Cartier Cohomology Frameworks

• Cartier cohomology is a framework that uses Frobenius-twisted differential forms to translate de Rham data into robust cohomological invariants.
• It extends classical theories through logarithmic, p-adic, cyclotomic, and prismatic reinterpretations, connecting algebraic, topological, and categorical approaches.
• The theory underpins key isomorphisms and transforms, facilitating comparisons between differential, crystalline, and coalgebraic cohomology in modern research.

Searching arXiv for recent and foundational papers on Cartier cohomology and related Cartier-theoretic frameworks. Cartier cohomology denotes a family of cohomological formalisms organized around the Cartier operator, the inverse Cartier isomorphism, Frobenius-semilinear structures, and their modern extensions. In the literature represented here, the term ranges from the classical characteristic-$p$ comparison between differential forms and de Rham cohomology, through logarithmic and $p$-adic de Rham–Witt theories, to cyclotomic and prismatic reinterpretations, and also to coalgebraic, module-theoretic, and crystal-theoretic constructions. A common feature is that Cartier-type operators convert differential or Hochschild-style data into cohomological invariants while retaining integral or Frobenius-twisted structure (Aoki, 2023, Antieau et al., 2018, Lindell, 14 Aug 2025).

1. Classical Cartier theory and its cohomological meaning

The classical point of departure is the Cartier isomorphism for a smooth $k$-scheme in characteristic $p$,

$$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$

together with its logarithmic analogue for log smooth morphisms of Cartier type,

$$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$

In this form, Cartier cohomology is the identification of de Rham cohomology sheaves with Frobenius-twisted differential forms, and in semistable situations the relevant objects are logarithmic rather than ordinary differentials (Aoki, 2023).

For smooth projective curves over a perfect field $k$ of characteristic $p$, the classical cohomological pairing

$$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$

puts the Cartier operator $\mathcal C$ on $p$0 and Frobenius $p$1 on $p$2 in adjunction. The exact relation is

$p$3

so Cartier–Manin and Hasse–Witt matrices are related by twisted transpose,

$p$4

This makes Cartier cohomology on curves inseparable from semilinear algebra: iterates are twisted products rather than ordinary powers,

$p$5

The curve case is also where errors in the literature have often arisen from conflating Cartier and Frobenius, or from forgetting the transpose and semilinear twists (Achter et al., 2017).

Several later developments generalize this classical pattern rather than replacing it. This suggests that “Cartier cohomology” is best viewed as a structural theme: a cohomology theory becomes Cartier-theoretic when its basic cohomology sheaves are described by Frobenius-twisted forms, or when its operators are related by Cartier-type adjunction or descent.

2. Logarithmic and $p$6-adic Cartier cohomology

In integral $p$7-adic Hodge theory, the most direct extension of classical Cartier theory is the $p$8-adic Cartier isomorphism. For semistable formal schemes over $p$9, Aoki proves a logarithmic semistable analogue of the Bhatt–Morrow–Scholze smooth comparison. If $k$0 is semistable, with local model

$k$1

and $k$2 is the semistable log structure, then for every $k$3 and $k$4,

$k$5

This is Theorem 4.25, the semistable/logarithmic $k$6-adic Cartier isomorphism. The right-hand side is Matsuue’s logarithmic de Rham–Witt complex with the Breuil–Kisin–Fargues twist $k$7, and the theorem is formulated at the level of cohomology groups rather than as an a priori quasi-isomorphism of complexes (Aoki, 2023).

The construction uses Fontaine’s ring

$k$8

the semistable $k$9-cohomology complex

$p$0

and its $p$1-specialization

$p$2

The décalage functor $p$3 is indispensable because it removes the almost torsion produced by perfectoid or pro-étale constructions and extracts integral complexes that compare to log de Rham–Witt objects. Locally, Aoki identifies these complexes with a semistable $p$4-de Rham complex,

$p$5

which is the semistable analogue of the BMS coordinate calculation (Aoki, 2023).

The $p$6 specialization recovers the semistable Hodge–Tate comparison,

$p$7

and on the special fiber Aoki proves a genuine Cartier isomorphism for truncated logarithmic de Rham–Witt complexes,

$p$8

When $p$9, the log structures become trivial and the theorem recovers the Bhatt–Morrow–Scholze smooth $$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$0-adic Cartier isomorphism (Aoki, 2023).

A parallel mixed-characteristic deformation of Cartier theory appears in $$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$1-de Rham cohomology. Pridham constructs a functorial lift of the Cartier isomorphism for smooth formal schemes over $$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$2, with Frobenius lift $$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$3, in the form

$$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$4

Under the identification

$$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$5

the Adams operation satisfies

$$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$6

which is the mixed-characteristic Cartier-type formula governing the comparison (Pridham, 2016).

