Ogus’ Comparison Theorem

• Ogus’ Comparison Theorem is a collection of canonical isomorphisms that relate de Rham and crystalline realizations, nonabelian Hodge theory, and the coniveau spectral sequence.
• In the context of 1-motives, it connects the de Rham realization with the crystalline framework via Frobenius actions, weight filtrations, and logarithmic splittings.
• The theorem further extends to characteristic p by equating inverse Cartier transforms with de Rham–Higgs comparisons, underpinning full faithfulness and algebraicity in complex arithmetic settings.

“Ogus’ Comparison Theorem” denotes several comparison statements associated with Ogus, and the term is used in materially different settings. For 1-motives over a number field, it refers to canonical isomorphisms

$$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$

at almost all good finite places, transporting crystalline Frobenius to a $\sigma_v$-semilinear $F_v$ on the de Rham realization and producing the filtered Ogus object $T_{Og}(M)\in FOg(K)(1)$; the associated functor $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$ is fully faithful (Andreatta et al., 2016). In characteristic $p$, the Ogus–Vologodsky–Schepler comparison gives a canonical quasi-isomorphism

$$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$

for a nilpotent Higgs bundle and its inverse Cartier flat bundle (Zebao, 21 Jul 2025). In the Bloch–Ogus framework, Ogus’ comparison identifies the coniveau spectral sequence with the hypercohomology spectral sequence of the Gersten complex (Deshmukh et al., 2020).

1. Terminological scope

Within the literature represented here, the expression covers arithmetic realizations of 1-motives, characteristic-$p$ nonabelian Hodge theory, and coniveau–Gersten comparison for étale cohomology.

Setting	Comparison statement	Source
1-motives over a number field	$$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$	(Andreatta et al., 2016)
Nonabelian Hodge in characteristic $p$	$\sigma_v$0	(Zebao, 21 Jul 2025)
Bloch–Ogus framework	coniveau spectral sequence $\sigma_v$1 hypercohomology spectral sequence of the Gersten complex	(Deshmukh et al., 2020)

The common pattern is a comparison between structures defined a priori in different languages: de Rham and crystalline realizations for 1-motives, de Rham and Higgs complexes after Frobenius pushforward in characteristic $\sigma_v$2, and coniveau and Gersten constructions in étale cohomology. A plausible implication is that “comparison” is not a single theorem but a family of statements in which Frobenius, weights, filtrations, or residue maps convert one formalism into another.

2. Ogus realization for 1-motives over number fields

A 1-motive $\sigma_v$3 over a number field $\sigma_v$4 consists of a lattice $\sigma_v$5, a semi-abelian variety $\sigma_v$6, and a morphism of $\sigma_v$7-group schemes $\sigma_v$8. Deligne’s universal $\sigma_v$9-extension of $F_v$0 is

$F_v$1

where $F_v$2 is an extension

$F_v$3

by a vector group $F_v$4 canonically isomorphic to the sheaf of invariant differentials $F_v$5 of the Cartier dual semi-abelian $F_v$6 of the abelian quotient (Andreatta et al., 2016).

The de Rham realization is defined by

$F_v$7

a finite-dimensional $F_v$8-vector space. Under the identification $F_v$9, one obtains a natural decreasing Hodge filtration on $T_{Og}(M)\in FOg(K)(1)$0 characterized by

$T_{Og}(M)\in FOg(K)(1)$1

Equivalently, $T_{Og}(M)\in FOg(K)(1)$2 may be identified with the $T_{Og}(M)\in FOg(K)(1)$3-linear dual of $T_{Og}(M)\in FOg(K)(1)$4.

The target category is the enriched Ogus category $T_{Og}(M)\in FOg(K)(1)$5. Its underlying category $T_{Og}(M)\in FOg(K)(1)$6 consists of finite-dimensional $T_{Og}(M)\in FOg(K)(1)$7-vector spaces equipped, for almost all unramified finite places $T_{Og}(M)\in FOg(K)(1)$8 of $T_{Og}(M)\in FOg(K)(1)$9, with $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$0-adic completions $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$1, bijective $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$2-semilinear endomorphisms $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$3, and comparison isomorphisms $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$4. The enrichment is an increasing finite exhaustive weight filtration $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$5 such that $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$6 is pure of weight $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$7 for almost all $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$8. Morphisms in $T_{Og}: \mathcal M_{1,\mathbb Q}\to FOg(K)$9 respect $p$0 and are strict. There is also a twist $p$1 defined by multiplying Frobenius by $p$2 and shifting weights by $p$3.

For a 1-motive $p$4 with model over $p$5, one sets

$p$6

Using the comparison

$p$7

one defines $p$8 on the de Rham side by transporting $p$9, where $$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$0 is the crystalline Frobenius. The Ogus realization is then

$$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$1

The weight filtration has graded pieces

$$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$2

where $$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$3 and $$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$4. At good unramified $$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$5, $$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$6 acts as $$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$7 on the lattice part, as the usual crystalline Frobenius on the abelian part, and as $$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$8 on the toric part. The essential image lies in $$\tau_{<p-\ell}F_*\Omega^*(H,\nabla)\cong \tau_{<p-\ell}\Omega^*(E,\theta)$$9, the subcategory of e-effective objects of weights $p$0 with the stated Artin–Lefschetz and l-effectivity conditions.

