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

Maxime Ramzi, Maria Yakerson

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.

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

Abstract

We study -monoids on which a prime acts invertibly, which we call -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 -perfection functor, and describe it in terms of Quillen's -construction, similarly to group-completion. This gives an alternative description of the -inverted higher algebraic -theory of a ring.
Paper Structure (5 sections, 28 theorems, 32 equations)

This paper contains 5 sections, 28 theorems, 32 equations.

Table of Contents

  1. Homotopy groups of $p$-perfect monoids
  2. Comparison of $p$-perfection and group completion
  3. $p$-perfection via $+$-construction
  4. Comparison with an alternative construction
  5. The bialgebra Frobenius and colimit perfections

Key Result

Theorem 1

Let $M$ be a $p$-perfect commutative monoid. Then:

Theorems & Definitions (69)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark
  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Proposition 1.5
  • ...and 59 more