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arxiv: 2412.10226 · v4 · submitted 2024-12-13 · 🧮 math.NT · math.AG

An algebraicity conjecture of Drinfeld and the moduli of p-divisible groups

Zachary Gardner , Keerthi Madapusi This is my paper

Pith reviewed 2026-05-23 07:28 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords moduli stacksp-divisible groupsprismatic technologyalgebraicityp-adic basesF-gaugesalgebraic geometry
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The pith

A uniform group-theoretic construction produces smooth stacks for p-divisible groups and verifies an algebraicity conjecture.

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

The paper gives a uniform construction of smooth stacks attached to smooth affine group schemes over the p-adic integers and bounded cocharacters using stacky prismatic technology. This verifies a conjecture on the algebraicity of such stacks. For general linear groups with minuscule cocharacters, these stacks are shown to be isomorphic to the moduli stack of truncated p-divisible groups of given height and dimension. This provides a linear algebraic classification over general p-adic bases. The work also establishes algebraicity for the stack of perfect F-gauges with Hodge-Tate weights zero and one at level n.

Core claim

Using stacky prismatic technology and an animated variant of higher frames, the paper constructs smooth stacks attached to a smooth affine group scheme over the p-adic integers and a one-bounded cocharacter. These stacks are representable when their tangent complexes are one-bounded. In the special case of the general linear group and a minuscule cocharacter, the stacks coincide with the moduli stack of truncated p-divisible groups, yielding a classification of such groups over very general p-adic bases. As an application, algebraicity is proved for the stack of perfect F-gauges of Hodge-Tate weights zero and one and level n.

What carries the argument

The stacky prismatic technology applied to animated higher frames, which constructs the smooth stacks and proves their representability properties.

If this is right

  • The constructed stacks are smooth for arbitrary smooth affine group schemes and one-bounded cocharacters.
  • The stacks classify truncated p-divisible groups when specialized to the general linear group with minuscule cocharacters.
  • Algebraicity holds for the stack of perfect F-gauges of Hodge-Tate weights 0 and 1 at level n.
  • The methods apply to a wide range of stacks with one-bounded tangent complexes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may allow uniform treatment of other moduli problems involving p-divisible groups.
  • Extensions could connect this classification to the geometry of local Shimura varieties.
  • The representability results might apply to stacks arising in global arithmetic geometry.

Load-bearing premise

The stacky prismatic technology can be applied verbatim together with the animated variant of higher frames to produce the required smooth stacks and their representability properties for arbitrary smooth affine group schemes and one-bounded cocharacters.

What would settle it

A computation showing that for some smooth affine group scheme over the p-adic integers and one-bounded cocharacter the associated stack is not smooth or fails to be isomorphic to the moduli stack of truncated p-divisible groups in the general linear case.

read the original abstract

We use the newly developed stacky prismatic technology of Drinfeld and Bhatt-Lurie to give a uniform, group-theoretic construction of smooth stacks $\mathrm{BT}^{G,\mu}_{n}$ attached to a smooth affine group scheme $G$ over $\mathbb{Z}_p$ and $1$-bounded cocharacter $\mu$, verifying a recent conjecture of Drinfeld. This can be viewed as a refinement of results of B\"ultel-Pappas, who gave a related construction using $(G,\mu)$-displays defined via rings of Witt vectors. We show that, when $G = \mathrm{GL}_h$ and $\mu$ is a minuscule cocharacter, these stacks are isomorphic to the stack of truncated $p$-divisible groups of height $h$ and dimension $d$ (the latter depending on $\mu$). This gives a generalization of results of Ansch\"utz-Le Bras, yielding a linear algebraic classification of $p$-divisible groups over very general $p$-adic bases, and verifying another conjecture of Drinfeld. The proofs use deformation techniques from derived algebraic geometry, combined with an animated variant of Lau's theory of higher frames and displays, and -- with a view towards applications to the study of local and global Shimura varieties -- actually prove representability results for a wide range of stacks whose tangent complexes are $1$-bounded in a suitable sense. As an immediate application, we prove algebraicity for the stack of perfect $F$-gauges of Hodge-Tate weights $0,1$ and level $n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs smooth stacks BT^{G,μ}_n attached to a smooth affine group scheme G over Z_p and a 1-bounded cocharacter μ, using the stacky prismatic formalism of Drinfeld-Bhatt-Lurie together with an animated variant of Lau's theory of higher frames and displays. This yields a uniform group-theoretic construction that verifies a recent algebraicity conjecture of Drinfeld. When G=GL_h and μ is minuscule, the stacks are shown isomorphic to the stack of truncated p-divisible groups of height h and dimension d (depending on μ), generalizing Anschütz-Le Bras. The work also establishes representability for a broader class of stacks whose tangent complexes are 1-bounded and applies this to prove algebraicity of the stack of perfect F-gauges of Hodge-Tate weights 0,1 and level n.

