Smith ideals of structured ring spectra

TL;DR

This paper develops a homotopy theory for Smith ideals of monoids in symmetric monoidal model categories, extending ideas of Jeff Smith to structured ring spectra and differential graded algebras.

Contribution

It introduces a new framework for Smith ideals that are not subobjects, linking them to monoid homomorphisms via a Quillen equivalence in the stable case.

Findings

Smith ideals form a homotopy theory in symmetric monoidal model categories

Quillen equivalence between Smith ideals and monoid homomorphisms in the stable case

Framework applies to structured ring spectra and differential graded algebras

Abstract

Pursuing ideas of Jeff Smith, we develop a homotopy theory of ideals of monoids in a symmetric monoidal model category. This includes Smith ideals of structured ring spectra and of differential graded algebras. Such Smith ideals are NOT subobjects, and as a result the theory seems to require us to consider all Smith ideals of all monoids simultaneously, rather then restricting to the Smith ideals of one particular monoid. However, we can take a quotient by a Smith ideal and get a monoid homomorphism. In the stable case, we show that this construction is part of a Quillen equivalence between a model category of Smith ideals and a model category of monoid homomorphisms.