Classification of fully dualizable linear categories

Abstract

We prove that if $R$ is a G-ring then every fully dualizable $R$-linear cocomplete category is equivalent to a twist by a ${\mathbb{G}}_m$-gerbe of the category of modules over a finite étale $R$-algebra. We also show that this holds more generally over an arbitrary commutative ring under an additional compact generation hypothesis. We include variants of these results that apply to $R$-linear graded categories, and to the context of $\infty$-categories linear over connective commutative ring spectra.

The theorem on invertible stable $\infty$-categories is now proven over any truncated connective $E_{\infty }$-ring