Some examples of nontrivial homotopy groups of modules (Q1599716)

In the 1950s, Hilton developed an analogue of homotopy theory that applied to modules rather than spaces. This theory extends some of the classical long exact sequences of homological algebra using derived functors in a slightly different way. For instance, given modules \(A, B\) over a ring \(\Lambda\), \(\text{Ext}^n_\Lambda(A,B)\) is the value of the \(n\)th right derived functor of \(\Hom_{\Lambda(A,-)}\) on \(B\). The \(n\)th projective homotopy group \(\underline{\pi}_n(A,B)\) is that of the \(n\)th left derived functor of \(\Hom_{\Lambda(A,-)}\) on \(B\). Similarly one has injective homotopy groups. These are dual but not isomorphic. Although they are closely linked with ideas in several areas of algebra, notably representation theory, these homotopy groups are relatively little studied and, for instance, examples of non-trivial homotopy groups, say, for modules over group rings are hard to find in the literature. In this paper a start on developing methods for calculating such groups is made. The arguments used are fairly `classical' in their methods, but it is interesting to see their limitations as whilst homotopy groups of modules over the cyclic group of order \(k\) can be calculated if the module structures are trivial, life is not so simple if they are not.