Classifying representations by way of Grassmannians

Abstract:

Let $\Lambda$ be a finite-dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine 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 $\Lambda$. 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 $\operatorname {\mathfrak {Grass}}^T_d$ of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety $\operatorname {\mathfrak {Grass}}^T_d$ 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 $\bigl ( J^{l}M/ J^{l+1}M \bigr )_{l \ge 0}$ 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 $M$.