Classifying representations by way of Grassmannians

摘要

Let Lambda be a finite-dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a. ne or coarse moduli space classifying, up to isomorphism, the representations of Lambda with fixed dimension d and fixed squarefree top T. Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations of.. In the case of existence of a moduli space - unexpectedly frequent in light of the stringency of fine classification - this space is always projective and, in fact, arises as a closed subvariety Grass(d)(T) of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety Grass(d)(T) is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of 'finite local representation type at a given simple T', the radical layering (J(l) M/J(l+1)M)(l >= 0) is shown to be a classifying invariant for the modules with top T. This relies on the following general fact obtained as a byproduct: proper degenerations of a local module M never have the same radical layering as M.