Higher Cartier Operators

• Higher Cartier operators are categorified forms of the classical Cartier operator in characteristic p, unifying iterative Frobenius structures across various settings.
• They enable coherent control of structural maps in derived, logarithmic, and topological contexts by packaging iterated operations through monadic and cyclotomic frameworks.
• Applications span from toric fixed ideals and noncommutative Hochschild homology to explicit differential operators, influencing modern algebraic geometry.

Searching arXiv for recent and foundational papers on higher Cartier operators and related Cartier frameworks. {"query":"higher Cartier operators Cartier Raynaud ring topological Cartier modules derived infinity-category log local Cartier transform arXiv", "max_results": 10} arxiv_search(query="higher Cartier operators Cartier Raynaud ring topological Cartier modules derived infinity-category log local Cartier transform", max_results=10) Higher Cartier operators are higher, iterated, or categorified forms of the Cartier operator in characteristic $p$. In the classical scheme-theoretic setting, they arise from a structural map $\kappa:F_*M\to M$ on a Cartier module; in derived and $\infty$-categorical settings, they are encoded by lax equalizers and a monad $\coprod_{n\ge 0}F^n$; in topological settings, they are generated by Frobenius, Verschiebung, and a degree-$1$ operator $d$ inside the topological Cartier--Raynaud ring; in logarithmic geometry, they appear through higher-level differential operators, $p^{m+1}$-curvature, and logarithmic Cartier transforms; in relative settings, they are the natural transformations $\chi_e$ attached to iterates of relative Frobenius; and in noncommutative settings, they are cyclotomic conjugation operators on topological Hochschild homology (Mattis et al., 2024, Bals, 2024, Ohkawa, 2014, Carvajal-Rojas et al., 20 Apr 2026, Rezchikov, 6 Jul 2026). The literature therefore uses the term for a family of closely related Frobenius-governed operations rather than for a single universal construction.

1. Classical Cartier operators and their iterates

For an $\mathbb F_p$-scheme $X$, a classical Cartier module is a quasi-coherent $\kappa:F_*M\to M$0-module $\kappa:F_*M\to M$1 equipped with an $\kappa:F_*M\to M$2-linear Cartier operator

$\kappa:F_*M\to M$3

where $\kappa:F_*M\to M$4 is the absolute Frobenius (Mattis et al., 2024). In this form, the operator is a $\kappa:F_*M\to M$5-linear structure map, and its higher versions are the coherent iterates

$\kappa:F_*M\to M$6

or, on a module $\kappa:F_*M\to M$7,

$\kappa:F_*M\to M$8

(Mattis et al., 2024, Carvajal-Rojas et al., 20 Apr 2026).

On a smooth scheme over a perfect field, the classical Cartier isomorphism is expressed on the de Rham complex by

$\kappa:F_*M\to M$9

with inverse $\infty$0, and in Grothendieck-dual form by the Katz operator

$\infty$1

(Carvajal-Rojas et al., 20 Apr 2026). In the logarithmic smooth case, Kato’s logarithmic Cartier isomorphism gives a quasi-isomorphism

$\infty$2

under the lifting hypotheses stated in the log-smooth setting (Fersi, 12 Dec 2025). At the operator level, the familiar formulas

$\infty$3

remain the model for later higher constructions (Fersi, 12 Dec 2025).

A persistent point in the literature is that “higher” may mean either iterated Frobenius-compatible maps $\infty$4 or operator algebras built from Cartier structures. The classical theory already contains both viewpoints: repeated application of $\infty$5, and structural control of differential forms or dualizing modules via Frobenius.

2. Derived and $\infty$6-categorical Cartier categories

A systematic higher-categorical framework is obtained by defining the $\infty$7-category of Cartier modules as a lax equalizer. For an endofunctor $\infty$8,

$\infty$9

so that an object is a pair $\coprod_{n\ge 0}F^n$0 with $\coprod_{n\ge 0}F^n$1 and a structural map

$\coprod_{n\ge 0}F^n$2

(Mattis et al., 2024). Morphisms are maps equipped with a homotopy-coherent commutative square

$\coprod_{n\ge 0}F^n$3

and the mapping spaces are computed as equalizers in $\coprod_{n\ge 0}F^n$4 (Mattis et al., 2024).

This formulation is stable under standard categorical hypotheses. If $\coprod_{n\ge 0}F^n$5 is stable and $\coprod_{n\ge 0}F^n$6 is exact, then $\coprod_{n\ge 0}F^n$7 is stable; if $\coprod_{n\ge 0}F^n$8 is presentable and $\coprod_{n\ge 0}F^n$9 preserves colimits, then $1$0 is presentable; and if $1$1 is Grothendieck abelian with exact, colimit-preserving $1$2, then $1$3 is Grothendieck abelian (Mattis et al., 2024). The induced $1$4-structure is characterized by the forgetful functor being $1$5-exact, and its heart is the lax equalizer of the induced functors on hearts (Mattis et al., 2024).

