Ogus-Vologodsky equivalence via stacks

Abstract: Using the relative de Rham stack for a family $X \to S$ in characteristic $p,$ we reprove the (local and global) Ogus-Vologodsky equivalence. Moreover, we observe that a lift of $S$ is not necessary. Instead, we use a lift of $X$ to the second Witt vectors of $S.$ The main ingredient is that, for a quasi-syntomic family $X/S,$ the relative de Rham stack admits a structure of a torsor over $X'$ which is the analogue of the Azumaya property of the algebra of differential operators. This can be applied to families of (reasonable) algebraic stacks, which gives rise to a logarithmic version of the Cartier equivalence. Along the way, we also obtain a decompleted version of the global Cartier equivalence.

• The paper presents a novel stack-theoretic construction of the de Rham space, replacing the classical Azumaya splitting with a geometric torsor structure.
• The paper shows that lifting the scheme X alone to the second Witt vectors is sufficient for the equivalence, significantly widening its applicability.
• The paper develops both global and local Fourier-Mukai-type equivalences and extends the framework to logarithmic and stack settings, enhancing nonabelian Hodge theory.

Ogus-Vologodsky Equivalence via Stacks

Introduction and Context

This paper reconsiders the Ogus-Vologodsky (OV) equivalence, providing a new stack-theoretic approach that circumvents some of the traditional lifting constraints. The OV equivalence, a pivotal result in positive characteristic nonabelian Hodge theory, constructs a categorical bridge between flat connections (modules with integrable connection) and Higgs modules in characteristic $p$, generalizing the complex-analytic nonabelian Hodge correspondence. Classically, the proof relies on the Azumaya property of the sheaf of differential operators and the ability to lift Frobenius morphisms or the base scheme. This work replaces the Azumaya splitting machinery with a geometric construction involving relative de Rham stacks, thereby facilitating a uniform extension to algebraic stacks and logarithmic geometries.

Main Contributions

Stack-Theoretic Construction of the De Rham Space

The central innovation is recasting the OV correspondence in terms of the relative de Rham stack $(X/S)^{dR}$. Given a family $X \to S$ over a perfect field of characteristic $p$, $(X/S)^{dR}$ is modeled as a geometric realization of a derived stack representing the relative de Rham complex. The morphism $\nu_{X/S}\colon (X/S)^{dR} \to X'$ (the Frobenius twist of $X$) is shown to be a torsor over $X'$, with the group stack corresponding to the shifted tangent bundle $T_{X'/S}^{\sharp}$.

Torsor Structure and the Azumaya Analogy

A crucial result is that for quasi-syntomic $X/S$, the relative de Rham stack $(X/S)^{dR}$0 carries a natural torsor structure over $(X/S)^{dR}$1, mirroring the Azumaya property of the sheaf of differential operators. This geometric torsor interpretation refines the classical viewpoint, allowing systematic descent arguments and extension to families and stacks.

Lifting Criteria and Replacement of the Base Lift

Classically, OV equivalence required lifting the base $(X/S)^{dR}$2 to $(X/S)^{dR}$3 or constructing a Frobenius lift. The paper shows that lifting $(X/S)^{dR}$4 alone (as opposed to $(X/S)^{dR}$5) to the second Witt vectors $(X/S)^{dR}$6 suffices to construct the desired equivalences—a strictly weaker condition. The torsor $(X/S)^{dR}$7 admits local sections without lifting $(X/S)^{dR}$8, vastly broadening the arsenal of applicable situations (beyond formal or infinitesimal liftings accessible by the original proof).

Global Fourier-Mukai-Type Equivalences

For families $(X/S)^{dR}$9 admitting a flat lift $X \to S$0, the paper identifies $X \to S$1, a "refined" or "pushed-out" de Rham stack, as a $X \to S$2-gerbe over $X \to S$3. The gerbe of splittings is explicitly identified with the gerbe of $X \to S$4-liftings. This leads to an explicit equivalence of abelian categories: $X \to S$5 where $X \to S$6 denotes the category of locally nilpotent modules over a completed PD-differential operator algebra, and $X \to S$7 the corresponding Higgs objects.