3. Full faithfulness and the mixed structure

The main theorem states that

$p$1

is fully faithful (Andreatta et al., 2016). Faithfulness is reduced to the de Rham realization: if $p$2, then $p$3; over $p$4, the Hodge realization detects $p$5 up to isogeny, so an integral multiple of $p$6 is zero, hence $p$7 in $p$8.

The fullness argument separates the semi-abelian and lattice parts. Given

$p$9

in $$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$0, one applies the “Bost–Ogus” reduction functor

$$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$1

to obtain a morphism on semi-abelian parts. Here $$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$2 identifies with $$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$3 with its Frobenius structure. By Bost’s algebraicity theorem,

$$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$4

is fully faithful, so the $$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$5-morphism comes from a $$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$6-morphism $$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$7, unique up to isogeny. The weight-$$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$8 part is handled separately: fullness on pure $$T_{dR}(M/K)\otimes_K K_v \cong T_{cris}(M_{k(v)})\otimes_{W(k(v))}K_v$$9-motives yields a lattice morphism $p$0 up to isogeny.

The mixed compatibility condition

$p$1

is obtained through a Frobenius-equivariant logarithmic splitting. For each good $p$2, one constructs a canonical $p$3-semilinear section

$p$4

of the map induced by the universal extension $p$5. This section is the unique Frobenius-equivariant section in the category of $p$6–$p$7-isocrystals. Its uniqueness forces any $p$8-morphism to be compatible with the logarithmic splitting, which yields the required 1-motive compatibility up to a positive integer multiple. That proves fullness.

The mixed example $p$9, $\sigma_v$00, with $\sigma_v$01, $\sigma_v$02, makes the point sharply. Here $\sigma_v$03, with weights $\sigma_v$04 and $\sigma_v$05. At a good unramified $\sigma_v$06, the Frobenius-equivariant splitting is computed via the $\sigma_v$07-adic logarithm: $\sigma_v$08 An $\sigma_v$09 morphism that is the identity on the underlying $\sigma_v$10 exists only if

$\sigma_v$11

Thus not every filtration-preserving, graded Frobenius-compatible $\sigma_v$12-linear map arises from a 1-motive morphism; the Ogus structure detects the obstruction through the logarithm and Frobenius equivariance.

4. De Rham–Higgs comparison in characteristic $\sigma_v$13

For a perfect field $\sigma_v$14 of characteristic $\sigma_v$15, a smooth pair $\sigma_v$16 that is $\sigma_v$17-liftable, and a Higgs sheaf $\sigma_v$18 on $\sigma_v$19 that is nilpotent of level $\sigma_v$20, the inverse Cartier transform

$\sigma_v$21

is an equivalence of categories (Zebao, 21 Jul 2025). Writing $\sigma_v$22, the classical Ogus–Vologodsky–Schepler comparison takes the form

$\sigma_v$23

in $\sigma_v$24. This is naturally phrased after Frobenius pushforward, and the truncation $\sigma_v$25 is necessary in general.

The case $\sigma_v$26 and $\sigma_v$27 recovers the Deligne–Illusie decomposition

$\sigma_v$28

The later mixed-Hodge-module refinement extends the comparison to natural subcomplexes: full complexes and weight-$\sigma_v$29 subcomplexes, intersection subcomplexes, Kontsevich subcomplexes, and, for $\sigma_v$30, piecewise “patched” complexes near semistable fibers.

A major refinement is the twisted local comparison. For any $\sigma_v$31 and $\sigma_v$32, after completing at $\sigma_v$33,

$\sigma_v$34

This is obtained by altering $\sigma_v$35 by a formal sum $\sigma_v$36 using a variant of the $\sigma_v$37-transform and the Artin–Hasse exponential, so that the inverse Cartier transform matches $\sigma_v$38. A consequence is

$\sigma_v$39

for all $\sigma_v$40; if these supports are finite, one gets a global isomorphism in $\sigma_v$41 without completing.

The same paper also isolates two systematic improvements over the classical truncated comparison. If $\sigma_v$42 splits as $\sigma_v$43 with $\sigma_v$44, then

$\sigma_v$45

in $\sigma_v$46, so the truncation can be removed. Without any splitting, if $\sigma_v$47, then for all $\sigma_v$48,

$\sigma_v$49

The canonical quasi-isomorphisms are compatible with products, and under the stated hypotheses one obtains equality of twisted hypercohomology dimensions and $\sigma_v$50-degeneration statements for Fontaine–Faltings modules.

5. Local Cartier transform and level-raising generalizations

A complementary formulation treats Ogus’ comparison as a categorical equivalence between Higgs modules and modules with integrable connections, together with explicit local formulas derived from a Frobenius lift (Shiho, 2012). In the local Ogus–Vologodsky correspondence, quasi-nilpotent Higgs modules on $\sigma_v$51 are equivalent to modules on $\sigma_v$52 carrying integrable connections whose $\sigma_v$53-curvature is nilpotent.

