On the injective dimension of unit Cartier and Frobenius modules

Alexandria Wheeler; Daniel Fink; Manuel Blickle; Wenliang Zhang

REVIEW 3 major objections 4 minor 28 references

On the injective dimension of unit Cartier and Frobenius modules

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that the injective dimension of every unit Cartier module over a noetherian F-finite ring of prime characteristic p is at most the dimension of its support plus one, and similarly for unit Frobenius modules over regular…

desk verdict Nice short paper on sharp injective dimension bounds for unit Cartier/Frobenius modules; the regular case is solid, but the singular case rests on a mis-stated, unproven proposition deferred to an unpublished preprint. read the letter →

arxiv 2412.08423 v1 pith:UTHU6LNH submitted 2024-12-11 math.AC math.AG

Manuel Blickle , Daniel Fink , Alexandria Wheeler , Wenliang Zhang This is my paper · ORCID

classification math.ACmath.AG MSC 13D0513A3514B15

keywords unitCartiermodulesFrobeniusinjectivedimensionF-finiteringsprimecharacteristiccrystalsglobalendomorphism

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a sharp, support-sensitive bound on how many injective steps are needed to resolve modules carrying a Frobenius action. For unit Cartier modules over any noetherian F-finite ring of prime characteristic p, the injective dimension in the category of unit Cartier modules is at most the dimension of the module's support plus one. Over a regular F-finite ring, the analogous statement holds for unit Frobenius modules, with the extra refinement that their injective dimension is at most the ordinary injective dimension of the underlying module plus one. Because the support dimension of a module is often much smaller than the dimension of the ambient ring, this converts an earlier ambient-ring bound into an intrinsic one.

What carries the argument

The main mechanism is the category of unit Cartier modules as right modules over the twisted polynomial ring $A[F]$, together with the unitalization functor $u(M)=\operatorname{colim}(M \to F^\flat M \to F^{\flat 2}M \to \cdots)$ that cuts Cartier modules down to unit ones. Over a regular ring, tensoring with the canonical module $\omega_R$ identifies unit Cartier modules with unit Frobenius modules, so bounds transfer between the two settings. The decisive step for singular A is a Kashiwara-type transfer statement, Proposition 4.1: for a surjection $\psi:R \to A$ with R regular F-finite, the derived unit of adjunction $\mathrm{id} \to R\psi^\flat_{\mathrm{unit}} \circ \psi^{\mathrm{unit}}_*$ is an isomorphism, so an injective resolution of the pulled-back module over R can be pushed back down to give an injective resolution over A. The bound then follows by comparing supports under pullback and using the regular case.

What would settle it

Find a noetherian F-finite ring A and a unit Cartier module N with $\operatorname{inj.dim}_{\mathrm{Cart}^{\mathrm{unit}}_A}(N) > \dim(\mathrm{Supp}_A\, N)+1$. A concrete check would be to compute this category-level injective dimension for a non-Cohen-Macaulay F-finite ring, since the paper's Remark 4.6 already gives equality $\dim A+1$ for the canonical module over equidimensional Cohen-Macaulay rings.

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Extended reading notes

Core claim

The paper's central claim is Theorem B: for every noetherian F-finite ring A of prime characteristic p, every unit Cartier module N satisfies $\operatorname{inj.dim}_{\mathrm{Cart}^{\mathrm{unit}}_A}(N) \leq \dim(\mathrm{Supp}_A\, N)+1$. Consequently the global injective dimension of the category of unit Cartier modules over A is at most $\dim A+1$, and this is optimal: for an equidimensional Cohen-Macaulay A, the canonical module $\omega_A$ has injective dimension exactly $\dim A+1$ in that category. The paper also proves Theorem A for regular F-finite rings R: every unit Frobenius module M satisfies $\operatorname{inj.dim}_{\mathrm{Frob}^{\mathrm{unit}}_R}(M) \leq \operatorname{inj.dim}_R(M)+1 \leq \dim(\mathrm{Supp}_R\, M)+1$. The first inequality sharpens the known bound $\operatorname{inj.dim}_R(M) \leq \dim(\mathrm{Supp}_R\, M)$ for the underlying module, and the second shows that the bound depends only on the support.

