Nilpotent approximation and completion of $\mathbb{E}_\infty$-algebra objects of stable symmetric monoidal model categories

Abstract

We develop a nilpotent approximation theory for Smith ideals, extending adic completion for commutative rings to monoid objects in locally presentable symmetric monoidal abelian categories and to $\mathbb{E}_\infty$-algebra objects in stable symmetric monoidal model categories. The main result is a formal completeness theorem: finite generation of a Smith ideal forces completeness of its nilpotent approximation. This gives a categorical analogue of the finite generation completeness phenomenon in classical adic completion, while remaining distinct from ordinary adic completion of quotient rings. As applications, we construct an almost mathematics version of nilpotent approximation and prove a homotopical completeness theorem for weakly compact Smith ideals. We then apply the general theory to motivic spectra. For the canonical morphism from algebraic cobordism to algebraic K-theory, we construct the corresponding K-theoretic nilpotent approximation of algebraic cobordism, prove its homotopical completeness and Bott periodicity, and establish a mod-$\ell$ Gabber rigidity theorem for the analogous approximation of $\mathbf{MGL}/\ell$ by $\mathbb{K}/l$.