An arXiv version-one preprint sets out a stack-based route to rigid cohomology and the coefficient theories that accompany it. Its central proposal is an arithmetic de Rham stack: a formal setting in which the paper studies rigid cohomology, overconvergent crystals, arithmetic D-modules and related functorial operations.
The paper’s ambition is structural rather than experimental. It aims to construct stacky approaches to rigid cohomology and its coefficients, establish abstract properties and derive arithmetic applications. The objects under discussion are mathematical: schemes and fields in characteristic p, Gelfand stacks, derived categories and cohomological constructions, rather than an empirical population.
A framework built around gluing
The starting assumptions are specific. The general setup permits a possibly non-perfect field of characteristic p, a complete discretely valued mixed-characteristic field with that residue field, and quasi-compact, quasi-separated, separated schemes of finite type. These conditions define the setting in which the later categorical and cohomological statements are made.
At the construction’s core, the absolute arithmetic de Rham stack is obtained by sheafifying a pre-stack in qfd Gelfand stacks evaluated on norm-bounded quotients. Put less formally, the paper starts with a preliminary stack-like assignment and imposes the sheaf behavior needed for the construction it studies.
The author presents this stacky approach as a way to avoid frame-choosing difficulties in rigid cohomology and to give the theory a frame-independent form. The stack is then used as a common setting for cohomology, coefficient categories and the operations that connect them.
The first tests are formal
One early test is descent, the rule that lets a construction be recovered from compatible local pieces. For an h-hypercover of affine finite-type schemes over Fp, the arithmetic de Rham construction sends the cover to a !-equivalence of the associated stacks.
The paper then tracks three classes of morphisms through its absolute six-functor formalism, the package of cohomological operations used to move between spaces. Étale, proper and smooth morphisms are sent respectively to cohomologically proper, cohomologically étale and prim morphisms with invertible prim dual.
It also states strong A1-invariance, the specified closed-open decomposition and a duality statement for smooth maps. For those smooth maps, the prim dual is shifted by minus twice the relative dimension, so the duality conclusion remains tied to the theorem’s smoothness and dimension hypotheses.
Connections to established theories
A central bridge links the abstract construction to established coefficient systems. For a scheme over k, the relative arithmetic de Rham category is stated to be symmetrically monoidally equivalent to solid overconvergent crystals, with a corresponding perfect-complex equivalence to derived overconvergent isocrystals.
That bridge is paired with a direct cohomological identification: the structure-sheaf cohomology of the relative arithmetic de Rham stack is identified with rigid cohomology. The claim places rigid cohomology inside the same formal setting as the coefficient categories, rather than treating it as a separate endpoint.
Another part of the framework is presented as arithmetic D-modules. For liftable proper smooth schemes, the paper says that pullback and extraordinary pushforward in the relative arithmetic de Rham category match the stated D-module functors.
A separate theorem concerns derived pushforward. If the morphism is proper and smooth and runs between quasi-compact, quasi-separated, separated schemes of finite type over k, each degree of derived pushforward sends an overconvergent isocrystal on the source to an overconvergent isocrystal on the target.
From equivalence to arithmetic consequences
The preprint also states a Fourier–Mukai equivalence for the arithmetic affine line over L. Under the coefficient-field and kernel conditions described in the paper, the specified Fourier–Mukai kernel produces an equivalence between the corresponding kernel categories.
A Hyodo–Kato comparison supplies another specialized link. Under perfectness and Frobenius-lift assumptions, the comparison diagram identifies the relevant perfect-complex categories of the arithmetic Frobenius quotient and the Hyodo–Kato stack.
The final results turn toward finiteness. For a smooth variety with an F-overconvergent isocrystal, rigid cohomology is finite-dimensional in each degree. The smooth-variety and F-overconvergent conditions are part of the stated scope of that result.
In the Laurent-series setting, the induced Hyodo–Kato pushforward yields a finite-dimensional Frobenius-equivariant differential module over a bounded Robba ring. Together with the finite-dimensionality statement, this describes how the formalism is intended to carry arithmetic information into a more structured coefficient object.
The boundaries are part of the result
The conditions are central rather than incidental. Depending on the theorem, the framework assumes characteristic p, perfectness, Frobenius lifts, liftability, properness or smoothness, along with the required finiteness and separation conditions on schemes. The categorical equivalences and preservation results should therefore be read within their specified mathematical settings.
The work is an arXiv version-one preprint, and the supplied analysis identifies no empirical sample, statistical analysis or experimental validation. It also notes that the supplied document is truncated, so later arguments and applications cannot be fully assessed from the reviewed material, while several arguments depend on cited foundational results.
Further assessment of the broadest applications would require inspection of the missing later sections and references, restoration of extraction-corrupted notation and independent checking of the foundational results on Gelfand stacks, analytic de Rham stacks and overconvergent sites. For now, the preprint’s contribution is a formal framework and a set of theorem-level connections, not a measured effect or population finding.
Paper data and sources
Original title: The arithmetic de Rham stack
Authors: Junhui Qin
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text