Load-bearing premise

The proof relies on a transfer statement, quoted from an earlier preprint and not proved in this text: pulling a module up from a quotient ring to a regular ring and pushing it back down recovers it up to quasi-isomorphism. If that statement failed, the main inequality for singular rings would not follow from the argument given.

Editorial extensions

If this is right

The category of unit Cartier modules over any noetherian F-finite ring A has global injective dimension exactly $\dim A+1$ whenever A is equidimensional Cohen-Macaulay, and at most $\dim A+1$ in general.
For unit Frobenius modules over a regular F-finite ring, the injective dimension in their own category is bounded by $\operatorname{inj.dim}_R(M)+1$, which is strictly sharper than $\dim(\mathrm{Supp}_R\, M)+1$ whenever the underlying module is not already extremal.
Because unit Cartier modules form an abelian category for singular F-finite rings, the bound gives a finite, dimension-controlled resolution theory for modules arising from the Frobenius action, including injective hulls of residue fields.
The same bound holds for quasi-coherent Cartier crystals, since that category is equivalent to unit Cartier modules, as noted in Remark 4.5.
The result removes dependence on a chosen regular presentation: earlier bounds depended on a surjection $R \to A$, while the new bound is intrinsic to A.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if the bound is correct, injective dimension in the unit-Cartier category is essentially a dimension-of-support invariant, so this number alone cannot separate unit Cartier modules whose supports have the same dimension.
A natural extension not tested here would be to drop the F-finiteness hypothesis; the proof uses a structure result that presents F-finite rings as quotients of regular rings, and the same support-plus-one bound is not established without F-finiteness.
The transfer statement quoted from the earlier preprint could be tested directly on a concrete nonregular quotient, such as $A = k[x,y]/(xy)$, by checking whether the derived unit is a quasi-isomorphism on a known unit Cartier module; this would exercise the load-bearing step on a small example.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper proves the right bound, and the regular case is solid. The singular case has a hole that is probably patchable but as printed it does not close.

What's actually new: Theorem A improves Ma's bound from dim(R)+1 to dim(Supp M)+1 for unit Frobenius modules over regular F-finite R. Theorem B removes the auxiliary regular presentation from the BF22 bound and gives the same support-based bound for unit Cartier modules over any F-finite A. That is a genuine improvement, and the internal machinery – Proposition 3.1, Lemma 3.3, Corollary 3.4 – is clean and self-contained. The equivalence with Cartier crystals is standard, and the support argument in Theorem 4.4 is basically fine modulo a typo in the inclusion direction.

The soft spot is exactly Proposition 4.1. The statement as printed says the derived unit is an isomorphism on D^+(Cart_R), but the adjunction at issue is between Cart^unit_A and Cart^unit_R, so the unit should be on D^+(Cart^unit_A). Lemma 4.3 uses the A-sided statement. So the printed proposition is unusable as written. The proof is a one-liner citing [BF22, Prop 9], an unpublished arXiv preprint. That is a derivability gap in the load-bearing step, not a demonstrated error; the intended claim is plausible and likely follows from BF22's work. But the manuscript should either prove Proposition 4.1 in the text or cite a published, vetted version.

There are smaller issues: the support inclusion in the proof of Theorem 4.4 is stated backwards (should be ⊆), and the optimality remark is borrowed from the same preprint. None of these change the main arc.

Bottom line: for a specialist in prime-characteristic commutative algebra, this is a useful short paper and it should get a serious referee. I would not desk-reject it. I'd ask the referee to verify the intended form of Proposition 4.1 and insist the authors fix the statement and the proof before acceptance. If the authors can do that, the paper becomes a solid contribution.

Referee Report

3 major / 4 minor

Summary. The paper establishes upper bounds for injective dimension in categories of unit Frobenius and unit Cartier modules over F-finite rings of prime characteristic. Theorem A (Theorem 3.5) states that over a regular F-finite ring R, every unit Frobenius module M satisfies inj.dim_{Frob^unit_R}(M) ≤ inj.dim_R(M)+1 ≤ dim(Supp_R M)+1. Theorem B (Theorem 4.4) states that over an arbitrary noetherian F-finite ring A, every unit Cartier module N satisfies inj.dim_{Cart^unit_A}(N) ≤ dim(Supp_A N)+1, and hence the global injective dimension of the category of unit Cartier modules is at most dim A+1. The proof of Theorem A combines a general comparison between Cartier and underlying module injective dimensions (Proposition 3.1), an equality over regular rings (Lemma 3.3), and Lyubeznik's bound. The proof of Theorem B reduces the singular case to the regular case through a Kashiwara-type adjunction induced by a surjection ψ:R→A with R regular (Section 4). The main technical tool is Proposition 4.1, which asserts that a certain derived unit of adjunction is an isomorphism; the proof is deferred to the authors' unpublished preprint [BF22, Proposition 9].