Locally, if $\sigma_v$54 are coordinates on a lift $\sigma_v$55 and $\sigma_v$56 on $\sigma_v$57, and if a Frobenius lift satisfies

$\sigma_v$58

then

$\sigma_v$59

For a Higgs module $\sigma_v$60 with

$\sigma_v$61

the induced connection $\sigma_v$62 on $\sigma_v$63 is given by

$\sigma_v$64

Under this transform, the $\sigma_v$65-curvature corresponds to the Higgs field.

Shiho generalizes this construction by introducing $\sigma_v$66-connections,

$\sigma_v$67

and defining the level-raising inverse image

$\sigma_v$68

For fixed $\sigma_v$69, successive Frobenius lifts yield a composite functor

$\sigma_v$70

At level $\sigma_v$71, this negative-level Frobenius descent is an equivalence on quasi-nilpotent objects: $\sigma_v$72 For general $\sigma_v$73, under the strong Frobenius lift condition given by an étale torus chart and $\sigma_v$74, the level-raising functor induces cohomological isomorphisms

$\sigma_v$75

From this, one obtains full faithfulness on lf-nilpotent subcategories and a $\sigma_v$76-linearized equivalence on nilpotent subcategories. A Witt-vector analogue, formulated with $\sigma_v$77-Witt-connections, gives the same pattern without liftability assumptions on Frobenius.

6. Bloch–Ogus comparison and the Nisnevich form

In the Bloch–Ogus framework, Ogus’ comparison theorem identifies the coniveau spectral sequence with the hypercohomology spectral sequence of the Gersten complex resolving the unramified sheaves associated to étale cohomology (Deshmukh et al., 2020). For a regular scheme $\sigma_v$78, the $\sigma_v$79-page is

$\sigma_v$80

and the $\sigma_v$81-differential is described by residue maps attached to codimension-$\sigma_v$82 specializations,

$\sigma_v$83

The Nisnevich analogue proved over a general base uses the following hypotheses: $\sigma_v$84 is a $\sigma_v$85-2, Noetherian, irreducible, regular scheme of finite type; $\sigma_v$86 is smooth of finite type of pure dimension $\sigma_v$87; $\sigma_v$88 with $\sigma_v$89 invertible on $\sigma_v$90; and $\sigma_v$91 is an l.c.c. complex. Under these conditions, the Nisnevich Gersten complex

$\sigma_v$92

is exact; equivalently, $\sigma_v$93 is a flasque resolution of the Nisnevich sheafification $\sigma_v$94.

Consequently, one obtains the Nisnevich coniveau spectral sequence

$\sigma_v$95

again with $\sigma_v$96. The purity input is Gabber’s absolute purity: $\sigma_v$97 for a closed immersion $\sigma_v$98 of regular schemes of pure codimension $\sigma_v$99, and in particular

$F_v$00

for $F_v$01. Combining localization with purity gives the residue maps. The Nisnevich refinement uses henselization arguments, Nisnevich distinguished squares, and absolute purity to extend the comparison over a general base.

7. Stack-theoretic reformulation and logarithmic variants

A recent reformulation reproves the Ogus–Vologodsky equivalence through the relative de Rham stack in characteristic $F_v$02 and shows that a lift of $F_v$03 is not necessary; instead, one uses a lift of $F_v$04 to the second Witt vectors of $F_v$05 (Terentiuk, 5 Apr 2026). For a morphism $F_v$06, the relative Frobenius is

$F_v$07

The relative de Rham stack is

$F_v$08

and for schemes the morphism

$F_v$09

is affine, represented by

$F_v$10

If $F_v$11 is smooth, $F_v$12 realizes $F_v$13 as a $F_v$14-gerbe over $F_v$15. The action of $F_v$16 encodes the $F_v$17-curvature: for a crystal $F_v$18,

$F_v$19

so the operator is the $F_v$20-curvature $F_v$21. This replaces the classical Azumaya-algebra viewpoint by a torsor/gerbe structure on the relative de Rham stack.

The modified de Rham gerbe

$F_v$22

is the global object relevant for the completed equivalence. The key result states that, for $F_v$23 representable quasi-syntomic, the gerbe of splittings of $F_v$24 is the gerbe of flat lifts of $F_v$25 to $F_v$26. Consequently, whenever $F_v$27 lifts flatly, one gets a symmetric monoidal equivalence

$F_v$28

and in particular

$F_v$29

If, in addition, there is a strong Frobenius lift $F_v$30, then $F_v$31 itself splits and one obtains the local Cartier equivalence

$F_v$32

On underlying bundles, $F_v$33 has $F_v$34 as underlying sheaf, and the connection is

$F_v$35

where $F_v$36 is the $F_v$37-connection correction associated to the Frobenius lift. The same framework extends to representable quasi-syntomic morphisms of algebraic stacks, to logarithmic pairs $F_v$38 with equivalences

$F_v$39

and to equivariant settings. A plausible implication is that the stack-theoretic formulation isolates the geometric mechanism behind the classical correspondence: the relevant comparison is governed by splittings of a de Rham gerbe, with Frobenius and $F_v$40-curvature built into the action.