Under countable coproduct hypotheses, the forgetful functor is monadic, with left adjoint determined by

$1$6

and monad

$1$7

(Mattis et al., 2024). In this description, a single structural map $1$8 canonically generates all coherent higher Cartier operators $1$9. The point is not merely that one can iterate $d$0, but that the monad packages the higher coherence conditions automatically.

The central derived comparison theorem states that, for a Grothendieck abelian category $d$1 and exact, colimit-preserving $d$2,

$d$3

as stable $d$4-categories, and this equivalence is $d$5-exact (Mattis et al., 2024). For Noetherian $d$6-schemes with finite absolute Frobenius, this yields

$d$7

and the perverse $d$8-structure on $d$9 induces the corresponding perverse $p^{m+1}$0-structure on $p^{m+1}$1 (Mattis et al., 2024).

In this framework, higher Cartier operators are no longer ad hoc iterates. They are part of the intrinsic monadic and derived structure of the category itself.

3. Topological Cartier modules and the topological Cartier--Raynaud ring

A spectrum-level incarnation is given by $p^{m+1}$2-typical topological Cartier modules. Such an object $p^{m+1}$3 is a spectrum with $p^{m+1}$4-action together with an $p^{m+1}$5-equivariant factorization of the $p^{m+1}$6-norm map

$p^{m+1}$7

equivalently maps

$p^{m+1}$8

and a degree-$p^{m+1}$9 operator $\chi_e$0 on $\chi_e$1 coming from the generator in $\chi_e$2 (Bals, 2024). The resulting $\chi_e$3-category $\chi_e$4 admits a compact generator $\chi_e$5, and for $\chi_e$6 one defines the topological Cartier--Raynaud ring spectrum

$\chi_e$7

The main equivalence is

$\chi_e$8

(Bals, 2024).

Its graded homotopy ring has a concrete presentation: $\chi_e$9 with $\mathbb F_p$0, $\mathbb F_p$1, and relations

$\mathbb F_p$2

(with $\mathbb F_p$3 for $\mathbb F_p$4),

$\mathbb F_p$5

$\mathbb F_p$6

for $\mathbb F_p$7 (Bals, 2024). In discrete situations such as $\mathbb F_p$8, the Hopf element $\mathbb F_p$9 maps to $X$0, so one recovers $X$1 and the classical Cartier--Raynaud relations (Bals, 2024).

Within this ring, higher Cartier operators are explicitly the higher composites generated by $X$2, $X$3, and $X$4. A standard family is

$X$5

Using $X$6, one gets the reduction rule

$X$7

The scaling relations

$X$8

govern how $X$9 moves past $\kappa:F_*M\to M$00 and $\kappa:F_*M\to M$01 and encode the $\kappa:F_*M\to M$02-power behavior of these higher operators (Bals, 2024).

A further structural theorem identifies all natural operations on homotopy groups. The graded ring of natural endomorphisms of

$\kappa:F_*M\to M$03

is canonically isomorphic to $\kappa:F_*M\to M$04 (Bals, 2024). Thus every natural operation on homotopy groups of $\kappa:F_*M\to M$05-typical topological Cartier modules is an $\kappa:F_*M\to M$06-linear combination of composites of $\kappa:F_*M\to M$07, $\kappa:F_*M\to M$08, and $\kappa:F_*M\to M$09, modulo the displayed relations. For $\kappa:F_*M\to M$10, $\kappa:F_*M\to M$11 recovers the classical Cartier--Raynaud ring $\kappa:F_*M\to M$12, and on examples such as the de Rham--Witt complex the operators reproduce Verschiebung, Frobenius, and the de Rham differential (Bals, 2024).

4. Logarithmic and higher-level Cartier transforms

In logarithmic geometry, higher Cartier operators arise from sheaves of log differential operators of higher level, divided Frobenius maps, and logarithmic Higgs--connection correspondences. For an integral log smooth morphism $\kappa:F_*M\to M$13 of fine log schemes in characteristic $\kappa:F_*M\to M$14, Ohkawa studies the higher-level sheaf $\kappa:F_*M\to M$15 built from the log $\kappa:F_*M\to M$16-PD envelope of the diagonal (Ohkawa, 2014). The higher $\kappa:F_*M\to M$17-curvature map

$\kappa:F_*M\to M$18

identifies the center as

$\kappa:F_*M\to M$19

and $\kappa:F_*M\to M$20 is Azumaya over its center with rank $\kappa:F_*M\to M$21 (Ohkawa, 2014).