Significance. If the arguments are correct, the results are natural and valuable: Theorem B removes the dependence on a chosen regular presentation in the prior bound of [BF22, Theorem 1], giving an intrinsic bound dim(Supp N)+1, and Remark 4.6 indicates that the bound is optimal for equidimensional Cohen-Macaulay rings. The statements improve on Ma's bound in the regular case and give a clean category-level injective dimension bound over all F-finite rings. The internal arguments in Section 3 are concise and mostly self-contained, and the overall strategy is clear and plausible. However, the central reduction in Section 4 rests on a statement that is both misprinted and not proved within the paper, so the current version is not fully verifiable as written.

major comments (3)

[§4, Proposition 4.1] The displayed unit is written as η : id_{D^+(Cart_R)} → Rψ^♭_unit ∘ ψ^unit_*. For the adjunction ψ^unit_* : Cart^unit_A ⇄ Cart^unit_R : ψ^♭_unit, the unit should be id_{D^+(Cart^unit_A)} → Rψ^♭_unit ∘ ψ^unit_*; the printed domain is the wrong category and, interpreted literally, the statement is not the unit of this adjunction. Lemma 4.3 applies the proposition to N ∈ Cart^unit_A and needs exactly the A-sided unit, so this is not a cosmetic typo: Theorem 4.4's reduction to the regular ring R depends on this equivalence. In addition, the proof is only 'Immediate from [BF22, Proposition 9]', where [BF22] is an unpublished preprint by two of the authors, and no statement or argument is reproduced. I ask the authors to correct the statement and provide a self-contained proof, or at least a precise statement and proof of the needed result from [BF22] within the paper.
[§4, Lemma 4.3] The lemma is the load-bearing transfer step, but its proof is not complete as written. The proof begins with an injective resolution ψ^unit_* N → I^• of length ≤ k in Cart^unit_R and then uses 'the equivalence N ≃ Rψ^♭_unit ψ^unit_* N of Proposition 4.1'. Because Proposition 4.1 is misstated and its proof is deferred, this equivalence is not established in the manuscript. Moreover, the step from the quasi-isomorphism N ≃ ψ^♭_unit I^• to the conclusion that N has an injective resolution of length ≤ k in Cart^unit_A requires that ψ^♭_unit I^• is a bounded complex of injectives and that the quasi-isomorphism is a genuine resolution of N; the text asserts this in one sentence. The argument is likely standard, but given that this lemma is exactly what makes Theorem 4.4 follow, the details should be spelled out.
[§4, Theorem 4.4 proof] The support equality Supp_A N = Supp_R ψ_* N = Supp_R uψ_* N is used to convert the regular-ring bound into dim(Supp_A N)+1, but the proof is abbreviated. The injectivity of ψ_* N → F^♭ψ_* N is shown, and the equality of supports of ψ_* N and F^♭ψ_* N follows from the localization isomorphism and the support containment; however, the final equality with Supp_R uψ_* N depends on the observation that the colimit u(ψ_* N) contains ψ_* N as a submodule because all transition maps are injective. This is true, but it should be stated explicitly, since it is needed for the dimension estimate.

minor comments (4)

[§4, Theorem 4.4 proof] In the last display of the proof, '= Supp_A N + 1' should read '= dim Supp_A N + 1'.
[§4, opening sentence] The word 'breiﬂy' is a typo for 'briefly'.
[§4, Remark 4.2] The remark states that ψ^♭ preserves the unit property and concludes Rψ^♭_unit M = ψ^!M over regular R; this is used only implicitly later, and it would help to state explicitly that ψ^♭ I^• is then a unit Cartier complex, so that uψ^♭ I^• = ψ^♭ I^•.
[§1, Theorem 1.3] The phrase 'an noetherian F-ﬁnite ring' has an article disagreement; it should be 'a noetherian F-finite ring'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the target bounds are not assumed and the main deduction is genuinely new; the main caveat is a load-bearing self-citation to the authors' own preprint, which is a self-containment and verification risk rather than a circular reduction.

