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Unstable $p$-completion in motivic homotopy theory

Klaus Mattis

Abstract

We define unstable $p$-completion in general $\infty$-topoi and the unstable motivic homotopy category, and prove that the $p$-completion of a nilpotent sheaf or motivic space can be computed on its Postnikov tower. We then show that the ($p$-completed) homotopy groups of the $p$-completion of a nilpotent motivic space $X$ fit into short exact sequences $0 \to \mathbb L_0 π_n(X) \to π_n^p(X_p^\wedge) \to \mathbb L_1 π_{n-1}(X) \to 0$, where the $\mathbb L_i$ are (versions of) the derived $p$-completion functors, analogous to the classical situation.

Unstable $p$-completion in motivic homotopy theory

Abstract

We define unstable -completion in general -topoi and the unstable motivic homotopy category, and prove that the -completion of a nilpotent sheaf or motivic space can be computed on its Postnikov tower. We then show that the (-completed) homotopy groups of the -completion of a nilpotent motivic space fit into short exact sequences , where the are (versions of) the derived -completion functors, analogous to the classical situation.
Paper Structure (27 sections, 184 theorems, 225 equations)

This paper contains 27 sections, 184 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Stable p-Completion
  3. Properties of the Stable p-Completion Functor
  4. The p-adic t-structure
  5. Comparison Results
  6. Unstable p-Completion in infinity-Topoi
  7. Definition of the p-Completion Functor
  8. Basic Properties of Unstable p-Completion
  9. Completions via Postnikov-towers
  10. Completions via Embeddings
  11. Completions of Presheaves
  12. Completions in the Nonabelian Derived Category
  13. Completions via Embeddings
  14. Comparison of the p-adic Hearts
  15. A Short Exact Sequence for Zariski Sheaves
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $X$ be a nilpotent pointed anima (resp. a spectrum). Then for every $n \ge 1$ (resp. any $n \in {\mathbb{Z}}$) there is a short exact sequence

Theorems & Definitions (439)

  • Theorem 1.1: Bousfield-Kan
  • Theorem 1.2
  • proof
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Proposition 1.5
  • proof
  • Theorem 1.6
  • proof
  • ...and 429 more