Equivariant Multiplicative Closure

TL;DR

This paper investigates the complexities of inverting elements in equivariant commutative rings, highlighting how their unique multiplicative structures affect localization and can lead to unexpected homotopy types.

Contribution

It introduces the concept of equivariant multiplicative closure and analyzes how localization impacts the homotopy type of equivariant commutative rings.

Findings

Localization can alter the homotopy type in equivariant settings

Indexed products require modified closure operations

Unexpected homotopy types can result from localization

Abstract

This paper describes an issue that arises when inverting elements of the homotopy groups of an equivariant commutative ring. Equivariant commutative rings possess an enhanced multiplicative structure arising from the presence of "indexed products" (products indexed by a set with a non-trivial action of the group). The formation of the "multiplicative closure" of a set must be altered in order to accomodate this structure, and the result of localizing an equivariant commutative ring can have an unexpected homotopy type.