Generators in formal deformations of categories

Abstract

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.