On minimal disjoint degenerations of modules over tame path algebras

For representations of tame quivers the degenerations are controlled by the dimensions of various homomorphism spaces. Furthermore, there is no proper degeneration to an indecomposable. Therefore, up to common direct summands, any minimal degeneration from M to N is induced by a short exact sequence 0 → U → M → V → 0 with indecomposable ends that add up to N . We study these ’building blocs’ of degenerations and we prove that the codimensions are bounded by two. Therefore, a quiver is Dynkin resp. Euclidean resp. wild iff the codimension of the building blocs is one resp. bounded by two resp. unbounded. We explain also that for tame quivers the complete classification of all the building blocs is a finite problem that can be solved with the help of a computer. Introduction If an algebraic group acts on a variety, the orbits are partially ordered by inclusion of their closures. Note that there are at least two general methods to determine the orbit closures, namely one method based on Gröbner bases [12] and another one proposed recently by Popov [18]. But both methods are quite impractical in the special case we are interested in. This is the action of G = Gld by conjugation on the variety Mod d A(k) of ddimensional representations of an associative finitely generated algebra. The points of this variety are the A-module structures on k and the orbits are in bijection with the isomorphism classes of d-dimensional modules. We write M ≤deg N and call N a degeneration of M resp. M a deformation of N iff the orbit to N lies in the closure of the orbit to M . Despite a nice representation theoretic characterization obtained by Zwara in [23], building on earlier work of Riedtmann in [19], it is in general a hard problem to determine the degeneration order. However, for tame quivers, i.e. quivers whose underlying graph is a Dynkin or an extended Dynkin diagram, the degeneration order on the representations coincides by [6, 4] with the partial order M ≤ N defined by [M,X ] ≤ [N,X ] for all modules X . Here and later on we abbreviate dimHom(X,Y ) by [X,Y ] and dimExt(X,Y ) by [X,Y ]. Since ∗E-mail:[email protected] E-mail:[email protected]