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A motivic construction of the de Rham-Witt complex

Junnosuke Koizumi, Hiroyasu Miyazaki

Abstract

The theory of reciprocity sheaves due to Kahn-Saito-Yamazaki is a powerful framework to study invariants of smooth varieties via invariants of pairs $(X,D)$ of a variety $X$ and a divisor $D$. We develop a generalization of this theory where $D$ can be a $\mathbb{Q}$-divisor. As an application, we provide a motivic construction of the de Rham-Witt complex, which is analogous to the motivic construction of the Milnor $K$-theory due to Suslin-Voevodsky.

A motivic construction of the de Rham-Witt complex

Abstract

The theory of reciprocity sheaves due to Kahn-Saito-Yamazaki is a powerful framework to study invariants of smooth varieties via invariants of pairs of a variety and a divisor . We develop a generalization of this theory where can be a -divisor. As an application, we provide a motivic construction of the de Rham-Witt complex, which is analogous to the motivic construction of the Milnor -theory due to Suslin-Voevodsky.
Paper Structure (22 sections, 59 theorems, 91 equations)

This paper contains 22 sections, 59 theorems, 91 equations.

Table of Contents

  1. Preliminaries
  2. Finite correspondences
  3. $\mathbb{Q}$-Cartier divisors
  4. $\mathbb{Q}$-modulus pairs
  5. $\mathbb{Q}$-Modulus pairs
  6. Cube invariance and reciprocity
  7. Cube invariance
  8. Reciprocity sheaves
  9. Modulus curves and motivic Hasse-Arf theorem
  10. Admissible rational functions
  11. Modulus curves
  12. Motivic Hasse-Arf theorem
  13. Motivic construction of the ring of Witt vectors
  14. Usual construction
  15. Motivic construction
  16. ...and 7 more sections

Key Result

Theorem 1

There is an isomorphism $h_0\mathbb{W}_n^+\xrightarrow{\sim}\mathbb{W}_n$ in $\mathop{\mathrm{PSh}}\nolimits(\operatorname{\mathrm{Cor}}_k^\mathrm{aff})$ which preserves the multiplication, Frobenius, and the Verschiebung.

Theorems & Definitions (136)

  • Theorem 1: Theorem \ref{['thm:ab-comparison']}, Proposition \ref{['prop:ring-comparison']}
  • Theorem 2: Corollary \ref{['BCKSgen']}
  • Theorem 3: Corollary \ref{['cor:p_typical_comparison']}
  • Theorem 4: Theorem \ref{['thm:rep-WOmega']}
  • Definition 1.1
  • Lemma 1.2
  • proof
  • Corollary 1.3
  • Definition 1.4
  • Definition 2.1
  • ...and 126 more