A new route through stacks
A preprint presents arithmetic de Rham stacks as a stack-based way to organize rigid cohomology and its coefficient theories. The paper constructs the stacks, studies their formal properties, and gives categorical links to overconvergent crystals, overconvergent isocrystals and arithmetic D-modules. It also develops related arithmetic applications.
How the construction is set up
The main setting is a scheme X of finite type over a field k of characteristic p, with the additional conditions that it is quasi-compact, quasi-separated and separated. The coefficient field K is complete and discretely valued, with residue field k. The organizing method is transmutation: the paper changes the way a scheme is represented by evaluating it on a chosen class of test rings.
In the arithmetic de Rham construction, each test ring contributes a quotient built from its norm: elements of norm at most one are taken modulo elements of norm less than one. The scheme's points over that quotient form a pre-stack, which is then sheafified to produce the arithmetic de Rham stack.
From local descent to global operations
One structural result is h-hyperdescent for affine schemes of finite type over Fp. This is a form of descent: the construction respects the paper's specified hypercovers, so compatible local pieces can be assembled into the corresponding global object. The statement is confined to that affine, finite-type characteristic-p setting.
The paper also builds a six-functor formalism, a package of cohomological operations used to move information between spaces. It gives the stated behavior for étale, proper and smooth morphisms, along with strong invariance under the affine line and closed-open excision. For a smooth morphism of relative dimension n, one prim-dual statement shifts the unit object by negative twice n.
The main identifications
The central categorical results make three identifications. The relative arithmetic de Rham category is identified with solid overconvergent crystals. Its perfect complexes correspond to the bounded derived category of overconvergent isocrystals, and the cohomology of its structure sheaf matches rigid cohomology. These statements apply to the specified categories and complexes in the theorem.
For schemes that are realizable and equipped with a proper smooth frame, the relative category can also be described through left modules over overconvergent arithmetic differential operators on the scheme's overconvergent tube. That gives the paper's arithmetic D-module realization, but only in the framed setting named in the theorem.
One application concerns pushforward. For a proper smooth morphism between quasi-compact, quasi-separated, separated schemes of finite type over k, derived pushforward in each degree sends overconvergent isocrystals on the source to overconvergent isocrystals on the target. The result is a preservation theorem under those hypotheses, not a statement about arbitrary morphisms or coefficient categories.
A second stack construction
The paper also develops Hyodo-Kato stacks as a complementary construction. It starts with pre-stack data equipped with Frobenius and forms a Frobenius quotient. The resulting formalism is stated to have the corresponding étale, proper and smooth behavior, together with affine-line invariance and excision for separated finite-type schemes over Fp.
Within that Hyodo-Kato framework, the comparison theorem gives a fully faithful derived embedding and identifies the relevant categories of perfect complexes when X is finite type over a perfect field and the coefficient field is perfect with a Frobenius lift. The paper also states a Fourier-Mukai equivalence between the relevant arithmetic de Rham categories using the specified kernel.
Another stated consequence is finite-dimensionality: for a smooth variety with an overconvergent F-isocrystal coefficient, each rigid-cohomology group is finite-dimensional over K.
Results tied to specific assumptions
Taken together, these are conditional mathematical theorems, not general claims without hypotheses. Their stated scope includes characteristic-p and finite-type conditions, perfectness, Frobenius lifts, realizability, and proper or smooth morphisms. The supplied document is an arXiv v2 preprint dated 31 August 2026, presenting a framework of constructions, equivalences and preservation results whose validity is tied to those assumptions.
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