The inverse syzygy problem in algebraic systems theory

Abstract

We present a constructive solution of the inverse syzygy problem over arbitrary coherent rings and show how it can be used to compute certain extension groups. The inverse syzygy problemIn algebraic systems theory, linear systems are mathematically modelled by modules over a ring D (typically a ring of linear differential or difference operators) and control theoretic properties of a system are related to homological properties of the corresponding module. In this note, we are mainly concerned with controllability and show that for a very large class of rings it is equivalent to torsionlessness of the system module. The latter property in turn can be effectively verified via an inverse syzygy problem provided that over D it is possible to solve effectively the direct syzygy problem.Due to lack of space, we can only present some results; for proofs and more details we refer to [1]. From a very theoretical point of view, the inverse syzygy problem was already solved by Auslander and Bridger [2]. An effective solution was first presented by Oberst [3] who also noticed the connection to controllability. Like all subsequent works (see e. g. [4]), our results are based on Oberst's algorithm, but significantly enlarge the class of admissible rings D. This algorithm can be extended to a construction of certain extension groups of the system module. In [5][6][7], one can find discussions of various notions of controllability and their relation to the vanishing of extension groups.In the sequel, we will only assume that D is a coherent ring, i. e. that any finitely generated left or right ideal of D can also be finitely presented (in other words, its syzygy module is also finitely generated). In particular, we do not assume that D is some sort of (possibly non-commutative) polynomial ring and we explicitly allow that D may contain zero divisors: in contrast to previous works, we do not need the existence of a quotient field of D. For a D-module M, we denote by M * = Hom D (M, D) its dual module (note that M * is a right D-module for a left D-module M). All considered modules are assumed to be finitely generated.Let M, N be two free left D-modules and β : M → N a module homomorphism. The direct syzygy problem consists of finding a free left D-module P and a homomorphism α : P → M such that im (α) = ker (β), whereas in the inverse syzygy problem we search for a free left D-module Q and homomorphism γ : N → Q such that im (β) = ker (γ).Since we are dealing with homomorphisms between free modules, these can be represented by matrices with entries in D: we write β(P) = PB with a matrix B of appropriate dimensions and β * (Q * ) = BQ * for the dual map β * : N * → M * . Solving the direct syzygy problem for β corresponds to computing the left syzygies of the rows of B. For many classes of rings, effective algorithms are known for this task; in particular, Gröbner bases can be used for polynomial rings. Oberst's algorithmObviously, the inverse syzygy problem corresponds to the direct problem with the arrows reverted and a dualisa...