On a Deformation Theory of Finite Dimensional Modules Over Repetitive Algebras (2021)
Let Λ be a basic finite dimensional algebra over an algebraically closed field k, and let Λ be the repetitive algebra of Λ. In this article, we prove that if V is a left Λ-module with finite dimension b over k, then V has a well-defined versal deformation ring R(Λ, V), which is a local complete Noetherian commutative k-algebra whose residue field is also isomorphic to k. We also prove that R(Λb, Vb) is universal provided that EndΛ (V) = k and that in this situation, R(Λb, Vb) 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 1 k.