The paper is devoted to the study of the theory of finite dimensional modules over a finitely generated associative algebra. In the first three sections basic definitions and recent results concerning degenerations of modules and related topics are discussed. Most of the material is contained in previous works of the author [see K. Bongartz, Adv. Math. 121, No. 2, 245-287 (1996; Zbl 0862.16007), Comment. Math. Helv. 69, No. 4, 575-611 (1994; Zbl 0832.16008)] and his followers [see e.g. G. Zwara, Colloq. Math. 72, No. 2, 281-303 (1997; Zbl 0890.16006)]. The next section is an account of well-known results from the theory of geometric quotients of varieties endowed with actions of algebraic groups. As an application in the final section the author considers in some detail the classical wild problem of pairs of \(n\times n\)-matrices. In particular, he classifies geometrically the orbits in the \(\text{PGL}_2\)-variety of pairs of \(2\times 2\)-matrices. In addition some pathologies occurring in the case of \(3\times 3\)-matrices are described. It should be noted that the paper under review contains many references (the list of references contains 67 citations), non-formal remarks and interesting examples.
For the entire collection see [Zbl 0897.00024].