full rationale

The claimed bounds are not assumed anywhere. Theorem 3.5 follows from Proposition 3.1, Lemma 3.3, and Corollary 2.7, and Theorem 4.4 follows from Corollary 3.4 together with Lemma 4.3. Neither step restates the conclusion as an input; the new intrinsic bound dim(Supp_A N)+1 is obtained by replacing dim(R) in the earlier regular-ring bound with the actual support dimension of N. The only load-bearing piece that is outsourced is Proposition 4.1, whose proof is 'Immediate from [BF22, Proposition 9]', a preprint by two of the present authors; moreover, the printed unit is written on D^+(Cart_R), whereas the adjunction ψ^unit_∗: Cart^unit_A ⇄ Cart^unit_R : ψ^♭_unit has its unit on Cart^unit_A, and Lemma 4.3 applies the statement to N ∈ Cart^unit_A. These are genuine self-containment and correctness concerns, and they may determine whether Theorem 4.4 is established as written, but they are not circularity: the cited proposition is a different statement, not the present theorem, and no parameter is fitted to the target inequality. No definitional, fitted-input, renaming, or uniqueness-imported circularity is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the established theory of unit Cartier and Frobenius modules (Lyu97, Ma14, BB13, BF22) and on Gabber's structural result on F-finite rings. The most consequential load-bearing ingredient is BF22 Proposition 9, imported as Proposition 4.1 without proof in this manuscript; it is a self-cited preprint by the first two authors. No free parameters or invented entities occur.

assumptions (5)

domain assumption Gabber's theorem: every F-finite noetherian ring of prime characteristic p is a homomorphic image of a regular F-finite ring.
Invoked in Theorem 4.4 to choose a regular resolution ring R -> A, and in Remark 2.1 to obtain a dualizing complex with Frobenius compatibility for F-finite rings.
domain assumption BF22, Proposition 9 (Kashiwara-type equivalence): for a surjective map R -> A with R regular, the derived unit id -> R(psi^flat_unit) composed with psi^unit_* is an isomorphism.
Stated as Proposition 4.1 in the present paper with proof deferred to the self-cited preprint BF22; this is the load-bearing transfer step on which Theorem B depends.
standard math Standard homological algebra machinery: injective dimensions, Ext long exact sequences, right adjoints preserve injectives, derived categories of abelian categories.
Used throughout Sections 3 and 4 in the proofs of Proposition 3.1, Lemma 3.3, and Lemma 4.3.
domain assumption Equivalence of categories Cart^unit_R and Frob^unit_R for regular F-finite R via tensoring with the canonical module omega_R, with injective dimension and support preserved.
Used to derive Corollary 2.7 and to pass from the unit Cartier bound (Corollary 3.4) to the unit Frobenius bound (Theorem 3.5).
domain assumption Unitalization u: Cart_A -> Cart^unit_A is exact, and the inclusion Cart^unit_A -> Cart_A is exact when A is regular; unit Cartier modules form an abelian category.
Used in Lemma 3.2 and Lemma 3.3 to relate injectivity in the category of unit Cartier modules to injectivity in the larger category of Cartier modules.

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Pith. "Pith review of On the injective dimension of unit Cartier and Frobenius modules." pith.science (2026). https://pith.science/paper/UTHU6LNH

@misc{pith2026241208423,
  author       = {Pith},
  title        = {Pith review of: On the injective dimension of unit Cartier and Frobenius modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTHU6LNH}},
  note         = {Machine review of arXiv:2412.08423}
}

abstract

Let $R$ be a regular $F$-finite ring of prime characteristic $p$. We prove that the injective dimension of every unit Frobenius module $M$ in the category of unit Frobenius modules is at most $\operatorname{dim}(\operatorname{Supp}_R(M))+1$. We further show that for unit Cartier modules the same bound holds over any noetherian $F$-finite ring $A$ of prime characteristic $p$. This shows that $\dim A+1$ is a uniform upper bound for the injective dimension of any unit Cartier module over a noetherian $F$-finite ring $A$.

