Generators in formal deformations of categories

In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a $k$-linear $\infty$-category for a field $k$. Our main result states that if $\mathcal{C}$ is a $k$-linear $\infty$-category which has a compact generator whose groups of self extensions vanish for sufficiently high positive degrees, then every formal deformation of $\mathcal{C}$ has zero curvature and moreover admits a compact generator.