Classifying representations by way of Grassmannians

Abstract

Let Λ \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 Λ \Lambda with fixed dimension d d and fixed squarefree top T T . Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations of Λ \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 G r a s s d T \operatorname {\mathfrak {Grass}}^T_d of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety G r a s s d T \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 T ’, the radical layering ( J l M / J l + 1 M ) l ≥ 0 \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 T . This relies on the following general fact obtained as a byproduct: proper degenerations of a local module M M never have the same radical layering as M M .