Obstructions to deformations of d.g. modules

Abstract:

Let $\mathbf {k}$ be a field and $n\geq 1$. There exist a differential graded $\mathbf {k}$-module $(V,d)$ and various approximations to a differential $d+td_{1}+t^{2}d_{2}+\cdots + t^{n}d_{n}$ on $V[[t]],$ one of which gives a non-trivial deformation, another is obstructed, and another is unobstructed at order $n$. The analogous problem in the category of $\mathbf {k}$-algebras in characteristic zero remains a long-standing open question.