Grzegorz Zwara

Degenerations of Finite-Dimensional Modules are Given by Extensions

AbstractLet A be a finite-dimensional k-algebra over algebraically closed field k and mod A be the category of finite-dimensional left A-modules. We show that a module M in mod A degenerates to another module N in mod A if and only if there is an exact sequence

Normality of orbit closures for Dynkin quivers¶of type ?n

0 \to N \to M \oplus Z \to Z \to 0

Degenerations for indecomposable modules and tame algebras

in mod A for some A-module Z. Moreover, we give a description of minimal degenerations of finite-dimensional A-modules.

STABLE EQUIVALENCE AND GENERIC MODULES

Let A be a representation-finite algebra. We show that a finite dimensional A-module M degenerates to another A-module N if and only if the inequalities dimK HomA (M, X) < dimK HomA (N, X) hold for all A-modules X. We prove also that if ExtA(X, X) 0 for any indecomposable A-module X, then any degeneration of A-modules is given by a chain of short exact sequences.

Degeneration-like orders on the additive categories of generalized standard Auslander-Reiten components

Abstract: It is known that the orbit closures for the representations of the equioriented Dynkin quivers ?n are normal and Cohen–Macaulay varieties with rational singularities. In the paper we prove the same for the Dynkin quivers ?n with arbitrary orientation.

Orbit closures and rank schemes

Abstract Let A be a finite dimensional associative algebra over an algebraically closed field such that there are, up to isomorphism, only finitely many indecomposable left A -modules. We show that the orbit closures in the associated module varieties are unibranch.

SINGULARITIES OF ORBIT CLOSURES IN MODULE VARIETIES AND CONES OVER RATIONAL NORMAL CURVES

Abstract Let A be a finite dimensional algebra over an algebraically closed field K. We investigate connection between the representation type of A and existence (and structure) of indecomposable A-modules N which are degenerations of other A-modules. We prove that if there is a common bound on the length of chains Mr

Singularities of orbit closures in module varieties

Let Λ and Γ be finite dimensional algebras. It is shown that any stable equivalence f [ratio ] mod Λ → mod Γ between the categories of finitely generated modules induces a bijection M [map ] M f between the sets of isomorphism classes of generic modules over Λ and Γ such that the endolength of M f is bounded by the endolength of M up to a scalar which depends only on f . Using Crawley-Boeveys characterization of tame representation type in terms of generic modules, one obtains as a consequence a new proof for the fact that a stable equivalence preserves tameness. This proof also shows that polynomial growth is preserved.