From the introduction: By a basic observation of M. Auslander [in Contemp. Math. 13, 27-39 (1982; Zbl 0529.16020)] two finitely generated modules \(M,N\) over a finite dimensional associative \(k\)-algebra \(A\) are isomorphic if and only if the equality \[ [X,M]:=\dim\operatorname{Hom}_A(X,M)=[X,N] \] holds for all finite dimensional modules \(X\). Therefore, one can define a partial order \(\leq_{\hom}\) on the set of isomorphism classes of modules with the same dimension vector, i.e. the same Jordan-Hölder-multiplicities, by writing \(M\leq_{\hom}N\) if and only if the inequality \([X,M]\leq [X,N]\) holds for all modules \(X\) of finite dimension. Ch. Riedtmann has shown [in Ann. Sci. Éc. Norm. Supér. (4) 19, No. 2, 275-301 (1986; Zbl 0603.16025)] that this partial order is always weaker than the degeneration order. Fortunately, there are some interesting and non-trivial classes of algebras where the two partial orders coincide, e.g. representation-finite algebras and tame hereditary algebras. Then one has a very good explicit description of the degeneration-order in all cases where the dimensions of the homomorphism spaces are easy to calculate.
However, in some situations (e.g. for self-injective algebras) the dimensions of the stable homomorphism spaces are easier to calculate than those of the homomorphism spaces. Thus it is natural to introduce a third relation \(\leq_{\text{stab}}\) on the set of isomorphism classes of modules with the same dimension vector and to compare it with the other relations. Here we define \(M\leq_{\text{stab}}N\) if and only if \(\underline{[X,M]}\leq\underline{[X,N]}\) holds for all \(X\), where \(\underline{[X,M]}\) denotes the dimension of the quotient of \(\operatorname{Hom}_A(X,M)\) by the subspace of homomorphisms factoring through a projective module.
The study of this new relation has been initiated by M. Kettler [in “Degenerations and stable homomorphisms”, Diss. Univ. Bern (2005)]. There it is shown that \(\leq_{\hom}\) is always stronger than the relation \(\leq_{\text{stab}}\), which in general is not a partial order (so that the chosen notation is somewhat misleading). Furthermore, it is proven that both relations coincide for modules over hereditary algebras or over \(k[T]/(T^2)\), and some evidence is collected that these two cases are the only ones where the relations are the same.
This is indeed true as our main result asserts. Theorem 1. Let \(A\) be a connected associative algebra of finite dimension over an algebraically closed field \(k\). Then the relations \(\leq_{\hom}\) and \(\leq_{\text{stab}}\) coincide if and only if \(A\) is hereditary or isomorphic to \(k[T]/(T^2)\).