$p$-perfection and group completion of $\mathbb{E}_\infty$-monoids

TL;DR

This paper investigates $p$-perfect $ ext{E}_ ext{infinity}$-monoids, their relation to group completion, and describes the $p$-perfection functor using Quillen's $+$-construction, offering insights into higher algebraic $K$-theory.

Contribution

It introduces the concept of $p$-perfection for $ ext{E}_ ext{infinity}$-monoids and relates it to group completion and Quillen's $+$-construction, providing a new perspective on $p$-inverted algebraic $K$-theory.

Findings

$p$-perfect $ ext{E}_ ext{infinity}$-monoids nearly embed into their group completion.

The $p$-perfection functor can be described via Quillen's $+$-construction.

Provides an alternative description of $p$-inverted higher algebraic $K$-theory.

Abstract

We study $\mathbb{E}_\infty$-monoids on which a prime $p$ acts invertibly, which we call $p$-perfect, in the non-group-complete situation. In particular, we prove that in many examples, they almost embed in their group-completion. We further study the $p$-perfection functor, and describe it in terms of Quillen's $+$-construction, similarly to group-completion. This gives an alternative description of the $p$-inverted higher algebraic $K$-theory of a ring.