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

TL;DR

This paper develops a homotopy-theoretic completion theory for $ ext{E}_ olinebreak ext{infty}$-algebras within symmetric monoidal model categories, extending classical algebraic concepts to a homotopical setting.

Contribution

It introduces a naive homotopy-theoretic completion of $ ext{E}_ olinebreak ext{infty}$-rings based on Smith ideals, generalizing adic completion to a homotopical context.

Findings

Completion is homotopically complete for weakly compact Smith ideals.

Formulates an approximation of algebraic cobordism by algebraic K-theory.

Inheritance of properties like Bott periodicity and Gabber rigidity in the approximation.

Abstract

This article mentions that Smith ideal theory generalizes the adic completion theory of commutative rings to monoid objects of locally presentable symmetric monoidal abelian categories. As an application, we provide an almost mathematics version of completion theory, and we prove that the project limit of the residue algebras by powers of almost finitely generated ideals of almost algebras are complete as well-known results of commutative algebra theory. Furthermore, we introduce a naive construction of the completion theory of $\mathbb{E}_\infty$-rings, which is a homotopy theory analogue of the adic completion theory of commutative algebra by Hovey's Smith ideal theory. We prove that the completion is homotopically complete for any weakly compact Smith ideal as the ordinal commutative algebra theory. Furthermore, by using the completion theory of Smith ideals of the symmetric monoidal category of motivic spectra due to Voevodsky, we formulate an approximation of algebraic cobordism by algebraic $K$-theory, and we prove that the approximation inherits properties of algebraic $K$-theory, like the Bott periodicity and the Gabber rigidity.