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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 givepresentations of $Tor_r^{A}(M,N)$, where $M$ is a finitely generatedcentralizing subbimodule of $A^m$, $m\geq 1$, and $N$ is a left $A$-submoduleof $A^l$, $l\geq 1$. 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 subbimoduleof $A^l$. As application of these computations, we test stably-freeness,reflexiveness, and we will compute also the torsion, the dual and the grade ofa given submodule of $A^m$. Skew $PBW$ extensions include many importantclasses of non-commutative rings and algebras arising in quantum mechanics, forexample, 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.

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