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Berthelot's conjecture via homotopy theory

Veronika Ertl, Alberto Vezzani

TL;DR

This work presents a motivic strategy to prove Berthelot's conjecture on relative rigid cohomology, constructing a de Rham realization $\mathrm{dR}_{\mathrm{rig}}^{\varphi}$ that associates to dualizable motives over a base $S$ canonical overconvergent $F$-isocrystals. By combining the six-functor formalism for motives with the Monsky–Washnitzer and overconvergent de Rham realizations, the authors show that for a smooth proper morphism $f:X\to S$ the push-forwards $R^q f_{\mathrm{rig}*}(X/(T,\overline{T},\mathfrak{P}))$ arise as realizations of overconvergent $F$-isocrystals on $S$, hence are vector bundles with Frobenius. The approach leverages dualizability, spreading out, and analytic descent to obtain finiteness and Frobenius structures in a frame-independent, coefficient-inclusive setting, providing a robust framework that unifies Berthelot's theory with motivic methods and suggesting compatibility with $\mathcal{D}$-module-type formalisms in a future development. The results yield a stronger, conceptual pathway to relative $p$-adic cohomology with coefficients of motivic origin and illuminate the structure of Gauß–Manin connections in the overconvergent regime.

Abstract

We use motivic methods to give a quick proof of Berthelot's conjecture stating that the push-forward map in rigid cohomology of the structural sheaf along a smooth and proper map has a canonical structure of overconvergent F-isocrystal on the base.

Berthelot's conjecture via homotopy theory

TL;DR

This work presents a motivic strategy to prove Berthelot's conjecture on relative rigid cohomology, constructing a de Rham realization that associates to dualizable motives over a base canonical overconvergent -isocrystals. By combining the six-functor formalism for motives with the Monsky–Washnitzer and overconvergent de Rham realizations, the authors show that for a smooth proper morphism the push-forwards arise as realizations of overconvergent -isocrystals on , hence are vector bundles with Frobenius. The approach leverages dualizability, spreading out, and analytic descent to obtain finiteness and Frobenius structures in a frame-independent, coefficient-inclusive setting, providing a robust framework that unifies Berthelot's theory with motivic methods and suggesting compatibility with -module-type formalisms in a future development. The results yield a stronger, conceptual pathway to relative -adic cohomology with coefficients of motivic origin and illuminate the structure of Gauß–Manin connections in the overconvergent regime.

Abstract

We use motivic methods to give a quick proof of Berthelot's conjecture stating that the push-forward map in rigid cohomology of the structural sheaf along a smooth and proper map has a canonical structure of overconvergent F-isocrystal on the base.
Paper Structure (17 sections, 25 theorems, 88 equations)

This paper contains 17 sections, 25 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. The conjecture
  3. The main result
  4. Why motives?
  5. Organization of the paper
  6. The overconvergent de Rham realization
  7. Overconvergent quasi-coherent sheaves
  8. Overconvergent motives
  9. Overconvergent de Rham cohomology
  10. Relative overconvergent rigid cohomology
  11. An overconvergent Monsky--Washnitzer realization
  12. The overconvergent motivic realization
  13. Frobenius structures
  14. Berthelot's conjecture
  15. Compatibility of the rigid realization with push-forward maps
  16. ...and 2 more sections

Key Result

Theorem 1

Let $f:X\rightarrow S$ be a smooth and proper morphism of algebraic varieties over $k$. By letting $(T,\overline{T},\mathfrak{P})$ vary among proper frames over $S$, the modules arise from an overconvergent $F$-isocrystal over $S$. Moreover, for any $(T,\overline{T},\mathfrak{P})$ as above, they are vector bundles on $]T[^\dagger_{\mathfrak{P}}$.

Theorems & Definitions (111)

  • Theorem
  • Proposition
  • Proposition
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 101 more