On a Deformation Theory of Finite Dimensional Modules Over Repetitive Algebras

Abstract

Let Λ be a basic finite dimensional algebra over an algebraically closed field \(\Bbbk \), and let \(\widehat {\Lambda }\) be the repetitive algebra of Λ. In this article, we prove that if \(\widehat {V}\) is a left \(\widehat {\Lambda }\)-module with finite dimension over \(\Bbbk \), then \(\widehat {V}\) has a well-defined versal deformation ring \(R(\widehat {\Lambda },\widehat {V})\), which is a local complete Noetherian commutative \(\Bbbk \)-algebra whose residue field is also isomorphic to \(\Bbbk \). We also prove that \(R(\widehat {\Lambda }, \widehat {V})\) is universal provided that \(\underline {\text {End}}_{\widehat {\Lambda }}(\widehat {V})=\Bbbk \) and that in this situation, \(R(\widehat {\Lambda }, \widehat {V})\) 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 \(\mathbb {P}^{1}_{\Bbbk }\).