Cellularization of structures in stable homotopy categories

TL;DR

This paper investigates how cellularization functors behave in stable homotopy categories, focusing on their effects on ring and module structures, and provides explicit calculations for Eilenberg-Mac Lane spectra.

Contribution

It characterizes the preservation properties of cellularization functors on rings and modules in stable homotopy categories, and computes cellularizations of Eilenberg-Mac Lane spectra.

Findings

Cellularization preserves modules over connective rings.

Cellularization does not generally preserve ring structures.

Explicit calculations of cellularizations of Eilenberg-Mac Lane spectra.

Abstract

We describe the formal properties of cellularization functors in triangulated categories and study the preservation of ring and module structures under these functors in stable homotopy categories in the sense of Hovey, Palmieri and Strickland, such as the homotopy category of spectra or the derived category of a commutative ring. We prove that cellularization functors preserve modules over connective rings but they do not preserve rings in general (even if the ring is connective or the cellularization functor is triangulated). As an application of these results, we describe the cellularizations of Eilenberg-Mac Lane spectra and compute all acyclizations in the sense of Bousfield of the integral Eilenberg-Mac Lane spectrum.