The correct relatively stable category for idempotent modules

TL;DR

This paper investigates the conditions under which Rickard modules in relatively stable categories exhibit idempotent behavior, revealing limitations and proposing modifications to recover idempotency.

Contribution

It answers an open question about the idempotent nature of Rickard modules and introduces a modification to achieve idempotent behavior in these categories.

Findings

Rickard modules are generally not idempotent in relatively stable categories.

Localization with respect to a tensor ideal subcategory does not ensure idempotency.

A specific modification can restore the idempotent property.

Abstract

We answer a question posed by Carlson, Peng, and Wheeler, and demonstrate that in general Rickard modules in relatively stable categories are not idempotent modules even if one localizes with respect to a tensor ideal subcategory. We also show that there is a modification one can make so as to recover the idempotent behaviour.