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Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs

Bruno Kahn, Hiroyasu Miyazaki, Shuji Saito, Takao Yamazaki

Abstract

We develop a theory of modulus sheaves with transfers, which generalizes Voevodsky's theory of sheaves with transfers. This paper and its sequel are foundational for the theory of motives with modulus, which is developed in [KMSY20].

Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs

Abstract

We develop a theory of modulus sheaves with transfers, which generalizes Voevodsky's theory of sheaves with transfers. This paper and its sequel are foundational for the theory of motives with modulus, which is developed in [KMSY20].

Paper Structure

This paper contains 45 sections, 92 theorems, 119 equations.

Table of Contents

  1. Modulus pairs and admissible correspondences
  2. Admissible correspondences
  3. Composition
  4. Categories of modulus pairs
  5. The functors $(-)^{(n)}$
  6. Changes of categories
  7. The closure of a finite correspondence
  8. Proof of Theorem \ref{['t2.1']}: case of $\omega$ and $\omega_s$
  9. Proof of Theorem \ref{['t2.1']}: case of $\tau$
  10. More on $\operatorname{\mathbf{\underline{M}Sm}}^{\operatorname{fin}}$ and $\operatorname{\mathbf{\underline{M}Cor}}^{\operatorname{fin}}$
  11. Fiber products and squarable morphisms
  12. Presheaf theory
  13. Modulus presheaves with transfers
  14. $\operatorname{\mathbf{MPST}}$ and ${\operatorname{\mathbf{PST}}}$
  15. $\operatorname{\mathbf{\underline{M}PST}}$ and ${\operatorname{\mathbf{PST}}}$
  16. ...and 30 more sections

Key Result

Theorem \oldthetheorem

The following assertions hold.

Theorems & Definitions (218)

  • Theorem \oldthetheorem: Voevodsky
  • Definition 1
  • Theorem \oldthetheorem
  • Remark 1
  • Definition 1.1.1
  • Lemma 1.1.3
  • proof
  • Lemma 1.2.1
  • Lemma 1.2.2
  • proof
  • ...and 208 more