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A note on higher almost ring theory

Fabian Hebestreit, Peter Scholze

TL;DR

This note generalises higher almost ring theory by showing that derived localisations along idempotent ideals can be classified and realized without a flatness hypothesis. It proves a dual classification: for connective ${\mathbb E_k}$-rings, the derived localisation data correspond to idempotent kernels $\ker(\pi_0\varphi)$ via the Amitsur complex, and extends the construction to animated commutative rings, yielding $R/I^\infty$ as a universal derived localisation with a recollement of module categories by $I$-almost modules. The results unify derived localisation in the ${\mathbb E_k}$-setting with Smith-ideals, provide explicit constructions and stability statements, and give concrete examples (including Frobenius-related phenomena) to illustrate when $R/I^\infty$ remains static or becomes more intricate. These insights broaden the applicability of almost ring theory, link to algebraic K-theory via Efimov’s fibre sequence, and clarify how smashing spectra behave under derived localisations.

Abstract

We explain a derived version of the basic construction of localisations of module categories by means of idempotent ideals, which lie at the heart of Faltings' almost ring theory. We use it to provide an example of a commutative algebra in positive characteristic whose Frobenius endomorphism does not induce an isomorphism on its smashing spectrum.

A note on higher almost ring theory

TL;DR

This note generalises higher almost ring theory by showing that derived localisations along idempotent ideals can be classified and realized without a flatness hypothesis. It proves a dual classification: for connective -rings, the derived localisation data correspond to idempotent kernels via the Amitsur complex, and extends the construction to animated commutative rings, yielding as a universal derived localisation with a recollement of module categories by -almost modules. The results unify derived localisation in the -setting with Smith-ideals, provide explicit constructions and stability statements, and give concrete examples (including Frobenius-related phenomena) to illustrate when remains static or becomes more intricate. These insights broaden the applicability of almost ring theory, link to algebraic K-theory via Efimov’s fibre sequence, and clarify how smashing spectra behave under derived localisations.

Abstract

We explain a derived version of the basic construction of localisations of module categories by means of idempotent ideals, which lie at the heart of Faltings' almost ring theory. We use it to provide an example of a commutative algebra in positive characteristic whose Frobenius endomorphism does not induce an isomorphism on its smashing spectrum.
Paper Structure (4 sections, 2 theorems, 27 equations)

This paper contains 4 sections, 2 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Smith ideals
  3. Proof of the main results
  4. Examples and Remarks

Key Result

Proposition 2.3

Let $(\mathcal{C},\otimes)$ be an $\mathbb E_k$-monoidal category. Assume that $\mathcal{C}$ admits all finite limits and colimits, $\mathcal{C}$ is pointed (i.e. the initial and final object agree), and that $\otimes$ commutes with finite colimits in each variable. For every $\mathbb E_k$-algebra $ If $\mathcal{C}$ is stable these are equivalences.

Theorems & Definitions (9)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Example 2.4
  • proof : Proof of Theorem B
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem A