On generically tame algebras over perfect fields. (Q436180)

The classical definition of tameness of an algebra over an algebraically closed field \(K\) refers to parametrizations of indecomposable (finite length) modules by families of modules over the polynomial algebra \(K[t]\) or its localizations. Generic tameness is a concept generalizing tameness to arbitrary rings, [see \textit{W. Crawley-Boevey}, Proc. Lond. Math. Soc., III. Ser. 63, No. 2, 241-265 (1991; Zbl 0741.16005)], however, the question how to ``parameterize'' indecomposable finite length modules in this general setting, even for algebras over fields, remained open for a long time.NEWLINENEWLINE Let \(\Lambda\) be a generically tame finite-dimensional algebra over an infinite perfect field. The main result of the paper asserts that, given a natural number \(d\), the \(d\)-dimensional indecomposable \(\Lambda\)-modules can be parameterized by a finite number of families of finite dimensional modules over bounded principal ideal domains. There are also results on Auslander-Reiten sequences of modules over generalically tame algebras, generalizing \textit{W. W. Crawley-Boevey}'s well-known theorem [in Proc. Lond. Math. Soc., III. Ser. 56, No. 3, 451-483 (1988; Zbl 0661.16026)] that almost all indecomposables (of fixed dimension) are \(\tau\)-stable in tame case over an algebraically closed field.NEWLINENEWLINE The proof is based on the reduction techniques introduced first by Kiev school and rephrased in the language of ditalgebras, developed in the book [\textit{R. Bautista} et al., Differential tensor algebras and their module categories. Lond. Math. Soc. Lect. Note Ser. 362. Cambridge: Cambridge University Press (2009; Zbl 1266.16007)].