3. Cyclotomic, topological, and prismatic reinterpretations

Modern homotopy-theoretic work recasts Cartier structures in terms of cyclotomic spectra, topological Hochschild homology, and prismatic geometry. Antieau–Nikolaus introduce $$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$7-typical topological Cartier modules, namely spectra with $$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$8-action and structure maps

$$C^{-1}:\Omega^i_{X^{(1)}/k}\xrightarrow{\sim}\mathcal H^i(\Omega^\bullet_{X/k}),$$9

They construct a cyclotomic $$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$0-structure whose heart is the category of derived $$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$1-complete $$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$2-typical Cartier modules,

$$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$3

and for a perfect field $$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$4 of characteristic $$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$5 and a smooth $$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$6-algebra $$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$7, they identify cyclotomic homotopy groups of $$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$8 with de Rham–Witt groups,

$$C^{-1}:\Omega^i_{(R_{k_0},N_0)^{(p)}/(k_0,Q_0)} \xrightarrow{\sim} \mathcal H^i\bigl(\Omega^\bullet_{(R_{k_0},N_0)/(k_0,Q_0)}\bigr).$$9

This makes de Rham–Witt complexes the basic Cartier-theoretic layers of cyclotomic homotopy theory (Antieau et al., 2018).

Bhatt–Lurie then geometrize absolute prismatic crystals through the Cartier–Witt stack $k$0. The crucial equivalence is

$k$1

with the limit running over bounded prisms. In this sense, quasi-coherent complexes on $k$2 are crystals of $k$3-complete complexes on the absolute prismatic site. The Hodge–Tate locus is described by

$k$4

and quasi-coherent complexes on $k$5 are classified by a $k$6-complete complex with an operator $k$7, subject to the condition that $k$8 acts locally nilpotently mod $k$9. This is explicitly presented as a form of Cartier duality (Bhatt et al., 2022).

A further noncommutative extension replaces differential forms by $p$0, polyvector fields by $p$1, and the Cartier operator by the cyclotomic Frobenius. For every $p$2-algebra $p$3, the basic noncommutative Cartier formula is the commutative square

$p$4

and, over $p$5, the Tate-fixed-point version recovers the classical Cartier compatibility formula for interior product. The same framework yields a $p$6-curvature formula for the Getzler–Gauss–Manin connection in terms of equivariant $p$7-fold cap product (Rezchikov, 6 Jul 2026).

These developments do not abolish classical Cartier cohomology; they enlarge its scope. The data block repeatedly presents cyclotomic Frobenius, Nygaard filtrations, and Tate diagonals as replacements for the classical Cartier operator in settings where $p$8, $p$9, prismatic cohomology, and periodic cyclic homology play the role previously held by de Rham or crystalline complexes.

4. Cartier smoothness, de Rham–Witt simplifications, and purity

A distinct but closely related usage arises for Cartier smooth rings. A commutative $$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$0-algebra $$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$1 is Cartier smooth if its cotangent complex is a flat ordinary $$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$2-module and the inverse Cartier map

$$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$3

is an isomorphism. For such rings, derived de Rham and de Rham–Witt theories degenerate to their underived forms: $$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$4 The Nygaard filtration on $$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$5 becomes explicit, and syntomic complexes are identified with logarithmic de Rham–Witt sheaves,

$$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$6

For local Cartier smooth rings one obtains the $$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$7-adic $$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$8-theoretic comparison

$$H^0(X,\Omega_X^1)\times H^1(X,\mathcal O_X)\longrightarrow k$$9

while $\mathcal C$0 is controlled by the same logarithmic Hodge–Witt objects via its motivic filtration (Kim, 2023).

The mixed-characteristic refinement is $\mathcal C$1-Cartier smoothness. A morphism $\mathcal C$2 is $\mathcal C$3-Cartier smooth if it is $\mathcal C$4-discrete and its reduction $\mathcal C$5 is Cartier smooth. For a bounded prism $\mathcal C$6 and a $\mathcal C$7-cotangent smooth $\mathcal C$8-algebra $\mathcal C$9, the paper proves that $p$00-Cartier smoothness is equivalent to a full package of prismatic comparison statements, including

$p$01

being an equivalence, the Nygaard-completion map

$p$02

being an equivalence, and the Frobenius comparison

$p$03

being an equivalence. Over a perfect prism, $p$04-Cartier smoothness is also equivalent to $p$05-smoothness, and it holds for valuation ring extensions over perfectoid bases (Bouis, 2022).

Cartier-theoretic purity appears on discretely ringed adic spaces. For a smooth discretely ringed adic space $p$06, the paper identifies logarithmic differentials with bounded differential sheaves

$p$07

defines an adic Cartier operator

$p$08

and proves the exact sequence

$p$09

in the tame setting. This becomes the key input in the purity theorem

$p$10

for a smooth closed immersion of codimension $p$11. The novelty is that ordinary étale $p$12-torsion purity fails, whereas tame cohomology together with the Cartier exact sequence restores the expected behavior (Koubaa, 2024).

