights and permissions
About this article
Cite this article
Oberst, U. The computation of purity filtrations over commutative noetherian rings of operators and their application to behaviors. Multidim Syst Sign Process 26, 389–404 (2015). https://doi.org/10.1007/s11045-013-0253-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11045-013-0253-4