Communicated by Kent R. Fuller Keywords: Representations of finite dimensional algebras Irreducible components of parameterizing varieties Generic properties of representations Path algebras modulo relationsThe irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra Λ are explored, with regard to both their geometry and the structure of the modules they encode. Provided that the base field is algebraically closed and has infinite transcendence degree over its prime field, we establish the existence and uniqueness (not up to isomorphism, but in a strong sense to be specified) of a generic module for any irreducible component C, that is, of a module which displays all categorically defined generic properties of the modules parametrized by C; the crucial point of the existence statementa priori almost obvious -lies in the description of such a module in a format accessible to representation-theoretic techniques. Our approach to generic modules over path algebras modulo relations, by way of minimal projective resolutions, is largely constructive. It is explicit for large classes of algebras of wild type. We follow with an investigation of the properties of such generic modules in terms of quiver and relations. The sharpest specific results on all fronts are obtained for truncated path algebras, that is, for path algebras of quivers modulo ideals generated by all paths of a fixed length; this class of algebras extends the comparatively thoroughly studied hereditary case, displaying many novel features.
We provide two alternate settings for a family of varieties modeling the uniserial representations with fixed sequence of composition factors over a finite dimensional algebra. The first is a quasi-projective subvariety of a Grassmannian containing the members of the mentioned family as a principal affine open cover; among other benefits, one derives invariance from this intrinsic description. The second viewpoint re-interprets the 'uniserial varieties' as locally closed subvarieties of the traditional module varieties; in particular, it exhibits closedness of the fibres of the canonical maps from the uniserial varieties to the uniserial representations.
Dedicated to the memory of Maurice Auslander whose questions triggered this workWe develop criteria for deciding the contravariant finiteness status of a subcategory A ⊆ Λ -mod, where Λ is a finite dimensional algebra. In particular, given a finite dimensional Λ-module X, we introduce a certain class of modules -we call them A-phantoms of X -which indicate whether or not X has a right A-approximation: We prove that X fails to have such an approximation if and only if X has infinite-dimensional Aphantoms. Moreover, we demonstrate that large phantoms encode a great deal of additional information about X and A and that they are highly accessible, due to the fact that the class of all A-phantoms of X is closed under subfactors and direct limits.
Abstract. Let Λ be a finite dimensional algebra over an algebraically closed field, and S a finite sequence of simple left Λ-modules. Quasiprojective subvarieties of Grassmannians, distinguished by accessible affine open covers, were introduced by the authors for use in classifying the uniserial representations of Λ having sequence S of consecutive composition factors. Our principal objectives here are threefold: One is to prove these varieties to be 'good approximations'-in a sense to be made precise-to geometric quotients of the (very large) classical affine varieties Mod-Uni(S) parametrizing the pertinent uniserial representations, modulo the usual conjugation action of the general linear group. We show that, to some extent, this fills the information gap left open by the frequent non-existence of such quotients. A second goal is that of facilitating the transfer of information among the 'host' varieties into which the considered quasi-projective, respectively affine, uniserial varieties are embedded. For that purpose, a general correspondence is established, between Grassmannian varieties of submodules of a projective module P on one hand, and classical varieties of factor modules of P on the other. Our findings are applied towards the third objective, concerning the existence of geometric quotients. The main results are then exploited in a representation-theoretic context: Among other consequences, they yield a geometric characterization of the algebras of finite uniserial type which supplements existing descriptions, but is cleaner and more readily checkable.
Abstract. The two main theorems proved here are as follows: If A is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of A is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization Comp A d of the family of finite A-module complexes with fixed sequence d of dimensions and an "almost projective" complex X ∈ Comp A d , there exists a canonical vector space embedding, where G is the pertinent product of general linear groups acting on Comp A d , tangent spaces at X are denoted by T X (−), and X is identified with its image in the derived category D b (A-Mod).
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