Local Equivalences and Strong Frobenius Lifts

On the local level, a flat lift of $X \to S$8 to $X \to S$9 together with a "strong Frobenius lift" (a precise technical strengthening ensuring functoriality) suffices to split the de Rham torsor. This yields

$p$0

reproducing and generalizing the original local OV equivalence, but without the need to lift the base $p$1.

Decompletion and Powers of Frobenius

A critical technical extension constructs a decompleted version of the equivalence, by taking successive pushouts along powers of Frobenius and considering torsors under group schemes of the form $p$2. In the presence of a lift of some power of $p$3, this leads to symmetric monoidal equivalences between categories of (possibly infinitely supported) modules and Higgs sheaves: $p$4

Logarithmic and Stack Generalizations

By allowing $p$5 and $p$6 to be stacks or log pairs, the de Rham stack formalism yields variants of the nonabelian Hodge correspondence in logarithmic geometry. For logarithmic structures induced by effective divisors $p$7, the categories of logarithmic connections with nilpotent $p$8-curvature are correlated with Higgs modules in $p$9-parabolic sheaves.

Functoriality and Equivariance

For smooth group actions $(X/S)^{dR}$0 on $(X/S)^{dR}$1, versions of the OV equivalence for $(X/S)^{dR}$2-equivariant connections and Higgs modules are recovered by considering the de Rham stacks $(X/S)^{dR}$3 and the relevant classifying stacks.

Technical Highlights

• Explicit stack-theoretic description of de Rham and PD-stack torsors, derived from the animated (derived) algebra formalism.
• Identification of splittings of the de Rham torsor with homotopy fixed points and strong Frobenius lifts.
• Analysis of the interplay between the stacky de Rham approach and classical Azumaya splitting: the stack construction is, in a precise sense, Koszul dual to the traditional differential operator approach.
• Extension and validation for algebraic stacks and, importantly, families rather than isolated schemes, which is facilitated by the greater flexibility of stack-theoretic descent and torsor formalism.

Implications and Future Directions

The stack-theoretic approach exposes the geometric underpinnings of the OV equivalence, bringing to light the fundamental torsor structures subtending the classical results. Removing the necessity of lifting the base renders the positive characteristic nonabelian Hodge correspondence more robust and applicable, especially in families and moduli problems involving stacks.

Practically, this makes the $(X/S)^{dR}$4-adic Simpson correspondence far more flexible for arithmetic, logarithmic, and equivariant variants. The explicit identification of the de Rham gerbe splitting obstructions with classical deformation-theoretic obstructions provides a foundation for further study of the deformation theory of D-modules in characteristic $(X/S)^{dR}$5, fine-tuning the control of moduli of flat bundles and Higgs sheaves in various stacky contexts.

Notably, the categorical and functorial extension given here sets the stage for prismatic and integral $(X/S)^{dR}$6-adic Hodge-theoretic analogues, as well as for applications to the arithmetic and geometric Langlands programs in characteristic $(X/S)^{dR}$7. The methods may also interact with prismatic cohomology and derived algebraic geometry to produce fully integral versions of the Hodge correspondence in positive characteristic.

Open Problems and Prospective Research

• Classification of lifts of Frobenius in various geometric and arithmetic contexts—especially in the field of stacks and non-reduced base schemes.
• Incorporation of prismatic (and crystalline) cohomology frameworks for further integral and characteristic-free generalizations.
• Investigation of the restriction to decompleted (finitely supported) objects and explicit links with Fourier-Mukai-type transforms and Geometric Langlands duality.

Conclusion

This work supplies a rigorous, flexible, and highly geometric reformulation of the Ogus-Vologodsky equivalence in positive characteristic. By recasting the theory in terms of derived algebraic stacks and relative de Rham torsors, the need for base lifting is obviated and the correspondence is extended to a wider class of algebraic objects, including stacks, logarithmic schemes, and families. The formalism introduced here (via explicit stacky torsors and animated rings) equips the field with powerful tools for both structural and computational advances in positive characteristic nonabelian Hodge theory and beyond.