Computing finite presentations of Tor and Ext over skew PBW extensions and some applications

In this paper we compute the $Tor$ and $Ext$ modules over skew $PBW$ extensions. If $A$ is a bijective skew $PBW$ extension of a ring $R$, we give presentations of $Tor_r^{A}(M,N)$, where $M$ is a finitely generated centralizing subbimodule of $A^m$, $m\geq 1$, and $N$ is a left $A$-submodule of $A^l$, $l\geq 1$. In the case of $Ext_A^{r}(M,N)$, $M$ is a left $A$-submodule of $A^m$ and $N$ is a finitely generated centralizing subbimodule of $A^l$. As application of these computations, we test stably-freeness, reflexiveness, and we will compute also the torsion, the dual and the grade of a given submodule of $A^m$. Skew $PBW$ extensions include many important classes of non-commutative rings and algebras arising in quantum mechanics, for example, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others.