Injectivity failure in crystalline comparisons

TL;DR

The paper reveals a fundamental slope obstruction to the injectivity of the de Rham–crystalline comparison for smooth affine varieties in positive characteristic, providing explicit counterexamples that show injectivity can fail in general. It develops a robust framework using fractional Nygaard filtrations, fractional Tate twists, and overconvergent F-isocrystals to capture non-integral slopes and to relate rigid, convergent, and Monsky–Washnitzer cohomologies. On the positive side, the authors establish injectivity for slope-filtered subspaces and in cohomological degree one, and they prove extensive separation results for the affinoid topology, along with a new integral comparison for MW cohomology and a conceptual explanation for non-finite generation phenomena. The work also extends the theory to arithmetic D-modules, proving full faithfulness-type results that connect $p$-adic Hodge-theoretic phenomena with noncommutative geometry, and it provides a coherent, slope-driven picture of how de Rham, crystalline, rigid, and MW cohomologies interrelate in the presence of good compactifications. Overall, these results sharpen our understanding of when p-adic cohomology theories align and when slope data obstructs alignment, highlighting the indispensable role of rigid cohomology in motivic contexts.

Abstract

For smooth affine varieties in positive characteristic, we identify a slope obstruction to the injectivity of the comparison morphism from rigid cohomology to rationalised crystalline cohomology. This yields a negative answer to a question of Esnault--Kisin--Petrov concerning the injectivity of the de Rham-to-crystalline comparison map for smooth affine schemes over the Witt vectors that admit good compactifications. In contrast, we establish injectivity for certain subspaces defined by slope conditions as well as in cohomological degree one. For the latter case, we also prove the result with coefficients in $F$-able overholonomic $D$-modules leveraging a generalisation of Kedlaya's full faithfulness theorem. Beyond injectivity, we obtain various separation results for the affinoid topology on rigid and convergent cohomology. These results allow us to determine integral algebraic de Rham cohomology modulo torsion and to provide a more conceptual explanation for Ertl and Shiho's construction of varieties for which integral Monsky--Washnitzer cohomology modulo torsion is not finitely generated. Along the way, we prove a new integral comparison theorem between Monsky--Washnitzer cohomology and algebraic de Rham cohomology and we define fractional $p$-adic Tate twists, computing non-integral slopes of crystalline cohomology.

Theorems & Definitions (198)

• Theorem 1.1.2: Theorem \ref{['thm_counterexampleGHC']}
• Theorem 1.1.3: Theorem \ref{['thm_counterexample']}
• Proposition 1.1.4
• Theorem 1.1.5: Theorem \ref{['thm_InjectivitySlopei']} and Theorem \ref{['thm_InjectivitySlopeibis']}
• Theorem 1.1.6: Theorem \ref{['thm_RC']}
• Conjecture 1.3.1
• Theorem 1.3.2: Esn25
• Theorem 1.4.1: Theorem \ref{['thm_DeterminationAlgebraicdR']}
• Theorem 1.4.2: Theorem \ref{['thm_slopeiminus1-variant']} and Theorem \ref{['thm_sloperseparated']}
• Corollary 1.4.3: Corollary \ref{['cor_ES20']} and Corollary \ref{['cor_RigidSlopeRSeparated']}