A Nilpotence Theorem for Rational Rigid 2-Rings of Moderate Growth

Abstract

In this short note, we prove a general nilpotence theorem for a rational rigid 2-ring all of whose objects satisfy a certain ``moderate growth condition'' inspired from the theory of tensor categories. This applies in particular to the category of modules over a rational $E_{\infty}$-ring, to the derived category of any super-Tannakian category in characteristic zero, and conjecturally to Voevodsky's rational category of mixed motives over a field $DM_{\mathbb{Q}}$. In fact, we further prove that any such category has enough tt-fields, which can be chosen to be of the form Perf(L) for an even 2-periodic field L.