Generic Representation Theory of Quivers with Relations

Let Λ be a basic finite dimensional algebra over an algebraically closed field K. Tameness of the representation type of Λ – the only situation in which one can, at least in principle, meaningfully classify all finite dimensional representations of Λ – is a borderline phenomenon. However, even in the wild scenario, it is often possible to obtain a good grasp of the “bulk” of d-dimensional representations, for any dimension d, by understanding finitely many individual candidates. The underlying approach was initiated by Kac in 1982 for the hereditary case, refined by Schofield in 1992, and extended to arbitrary finitely generated K-algebras by Crawley-Boevey-Schröer in 2002 ([14], [15], [5]). The idea is to explore the generic behavior of the modules represented by the irreducible components of the affine variety, Modd(Λ), which parametrizes the d-dimensional left Λ-modules. Given an irreducible subvariety C of Modd(Λ), a certain property is said to be C-generic in case it is shared by all modules corresponding to the points in a dense open subset of C; as is common, we will, more briefly, speak of the modules in C, or in a certain subset of C. To date, generic numbers of indecomposable summands, combined with their generic dimension vectors, have been the main objects of study. Roughly, our goal is to expand this program by focusing attention on and broadly investigating the modules we call generic for C; these are modules in C that exhibit all categorically defined C-generic properties, i.e., all those generic properties which are invariant under self-equivalences of the category Λ-mod. (The modules we call generic for C actually satisfy a slightly stronger condition; see Definition 4.2. Existence and uniqueness, up to self-equivalences of Λ-mod, are guaranteed by Theorem 4.3; see also Theorem B below.) We start by establishing a general framework for this purpose, and then carry out the implied program for the core class of algebras we are targeting in this paper, the truncated path algebras. These are the algebras of the formKQ/I, where Q is an arbitrary quiver and I the ideal generated by all paths of a given fixed length. They include the hereditary algebras, but display a wealth of new phenomena. In particular, the subvarieties of Modd(Λ) corresponding to the modules with fixed dimension vector are no longer irreducible in general, but may have arbitrarily many irreducible components. Moreover, their generic homological properties range over a wide spectrum. We also point to the fact that the truncated path algebras play a prominent role relative to arbitrary basic finite