Under a log strong lifting of the $\kappa:F_*M\to M$22-st relative Frobenius modulo $\kappa:F_*M\to M$23, the divided Frobenius

$\kappa:F_*M\to M$24

produces a splitting module

$\kappa:F_*M\to M$25

and hence the log local Cartier transform

$\kappa:F_*M\to M$26

between suitable indexed $\kappa:F_*M\to M$27-modules and Higgs modules (Ohkawa, 2014). The higher-level Cartier operator on logarithmic $\kappa:F_*M\to M$28-forms is the $\kappa:F_*M\to M$29-linear map

$\kappa:F_*M\to M$30

with local formulas

$\kappa:F_*M\to M$31

$\kappa:F_*M\to M$32

(Ohkawa, 2014). When the strong lifting comes from the absolute Frobenius, the correction terms vanish and the formulas simplify to

$\kappa:F_*M\to M$33

(Ohkawa, 2014).

A parallel logarithmic development generalizes the Ogus--Vologodsky and Shiho--Oyama constructions to log smooth schemes. For a log smooth $\kappa:F_*M\to M$34 over a perfect field, the exact relative Frobenius

$\kappa:F_*M\to M$35

is log flat, and under a lifting of the exact relative Frobenius to Witt vectors one obtains a fully faithful functor from quasi-coherent modules on $\kappa:F_*M\to M$36 equipped with a quasi-nilpotent Higgs field to quasi-coherent modules on $\kappa:F_*M\to M$37 equipped with a quasi-nilpotent integrable connection (Fersi, 12 Dec 2025). The operator-level mechanism uses logarithmic differential operators $\kappa:F_*M\to M$38 on the PD-envelope of the diagonal, with composition rules

$\kappa:F_*M\to M$39

$\kappa:F_*M\to M$40

(Fersi, 12 Dec 2025). The $\kappa:F_*M\to M$41-curvature map

$\kappa:F_*M\to M$42

sends $\kappa:F_*M\to M$43 to $\kappa:F_*M\to M$44, and for a connection with vanishing $\kappa:F_*M\to M$45-curvature, operators $\kappa:F_*M\to M$46 vanish whenever some $\kappa:F_*M\to M$47 (Fersi, 12 Dec 2025).

Both theories make the same structural point: the logarithmic Cartier transform is mediated by explicit higher differential operators and by Azumaya-splitting techniques, not merely by a single first-order operator.

5. Relative Cartier operators and pullback along regular maps

For a regular $\kappa:F_*M\to M$48-finite morphism $\kappa:F_*M\to M$49 of locally noetherian schemes, Carvajal-Rojas and Stäbler construct a relative Cartier isomorphism and higher relative Cartier operators valid without a smoothness assumption. Writing $\kappa:F_*M\to M$50 and $\kappa:F_*M\to M$51, they define canonical isomorphisms

$\kappa:F_*M\to M$52

for all $\kappa:F_*M\to M$53, compatible in $\kappa:F_*M\to M$54 and recursively related through the base-change isomorphisms $\kappa:F_*M\to M$55 (Carvajal-Rojas et al., 20 Apr 2026). The associated higher relative Cartier operator is the natural transformation

$\kappa:F_*M\to M$56

and after pushforward along absolute Frobenius on the base one obtains

$\kappa:F_*M\to M$57

(Carvajal-Rojas et al., 20 Apr 2026).

On an affine chart admitting a $\kappa:F_*M\to M$58-basis $\kappa:F_*M\to M$59, the local formula is explicit: $\kappa:F_*M\to M$60 sends

$\kappa:F_*M\to M$61

(Carvajal-Rojas et al., 20 Apr 2026). This formula is the relative analogue of the classical Cartier operator, with the role of local coordinates played by a $\kappa:F_*M\to M$62-basis.

These operators induce pullback on Cartier modules. If $\kappa:F_*M\to M$63 is a degree-$\kappa:F_*M\to M$64 Cartier structure, then the pulled-back operator is

$\kappa:F_*M\to M$65

(Carvajal-Rojas et al., 20 Apr 2026). The resulting functor $\kappa:F_*M\to M$66 is compatible with finite covers and with factorizations through polynomial and pristine maps (Carvajal-Rojas et al., 20 Apr 2026).

The principal application is to invariants of Cartier modules. Test modules commute with regular pullback: $\kappa:F_*M\to M$67 and mixed test modules satisfy a higher-Cartier scaling relation

$\kappa:F_*M\to M$68

when $\kappa:F_*M\to M$69 is generated in degree $\kappa:F_*M\to M$70 (Carvajal-Rojas et al., 20 Apr 2026). Under the stated setup, the sets

$\kappa:F_*M\to M$71

are finite on boxes, and the constancy regions are $\kappa:F_*M\to M$72-fractal (Carvajal-Rojas et al., 20 Apr 2026). Here the higher relative Cartier operators are not auxiliary; they are the mechanism that transports Frobenius-linear structure through regular morphisms.

