Matrix methods for projective modules over σ-PBW extensions (2015)
In this monograph, we study finitely generated projective modules defined on a certain type of noncommutative rings, called σ−P BW extensions, also known as skew P BW extensions. This class of noncommutative rings of polynomial type include many important examples of algebras and rings of recent interest as Weyl algebras, enveloping algebras of Lie algebras of finite dimension, diffusion algebras, quantum algebras, quadratic algebras in three variables, among many others. The study of projective modules was developed from a constructive matrix approach that will allow us to make effective calculations using a powerful computational tool: noncommutative Gröbner bases. Specifically, we establish an equivalent constructive matrix interpretation for the notions of being a projective, stably free or free module. Because of the close relationship between these three kinds of modules, we investigate when a given finitely generated module belongs to one of these classes. In this regard, Stafford showed that any stably free module on the Weyl algebra D = An(k) or Bn(k), with rank ≥ 2, turns out to be free; in this direction, we present a constructive proof of such important theorem for arbitrary rings which satisfy the condition range. On the other hand, we present several matrix descriptions of Hermite rings, various - characterizations of PF rings, and some subclasses of Hermite rings. However, since there is a variety of noncommutative rings that have nontrivial stably free modules, we use the Stafford’s theorem, the stable range of a ring, and existing bounds for Krull dimension of a skew P BW extension, in order to set a value from which all stably free module are free. In the second part of this thesis, we develop the theory of Gröbner bases for arbitrary bijective skew P BW extensions. Specifically, we extend Gröbner theory of quasi-commutative bijective skew extensions to arbitrary bijective skew P BW extensions. We construct Buchberger’s algorithm for left (right) ideals and modules over these noncommutative rings, and we present elementary applications of this theory as the membership problem, calculation of the syzygy module, intersection of ideals and modules, the quotient ideal, presentation of a module, calculation of free resolutions and the kernel and image of a homomorphism. Finally, we use the constructive proofs established in the early chapters, in order to develop effective algorithms to compute the projective dimension of a given module, algorithms for testing stably-freeness, procedures for computing minimal presentations and bases for free modules.