Computing the ideal generated by Fox derivatives (Q1065145)

The Fox derivatives \(\partial v/\partial x_ i\) (1\(\leq i\leq n)\) of the word \(v\) in the group \(F\), free on the basis \(x_ 1,...,x_ n\), are defined by the equation \((v-1)=\sum^{n}_{i=1}(x_ i-1)\partial v/\partial x_ i\), in \(\mathbb ZF\). Let \(\mathcal V\) be a variety of groups defined by the set of identities \(V\subseteq F\). For a group \(B\), define \(\Delta'_ V\) to be the right ideal of \(\mathbb ZB\) generated by the elements \(\frac{\partial v}{\partial x_ j}(b_ 1,...,b_ n)\), where \(v\in V\) and \(b_ i\in B\), \(i,j=1,...,n\). The two-sided ideal \(\Delta'_ V\) is the annihilation of those right \(\mathbb ZB\)-modules \(A\) for which, if \(B\in {\mathcal V}\), then the semidirect product \(BA\in\mathcal V\); and the category of modules over \(\mathbb ZB/\Delta'_ V\) is that used in the construction of a homology theory for the variety \(\mathcal V\) [\textit{U. Stammbach}, Homology in group theory, Lect. Notes Math. 359. Berlin etc.: Springer (1973; Zbl 0272.20049)]. In this note \(\Delta'_ V\) is computed for an intersection, a commutator, and a product of varieties. \{There are some annoying misprints on the first side of the English translation: line 6: \(j\) should be \(i\); line 15: \(\Delta'_ H(V)\) should be \(\Delta'_ V(H)\); line 23: \(d_ j\) should be \(a_ j\), \(d\) should be \(a\), and \(d_ i\) should be \(a_ i\); line 26: ideal should be module.\}