6. Toric and analytic realizations

In toric geometry, higher Cartier operators are organized by graded Cartier algebras of $\kappa:F_*M\to M$73-linear maps. For an affine toric variety $\kappa:F_*M\to M$74 in characteristic $\kappa:F_*M\to M$75, an effective toric $\kappa:F_*M\to M$76-divisor $\kappa:F_*M\to M$77, and $\kappa:F_*M\to M$78 with

$\kappa:F_*M\to M$79

the toric map

$\kappa:F_*M\to M$80

is given on monomials by

$\kappa:F_*M\to M$81

Its iterates $\kappa:F_*M\to M$82 are the higher Cartier operators in this setting (Hsiao et al., 2010). For a toric ideal $\kappa:F_*M\to M$83 and rational $\kappa:F_*M\to M$84, the Cartier algebra generated by $\kappa:F_*M\to M$85 and $\kappa:F_*M\to M$86 fixes precisely the ideals

$\kappa:F_*M\to M$87

for some nonempty subset of faces $\kappa:F_*M\to M$88 of $\kappa:F_*M\to M$89 (Hsiao et al., 2010). These fixed ideals coincide with toric intermediate adjoint ideals defined from a log resolution, so the higher Cartier algebra has both combinatorial and birational descriptions (Hsiao et al., 2010).

A different realization appears in function-field analysis. On $\kappa:F_*M\to M$90, Jeong defines Cartier operators

$\kappa:F_*M\to M$91

and from them the two higher sequences $\kappa:F_*M\to M$92 and $\kappa:F_*M\to M$93 by

$\kappa:F_*M\to M$94

for $\kappa:F_*M\to M$95 with $\kappa:F_*M\to M$96 (Jeong, 2015). These operators are linked to Hasse derivatives by binomial inversion: $\kappa:F_*M\to M$97

$\kappa:F_*M\to M$98

(Jeong, 2015). The sequences $\kappa:F_*M\to M$99 and $\infty$00 form orthonormal bases of the space of continuous $\infty$01-linear functions on $\infty$02, admit $\infty$03-adic digit extensions to orthonormal bases of all continuous functions, and yield Wronskian criteria for linear independence (Jeong, 2015).

These examples show that the phrase “higher Cartier operators” also governs explicit operator algebras: in toric geometry, $\infty$04-linear monomial operators and their fixed ideals; in positive-characteristic analysis, continuous operators that act as substitutes for higher derivatives.

7. Cyclotomic and noncommutative Cartier formulae

A noncommutative version is formulated on topological Hochschild homology. For every $\infty$05-algebra $\infty$06, topological Hochschild homology $\infty$07 is an $\infty$08-spectrum with cyclotomic structure, and topological Hochschild cohomology $\infty$09 acts on it by cap product

$\infty$10

(Rezchikov, 6 Jul 2026). Using the Hill--Hopkins--Ravenel norm and Tate or geometric fixed points, one constructs a genuine $\infty$11-equivariant $\infty$12-fold cap product

$\infty$13

together with the cyclotomic Frobenius

$\infty$14

and the Tate diagonal $\infty$15 on $\infty$16 (Rezchikov, 6 Jul 2026).

The absolute noncommutative Cartier formula is

$\infty$17

for $\infty$18 and $\infty$19 (Rezchikov, 6 Jul 2026). A relative version over a base ring $\infty$20 has the same form,

$\infty$21

with the relative cyclotomic Frobenius and relative Tate diagonal (Rezchikov, 6 Jul 2026).

For smooth commutative algebras over $\infty$22, the Hochschild--Kostant--Rosenberg identifications recover the classical Cartier conjugation formula on differential forms: $\infty$23 (Rezchikov, 6 Jul 2026). In this sense, the cyclotomic Frobenius plays the role of the inverse Cartier map, and the $\infty$24-Tate diagonal plays the role of the induced higher operator on Hochschild cohomology.

The same formalism computes the $\infty$25-curvature of the Getzler--Gauss--Manin connection: $\infty$26 and, in the symplectic setting described in the paper, yields

$\infty$27

for the quantum connection and quantum Steenrod operations under the stated assumptions (Rezchikov, 6 Jul 2026). This extends higher Cartier operators from algebraic Frobenius-linear maps to cyclotomic operators governing the interaction between cap products, Frobenius, and equivariant structures.

Taken together, these developments indicate that higher Cartier operators now form a broad technical language spanning Cartier modules, logarithmic and derived geometry, spectrum-level algebra, and cyclotomic homotopy theory. The common structure is the coherent control of iterated Frobenius or cyclotomic operations, together with explicit operator algebras that organize their relations.