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Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Alvarez-Montaner, M

J. Alvarez-Montaner, M. Blickle, and G. Lyubeznik : Generators of D -modules in positive characteristic , Math. Res. Lett. 12 (2005), no. 4, 459--473. MR2155224

work page 2005
[2]

Bhatt, M

B. Bhatt, M. Blickle, G. Lyubeznik, A. K. Singh, and W. Zhang : Local cohomology modules of a smooth Z -algebra have finitely many associated primes , Invent. Math. 197 (2014), no. 3, 509--519. MR3251828

work page 2014
[3]

Bhatt, J

B. Bhatt, J. Lurie : A Riemann - Hilbert correspondence in positive characteristic , Camb. J. Math. 7 (2019), no. 1-2, 71--217. MR3922360

work page 2019
[4]

Blickle : The D -module structure of R[F] -modules , Trans

M. Blickle : The D -module structure of R[F] -modules , Trans. Amer. Math. Soc. 355 (2003), no. 4, 1647--1668. MR1946409

work page 2003
[5]

Blickle : Minimal --sheaves, Algebra and Number Theory 2 (2008), no

M. Blickle : Minimal --sheaves, Algebra and Number Theory 2 (2008), no. 3, 347--368. MR2407119

work page 2008
[6]

Blickle and G

M. Blickle and G. B\" o ckle : Cartier modules: finiteness results, J. Reine Angew. Math. 661 (2011), 85--123. MR2863904

work page 2011
[7]

Cartier Crystals

M. Blickle and G. B\" o ckle : Cartier Crystals, arXiv:1309.1035, 2013

work page Pith review arXiv 2013
[8]

Cartier Crystals have finite global dimension

M. Blickle and D. Fink : Cartier crystals have finite global dimension, arXiv:2211.11466, 2022

work page Pith review arXiv 2022

Show all 28 references

[9]

Bhatt, M

B. Bhatt, M. Blickle. K. Schwede and K. Tucker : F -finite schemes have a canonical dualizing complex, pre-print, 2024

2024
[10]

Emerton and M

M. Emerton and M. Kisin : The R iemann- H ilbert correspondence for unit F -crystals , Ast\' e risque (2004), no. 293, vi+257. MR2071510

2004
[11]

Gabber : Notes on some t -structures , Geometric aspects of D work theory

O. Gabber : Notes on some t -structures , Geometric aspects of D work theory. V ol. I , II , Walter de Gruyter, Berlin, 2004, pp. 711--734. MR2099084

2004
[12]

Hartshorne and R

R. Hartshorne and R. Speiser : Local cohomological dimension in characteristic p , Ann. of Math. (2) 105 (1977), no. 1, 45--79. MR0441962 (56 \#353)

1977
[13]

Hochster : Some finiteness properties of L yubeznik's F -modules , Algebra, geometry and their interactions, Contemp

M. Hochster : Some finiteness properties of L yubeznik's F -modules , Algebra, geometry and their interactions, Contemp. Math., vol. 448, Amer. Math. Soc., Providence, RI, 2007, pp. 119--127. MR2389238

2007
[14]

C. L. Huneke and R. Y. Sharp : Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), no. 2, 765--779. MR1124167 (93m:13008)

1993
[15]

Katzman, L

M. Katzman, L. Ma, I. Smirnov, and W. Zhang : D -module and F -module length of local cohomology modules , Trans. Amer. Math. Soc. 370 (2018), no. 12, 8551--8580. MR3864387

2018
[16]

Katzman and W

M. Katzman and W. Zhang : The support of local cohomology modules, Int. Math. Res. Not. IMRN (2018), no. 23, 7137--7155. MR3920344

2018
[17]

Kunz : Characterizations of regular local rings for characteristic p , Amer

E. Kunz : Characterizations of regular local rings for characteristic p , Amer. J. Math. 91 (1969), 772--784. MR0252389 (40 \#5609)

1969
[18]

Lyubeznik : F -modules: applications to local cohomology and D -modules in characteristic p>0 , J

G. Lyubeznik : F -modules: applications to local cohomology and D -modules in characteristic p>0 , J. Reine Angew. Math. 491 (1997), 65--130. MR1476089 (99c:13005)

