Chapter 16 Computing Tor and Ext In this chapter we compute the Tor and Ext modules over skew PBW ex- tensions. By computing we mean to give presentations of Tor A r (M,N ), where M is a finitely generated centralizing subbimodule of A m , m ≥ 1, and N is a left A-submodule of A l , l ≥ 1. For Ext r A (M,N ), M is a left A-submodule of A m and N is a finitely generated centralizing subbimodule of A l . The technique we will use for computing the modules Tor and Ext is very simple: we compute presentations of submodules of free modules using syzygies and Gr¨obner bases as we saw in the previous chapters, and then, we compute free resolutions and the corresponding homology modules. As applications, we will test stably-freeness, reflexiveness, and we will compute the torsion, the dual and the grade of a given submodule of a free module. 16.1 Centralizing Bimodules For the computations of the present chapter we have to consider a special type of subbimodule and also the following notion. Definition 16.1.1. Let R be a ring. We will say that R is Gr¨obnersoluble (GS) if R is both LGS and RGS. From now on in this chapter we will assume that R is a GS ring and A = σ(R)〈x 1 ,...,x n 〉 is a bijective skew PBW extension of R. Definition 16.1.2 ([74]). Let M be an A-bimodule. The centralizer of M is defined by Cen A (M ) := {f ∈ M | f a = af, for every a ∈ A}. We say that M is centralizing if M is generated as a left A-module (and, equivalently, as a right A-module) by its centralizer. M is a finitely gen- erated centralizing A-bimodule if there exists a finite set of elements in Cen A (M ) that generates M . 317 © Springer Nature Switzerland AG 2020 https://doi.org/10.1007/978-3-030-53378-6_16 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28,