On a deformation theory of finite dimensional modules over repetitive algebras

Abstract

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