On lower central series quotients of finitely generated algebras over \(\mathbb Z\). (Q479768)

Let \(A\) be a unital associative algebra over a commutative ring \(R\). Let \(L_1(A)=A\) and \(L_k(A)=[A,L_{k-1}(A)]\) for \(k>1\) be the lower central series of \(A\) considered as a Lie algebra and let \(M_k(A)\) be the two-sided ideal of \(A\) generated by \(L_k(A)\). Starting with \textit{B. Feigin} and \textit{B. Shoikhet} [Math. Res. Lett. 14, No. 5-6, 781-795 (2007; Zbl 1174.17020)], in the recent decade, the quotients \(B_k(A)=L_k(A)/L_{k+1}(A)\) and \(N_k(A)=M_k(A)/M_{k+1}(A)\) have been studied for different \(A\), with more intensive study of \(B_k(A)\). In the paper under review the authors study \(B_k(A)\) and \(N_k(A)\) for the factor ring \(A=\mathbb Z\langle X_n\rangle/(f)\) of the free \(n\)-generated associative ring \(A=\mathbb Z\langle X_n\rangle\) modulo a single homogeneous relation \(f=0\). The authors are especially interested in the rank and the additive torsion of these quotients. They consider in detail the cases of the \(q\)-polynomial ring \(\mathbb Z\langle x,y\rangle/(yx-qxy)\), and the rings \(\mathbb Z\langle x,y\rangle/(x^m+y^m)\) and \(\mathbb Z\langle x,y\rangle/(x^m)\). It has turned out that there is a difference in the behaviour of the rank of \(B_k(A)\) for \(f=x^m+y^m\) and \(f=x^m\). The difference is encoded in the information about the locus of non-reduced points in the spectrum associated with the abelianization of \(A=\mathbb Q\langle x,y\rangle/(f)\). The authors establish a general result which guarantees the finite dimensionality of \(B_k(A)\) and \(N_k(A)\) for finitely generated graded \(\mathbb Q\)-algebras which generalizes a result of \textit{D. Jordan} and \textit{H. Orem} [Int. Math. Res. Not. 2015, No. 15, 6330-6352 (2015; Zbl 1334.16029)]. Finally, in the appendix the authors present an experimental verification of the observations of \textit{A. Krasilnikov} [in J. Algebra 392, 10-22 (2013; Zbl 1296.16019)] about torsion in \(N_k(\mathbb Z\langle X_n\rangle)\).