Search

In order to make constructive the theory of projective modules, stably free modules, Hermite rings and extended modules studied in the previous chapters, in the present and next chapters we will study the theory of Gobner bases of left (right) ideals and modules for bijective skew PBW extensions.

Skew PBW extensions include wellknown classes of Ore algebras, operator algebras and also a lot of quantum rings and algebras. The universal property that characterizes these objects and its existence will be studied.

As we saw in Section 8.7, and also at the beginning of Chapter 9, all rings and algebras presented in Chapter 2 are PSF.

In this chapter we will investigate the localization of skew PBW extensions with respect to different kinds of multiplicative subsets.

Serre’s Theorem for bijective skew PBW extensions (Theorem 8.2.3) says that if M is an f.g. projective module over a bijective skew PBW extension A of a left noetherian, left regular PSF ring R, then M is stably free.

In particular, we will investigate the regularity of these rings and Serre’s theorem on the stably freeness of every finitely generated projective module. Moreover, we will compute the Quillen’s K-groups for these extensions.

There are some classical and elementary applications of Grobner theory that we will study in this chapter. We will consider the membership problem, and we will compute the syzygy module, free resolutions of modules, the intersection and quotient of ideals and submodules, the matrix presentation of a finitely presented module, and the kernel and the image of homomorphism between modules.

In this chapter we will make some matrix computations over skew PBW extensions using Grobner bases.

In this chapter R is a commutative domain and A is an R-algebra. In some places R = K is a field, or R = Z, i.e., we will study also the Zariski problem for rings.

The results presented in this chapter are an easy generalization of those of the previous chapter, i.e., taking m = 1 we get the theory of Grobner bases for the left ideals of A.

Motivated by Theorem 18.5.13, Example 18.5.15 and Theorem 18.6.8 of the previous chapter, we will introduce semi-graded versions of Koszul and Artin{ Schelter regular algebras. We will classify many of the skew PBW extensions considered in the present monograph according to these two types of algebras.

In this first chapter of applications we study some algebraic and categorical topics of the noncommutative algebraic geometry of skew PBW extensions. For this task we will follow the approach presented by M. Artin and J.J. Zhang in [28], by A.B. Verevkin in [398], [399] and by D. Rogalski in [344].

Now we will compute the global, Krull, Goldie and Gelfand{Kirillov dimensions for bijective skew PBW extensions. All of these computations will be done also for the skew quantum polynomials.