1997
[19]

Lyubeznik : On some local cohomology invariants of local rings, Math

G. Lyubeznik : On some local cohomology invariants of local rings, Math. Z. 254 (2006), no. 3, 627--640. MR2244370

2006
[20]

Lyubeznik : On the vanishing of local cohomology in characteristic p>0 , Compos

G. Lyubeznik : On the vanishing of local cohomology in characteristic p>0 , Compos. Math. 142 (2006), no. 1, 207--221. MR2197409 (2007b:13029)

2006
[21]

Lyubeznik, A

G. Lyubeznik, A. K. Singh, and U. Walther : Local cohomology modules supported at determinantal ideals, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 11, 2545--2578. MR3562351

2016
[22]

Ma : The category of F -modules has finite global dimension , J

L. Ma : The category of F -modules has finite global dimension , J. Algebra 402 (2014), 1--20. MR3160413

2014
[23]

Ma and W

L. Ma and W. Zhang : Eulerian graded D -modules , Math. Res. Lett. 21 (2014), no. 1, 149--167. MR3247047

2014
[24]

Peskine and L

C. Peskine and L. Szpiro : Dimension projective finie et cohomologie locale. Applications \` a la d\' e monstration de conjectures de M. Auslander, H. Bass et A. Grothendieck. Inst. Hautes \' E tudes Sci. Publ. Math. No. 42 (1973), 47--119. MR374130

1973
[25]

Switala and W

N. Switala and W. Zhang : A dichotomy for the injective dimension of F -finite F -modules and holonomic D -modules , Comm. Algebra 47 (2019), no. 6, 2525--2539. MR3957114

2019
[26]

Zhang : Lyubeznik numbers of projective schemes, Adv

W. Zhang : Lyubeznik numbers of projective schemes, Adv. Math. 228 (2011), no. 1, 575--616. MR2822240

2011
[27]

Zhang : On injective dimension of F -finite F -modules and holonomic D -modules , Bull

W. Zhang : On injective dimension of F -finite F -modules and holonomic D -modules , Bull. Lond. Math. Soc. 49 (2017), no. 4, 593--603. MR3725482

2017
[28]

Zhang : F^e -modules with applications to D -modules , J

W. Zhang : F^e -modules with applications to D -modules , J. Algebra 617 (2023), 340--351. MR4517004

2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Alvarez-Montaner, M

J. Alvarez-Montaner, M. Blickle, and G. Lyubeznik : Generators of D -modules in positive characteristic , Math. Res. Lett. 12 (2005), no. 4, 459--473. MR2155224

work page 2005

[2] [2]

Bhatt, M

B. Bhatt, M. Blickle, G. Lyubeznik, A. K. Singh, and W. Zhang : Local cohomology modules of a smooth Z -algebra have finitely many associated primes , Invent. Math. 197 (2014), no. 3, 509--519. MR3251828

work page 2014

[3] [3]

Bhatt, J

B. Bhatt, J. Lurie : A Riemann - Hilbert correspondence in positive characteristic , Camb. J. Math. 7 (2019), no. 1-2, 71--217. MR3922360

work page 2019

[4] [4]

Blickle : The D -module structure of R[F] -modules , Trans

M. Blickle : The D -module structure of R[F] -modules , Trans. Amer. Math. Soc. 355 (2003), no. 4, 1647--1668. MR1946409

work page 2003

[5] [5]

Blickle : Minimal --sheaves, Algebra and Number Theory 2 (2008), no

M. Blickle : Minimal --sheaves, Algebra and Number Theory 2 (2008), no. 3, 347--368. MR2407119

work page 2008

[6] [6]

Blickle and G

M. Blickle and G. B\" o ckle : Cartier modules: finiteness results, J. Reine Angew. Math. 661 (2011), 85--123. MR2863904

work page 2011

[7] [7]

Cartier Crystals

M. Blickle and G. B\" o ckle : Cartier Crystals, arXiv:1309.1035, 2013

work page Pith review arXiv 2013

[8] [8]

Cartier Crystals have finite global dimension

M. Blickle and D. Fink : Cartier crystals have finite global dimension, arXiv:2211.11466, 2022

work page Pith review arXiv 2022