Module varieties of finite dimensional algebras
Abstract: Given a finite-dimensional algebra A, the set of A-modules
of a fixed dimension d can be viewed as a variety. This variety
carries a group action whose orbits correspond to isomorphism classes
of A-modules. A natural problem is to characterize various properties
of an algebra A in terms of its module varieties.
For example, if A is assumed to have global dimension one, then it is
not difficult to show that A has finitely many indecomposable modules
(up to isomorphism) if and only if all of its module varieties have a
dense orbit, which is also if and only if all weight spaces of
semi-invariants in the coordinate rings of its module varieties have
dimension one. Our goal is to generalize these statements (with
modification) to higher global dimension. After explaining the
background, we present counterexamples to the naive generalizations,
along with plausible modifications and cases where these modifications
are correct. (Joint work with Calin Chindris, Piotr Dowbor, and Jerzy
Weyman)