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Motivic homotopy theory with ramification filtrations

Junnosuke Koizumi, Hiroyasu Miyazaki, Shuji Saito

TL;DR

This work develops a modulus-enhanced motivic framework that captures ramification phenomena beyond the classical $\mathbb{A}^1$-invariant setting. By replacing $\mathbb{A}^1$ with the cube ${\overline{\square}}$ and enforcing cube- and SNC-blow-up invariances, the authors construct the stable category $\operatorname{mSH}(S,\Lambda)$ and show its compatibility with existing motivic theories, including log motives and Annala–Iwasa’s non-$\mathbb{A}^1$ framework. They prove representability of ramification-filtered cohomology theories, notably Hodge and Hodge–Witt with modulus, and establish a mechanism to extend reciprocity sheaves to ramification-filtered cohomology via a modulus functor $(-)^{\mathrm{mod}}$, linking ramification data with motivic representations. The results yield a robust toolkit for studying ramification in a motivic context, with applications to $p$-adic cohomology and irregular singularities, and connect ramification theory to logarithmic and reciprocity perspectives within active motives.

Abstract

The aim of this paper is to connect two important and apparently unrelated theories: motivic homotopy theory and ramification theory. We construct motivic homotopy categories over a qcqs base scheme $S$, in which cohomology theories with ramification filtrations are representable. Every such cohomology theory enjoys basic properties such as the Nisnevich descent, the cube-invariance, the blow-up invariance, the smooth blow-up excision, the Gysin sequence, the projective bundle formula and the Thom isomorphism. In case $S$ is the spectrum of a perfect field, the cohomology of every reciprocity sheaf is upgraded to a cohomology theory with a ramification filtration represented in our categories. We also address relations of our theory with other non-$\mathbb{A}^1$-invariant motivic homotopy theories such as the logarithmic motivic homotopy theory of Binda, Park, and Østvær and the theory of motivic spectra of Annala-Iwasa.

Motivic homotopy theory with ramification filtrations

TL;DR

This work develops a modulus-enhanced motivic framework that captures ramification phenomena beyond the classical -invariant setting. By replacing with the cube and enforcing cube- and SNC-blow-up invariances, the authors construct the stable category and show its compatibility with existing motivic theories, including log motives and Annala–Iwasa’s non- framework. They prove representability of ramification-filtered cohomology theories, notably Hodge and Hodge–Witt with modulus, and establish a mechanism to extend reciprocity sheaves to ramification-filtered cohomology via a modulus functor , linking ramification data with motivic representations. The results yield a robust toolkit for studying ramification in a motivic context, with applications to -adic cohomology and irregular singularities, and connect ramification theory to logarithmic and reciprocity perspectives within active motives.

Abstract

The aim of this paper is to connect two important and apparently unrelated theories: motivic homotopy theory and ramification theory. We construct motivic homotopy categories over a qcqs base scheme , in which cohomology theories with ramification filtrations are representable. Every such cohomology theory enjoys basic properties such as the Nisnevich descent, the cube-invariance, the blow-up invariance, the smooth blow-up excision, the Gysin sequence, the projective bundle formula and the Thom isomorphism. In case is the spectrum of a perfect field, the cohomology of every reciprocity sheaf is upgraded to a cohomology theory with a ramification filtration represented in our categories. We also address relations of our theory with other non--invariant motivic homotopy theories such as the logarithmic motivic homotopy theory of Binda, Park, and Østvær and the theory of motivic spectra of Annala-Iwasa.

Paper Structure

This paper contains 45 sections, 106 theorems, 250 equations, 1 table.

Table of Contents

  1. Overview of the construction
  2. Relation with other motivic homotopy categories
  3. Representability of cohomology theories with ramification filtrations
  4. Constructions and basic properties
  5. Construction of the motivic homotopy categories with modulus
  6. Relative $\mathbb{Q}$-SNCD
  7. log-smooth $\mathbb{Q}$-modulus pairs
  8. Nisnevich topology on $\operatorname{\mathrm{mSm}}_S$
  9. Construction of the motivic homotopy categories
  10. Comparison with Morel-Voevodsky's categories
  11. Properties of motivic homotopy categories with modulus
  12. Motive of $(\mathbb{P}^n,\mathbb{P}^{n-1})$
  13. Smooth blow-up excision
  14. Tame Hasse-Arf theorem
  15. Relation with logarithmic motivic stable homotopy theory
  16. ...and 30 more sections

Key Result

Theorem 1

Let $\mathcal{C}\in \{\operatorname{\mathrm{Sp}},\operatorname{\mathrm{Mod}}_\Lambda\}$In fact, we prove similar statements in the unstable category, allowing the case $\mathcal{C}=\operatorname{\mathrm{Spc}}_*$ too. be as above and $\mathop{\mathrm{M}}\nolimits$ be as eq;Motivicfunctor. Let $\mathc

Theorems & Definitions (290)

  • Theorem 1
  • Example 2
  • Example 3
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Definition 1.6
  • ...and 280 more