Purity filtration of 2-dimensional linear systems

Alban Quadrat
The purpose of this paper is to show that every linear partial differential (PD) system defined by means of a matrix with entries in the noncommutative polynomial ring D = A∂1,. .. , ∂n of PD operators in ∂1 = ∂ ∂x 1 ,. .. , ∂n = ∂ ∂xn with coefficients in a differential ring A, which satisfies certain regularity conditions, is equivalent to a linear PD system defined by an upper triangular matrix of PD operators formed by three diagonal blocks: the first (resp., second) diagonal block defines
more » ... onal block defines a dim(D)-dimensional (resp., dim(D) − 1-dimensional) linear PD system and the third one defines a linear PD of dimension less or equal to dim(D)−2. In particular, if n = 2, then the equivalent upper triangular matrix corresponds to the purity filtration of the finitely presented left D-module M associated with the linear PD system. Moreover, repeating the same techniques with the linear PD system of dimension less or equal to dim(D) − 2, the purity filtration of M can be obtained in the general case (i.e., n ≥ 2). Finally, this equivalent form of the linear PD system can be used to obtain a Monge parametrization and for closed-form integration of linear PD systems. I. ALGEBRAIC ANALYSIS In this section, we shortly recall a few results on the algebraic analysis approach to linear systems theory ([5]). Theorem 1 ([4], [9]): Let D be a ring, R ∈ D q×p a q×p-matrix with entries in D, M = D 1×p /(D 1×q R) the left D-module finitely presented by R, {f j } j=1,...,p the standard basis of D 1×p (i.e., f j is defined by 1 at the j th entries and 0 elsewhere), π : D 1×p −→ M the canonical projection onto M , y j = π(f j) for j = 1,. .. , p, and F a left D-module. Then, the abelian group isomorphism χ : hom D (M, F) −→ ker F (R.) = {η ∈ F p | R η = 0} φ −→ (φ(y 1). .. φ(y p)) T , (1) holds, where hom D (M, F) is the abelian group of left D-homomorphisms (i.e., left D-linear maps) from M to F. Theorem 1 shows that there is a one-to-one correspondence between the elements of hom D (M, F) and the elements of the linear system (or behaviour) ker F (R.). Hence, ker F (R.) can be studied by means of the left D-modules M and F. In this paper, we shall study algebraic properties of M and particularly its so-called purity filtration ([3]). Definition 1 ([9]): Let D be a left noetherian domain (namely, a ring without zero-divisors and for which every This paper is dedicated to Prof. Ulrich Oberst on the occasion of his 70th birthday. His work on mathematical systems theory has alway been a source of inspiration for us and, we are sure, for the next generations. left ideal is finitely generated as a left D-module) and M a finitely generated left D-module. Then, we have: 1) M is free if there exists r ∈ N = {0, 1,. . .} such that M ∼ = D 1×r. Then, r is called the rank of M. 2) M is projective if there exist r ∈ N and a left D-module N such that M ⊕N ∼ = D 1×r , where ⊕ denotes the direct sum of left D-modules. 3) M is reflexive if the canonical left D-homomorphism ε : M −→ hom D (hom D (M, D), D) defined by ε(m)(f) = f (m) for all f ∈ hom D (M, D) and all m ∈ M , is bijective, i.e., ε is a left D-isomorphism. 4) M is torsion-free if the torsion left D-submodule of M , namely, t(M) = {m ∈ M | ∃ d ∈ D\{0} : d m = 0}, is trivial, i.e., if t(M) = 0. The elements of t(M) are called the torsion elements of M. We have the following short exact sequence of left D-modules 0 −→ t(M) i −→ M ρ −→ M/t(M) −→ 0, (2) i.e., i is injective, ρ is surjective and ker ρ = im i. 5) M is torsion if t(M) = M , i.e., if every element of M is a torsion element of M. If D is a left noetherian ring and M a finitely generated left D-module, then M admits a finite free resolution. .. .R3 − − → D 1×r2 .R2 − − → D 1×r1 .R1 − − → D 1×r0 π −→ M −→ 0, (3) where R i ∈ D ri×ri−1 and .R i : D 1×ri −→ D 1×ri−1 is defined by (.R i)(λ) = λ R i for all λ ∈ D 1×ri ([9]).