5. Algebraic, coalgebraic, and categorical Cartier cohomology

In another line of development, Cartier cohomology is the coalgebra-side analogue of Hochschild cohomology. For a $p$13-coring $p$14, Cartier cohomology is defined by

$p$15

and computed by the cobar complex

$p$16

If $p$17 is finitely generated projective as a left $p$18-module and $p$19 is the right algebra, then the paper proves

$p$20

and, more strongly,

$p$21

as $p$22-algebras. Thus Cartier cohomology becomes relative Hochschild cohomology of the dual algebra, up to opposite structure (Lindell, 14 Aug 2025).

The same terminology appears for associative coalgebras under the synonymous name coHochschild cohomology. In the dendriform setting, the paper defines a refined cohomology for dendriform coalgebras and constructs a natural cochain map

$p$23

from dendriform cochains to coHochschild, hence Cartier, cochains. This induces

$p$24

and for self-coefficients it is compatible with Gerstenhaber structures. Cartier cohomology is therefore the associative-coalgebra shadow of the split dendriform theory (Das, 2019).

A geometric module-theoretic incarnation is provided by Cartier sheaves and Cartier crystals on Noetherian $p$25-schemes. A Cartier sheaf is a quasi-coherent sheaf $p$26 with a right Frobenius action

$p$27

After localizing coherent Cartier sheaves by nilpotent objects one obtains Cartier crystals, and the paper constructs the basic cohomological operations

$p$28

proves that $p$29 preserves coherent cohomology up to nilpotence for finite type morphisms, and shows that $p$30 has bounded cohomological amplitude on crystals. This turns the abelian theory of Cartier modules into a geometric cohomology theory with exact triangles and adjunctions (Blickle et al., 2013).

A higher-categorical version replaces the abelian category of Cartier modules by a lax equalizer. For an endofunctor $p$31,

$p$32

If $p$33 is Grothendieck abelian and $p$34 is exact and colimit-preserving, the paper proves

$p$35

For $p$36 an $p$37-scheme this means derived Cartier modules are exactly complexes with a derived Frobenius map $p$38, and it yields a conceptual construction of the perverse $p$39-structure on coherent derived Cartier modules (Mattis et al., 2024).

6. Cartier transforms, crystals, and global comparison functors

The phrase “Cartier cohomology” also covers transform theories that exchange Frobenius-twisted objects with connection-type or crystal-type objects while preserving cohomology. In dimension one, the Cartier transform between crystals on the $p$40-crystalline and prismatic sites is

$p$41

and induces an equivalence between locally finite free $p$42-crystals and locally finite free prismatic crystals. The same transform is identified locally with the explicit $p$43-twisted Simpson correspondence

$p$44

and the corresponding cohomology theories are computed by two-term twisted de Rham complexes on the two sides. This is a prismatic $p$45-deformed Cartier comparison rather than a direct de Rham–Witt statement (Gros et al., 2022).

Xu lifts the Ogus–Vologodsky Cartier transform modulo $p$46. Starting from a smooth formal $p$47-scheme $p$48, with Frobenius twist $p$49, he constructs a global equivalence

$p$50

which becomes, after passing through stratifications, the desired transform from quasi-nilpotent $p$51-torsion modules with integrable $p$52-connection on $p$53 to quasi-nilpotent $p$54-torsion modules with integrable connection on $p$55. When a Frobenius lifting exists, the transform is compatible with Shiho’s explicit pullback construction. Xu then applies this machinery to relative Fontaine modules, giving a new interpretation of their divided Frobenius data and recovering cohomological results of Faltings (Xu, 2017).

These transform results show that Cartier cohomology is not only a collection of fixed cohomology groups. It also includes equivalences of categories whose purpose is cohomological: they transport connections, $p$56-connections, Higgs fields, crystals, and filtrations across Frobenius-twisted geometries in a way compatible with de Rham, crystalline, prismatic, or Fontaine-module cohomology.

In this broader sense, the literature presents Cartier cohomology as a unifying language for Frobenius-twisted descent. In characteristic $p$57, it begins with

$p$58

in logarithmic and $p$59-adic geometry it becomes a comparison with de Rham–Witt and $p$60-cohomology; in cyclotomic and prismatic settings it is recast by Frobenius, Nygaard, and the Cartier–Witt stack; and in coalgebraic or categorical settings it becomes a theory of cochains, crystals, and derived Frobenius modules. The recurring content is the same: Cartier operators and Cartier-type transforms identify cohomology with a more rigid Frobenius-linear object, and that identification then governs structure, comparison, and descent across a wide range of modern cohomological theories (Aoki, 2023, Bhatt et al., 2022, Lindell, 14 Aug 2025, Xu, 2017).