Our final chapter deals with a variety of themes at varying levels of detail. It provides a selection of results which, taken together, illustrate the diversity of the field and the broad range of techniques used.

In § 4.1–4.2 we continue to investigate the finite basis problem for identities of group representations. More specifically, we are concerned with questions of the following type: for a given multilinear identity, is a variety satisfying this identity finitely based?

It is well known that every identity of a linear algebra over a field of characteristic zero is equivalent to a system of multilinear identities, which can be effectively derived from the initial one. For group representations the corresponding statement is not true (otherwise, by Theorem 1.2.4, every variety would be homogeneous) and, in general, multilinear identities do not play here such a prevalent role. But if a variety does happen to be determined by multilinear identities, or at least satisfies some identity of this sort, then one can immediately derive significant consequences. For example, in many cases such a variety is finitely based.

The main result of §4.1 states that in certain respects the behavior of multilinear identities of group representations is closely related to the behavior of those of associative algebras. In particular, from this fact and a recent outstanding result of Kemer [40], we deduce that every system of multilinear identities of group representations over a field of characteristic zero is finitely based. In §4.2 we prove a rather old theorem of Cohen [10] which implies that every representation of an abelian group over a noetherian ring is finitely based. A far-going generalization of this theorem has been recently announced by Krasil'nikov [47].