On a deformation theory of finite dimensional modules over repetitive algebras

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