On a Deformation Theory of Finite Dimensional Modules Over Repetitive Algebras

Let be a basic finite dimensional algebra over an algebraically closed field k, and let (Lambda) double under bar be the repetitive algebra of Lambda. In this article, we prove that if (V) over cap is a left (Lambda) over cap -module with finite dimension over k, then (V) over cap has a well-defined versal deformation ring R((Lambda) over cap (V) over cap), which is a local complete Noetherian commutative k-algebra whose residue field is also isomorphic to k. We also prove that R((Lambda) over cap(V) over cap) is universal provided that End((Lambda) over cap)((V) over cap) = k and that in this situation, R((V) over cap) is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over P-k(1).