Significance. If the central claims hold, the paper supplies a significant uniform construction in p-adic geometry that refines Bültel-Pappas displays and verifies two conjectures of Drinfeld. The deformation-theoretic arguments in derived algebraic geometry, combined with the prismatic and animated-frame technology, produce representability results applicable to local and global Shimura varieties. The explicit comparison in the GL_h case and the algebraicity statement for F-gauges are concrete advances.

minor comments (2)
  1. [§1] §1, paragraph following the statement of the main theorem: the phrase '1-bounded in a suitable sense' is used without an immediate cross-reference to the precise definition of 1-bounded tangent complexes that appears later; adding the reference would improve readability.
  2. [Introduction] The comparison isomorphism for G=GL_h is stated in the abstract and introduction but the precise statement of the isomorphism (including the dependence of dimension d on μ) would benefit from an explicit equation or theorem label in the body.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance, and recommendation to accept. The report accurately captures the main results on the uniform construction of the stacks BT^{G,μ}_n, the verification of Drinfeld's algebraicity conjecture, the comparison with truncated p-divisible groups in the GL_h case, and the broader representability statements.

Circularity Check

0 steps flagged

No significant circularity; construction applies external prismatic technology

full rationale

The derivation applies the stacky prismatic formalism of Drinfeld-Bhatt-Lurie together with an animated variant of Lau's higher frames to define the stacks BT^{G,μ}_n for general smooth affine G and 1-bounded μ. Smoothness and representability follow from deformation-theoretic arguments on 1-bounded tangent complexes in derived algebraic geometry; the GL_h case recovers the truncated p-divisible group stack by explicit comparison of displays. No equations or steps reduce the output stacks to quantities defined from the input data by fitting, self-definition, or load-bearing self-citation. The central claim is therefore an application of independent external technology rather than a closed loop internal to the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work applies existing prismatic and derived-algebraic-geometry frameworks; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math The stacky prismatic site and its properties as developed by Drinfeld and Bhatt-Lurie are available and functorial for the required group schemes.
    Invoked to produce the stacks BT^{G,μ}_n.
  • standard math Deformation theory in derived algebraic geometry applies to the tangent complexes that are 1-bounded.
    Used to prove representability for a wide range of stacks.

pith-pipeline@v0.9.0 · 5822 in / 1345 out tokens · 28026 ms · 2026-05-23T07:28:38.974436+00:00 · methodology

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Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages · 3 internal anchors

  1. [1]

    Prismatic Dieudonn´ e theory

    Johannes Ansch¨ utz and Arthur-C´ esar Le Bras. “Prismatic Dieudonn´ e theory”. In:Forum Math. Pi 11 (2023), Paper No. e2, 92

    work page 2023

  2. [2]

    Duality in the flat cohomology of curves

    M. Artin and J. S. Milne. “Duality in the flat cohomology of curves”. In: Invent. Math. 35 (1976), pp. 111–129

    work page 1976

  3. [3]

    G-µ -displays and local shtuka

    Sebastian Bartling. “ G-µ -displays and local shtuka”. In: (2022). url: https://arxiv.org/abs/2206.13194

    work page arXiv 2022

  4. [4]

    Th´ eorie de Dieudonn´ e cristalline

    Pierre Berthelot, Lawrence Breen, and William Messing. Th´ eorie de Dieudonn´ e cristalline. II. Vol. 930. Lecture Notes in mathematics. Berlin: Springer-Verlag, 1982

    work page 1982

  5. [5]

    Notes on crystalline cohomology

    Pierre Berthelot and Arthur Ogus. Notes on crystalline cohomology . Princeton, N.J.: Princeton University Press, 1978

    work page 1978

  6. [6]

    PrismaticF -gauges

    Bhargav Bhatt. PrismaticF -gauges. url: https://www.math.ias.edu/~bhatt/teaching/mat549f22/lectures.pdf

    work page

  7. [7]

    Tannaka duality r evisited

    Bhargav Bhatt and Daniel Halpern-Leistner. “Tannaka duality r evisited”. In: Advances in Mathematics 316 (2017), pp. 576–612

    work page 2017

  8. [8]

    Absolute prismatic cohomology

    Bhargav Bhatt and Jacob Lurie. “Absolute prismatic cohomology ”. In: (2022). eprint: 2201.06120. url: https://arxiv.org/abs/2201.06120

    work page arXiv 2022

  9. [9]

    The prismatization of p-adic formal schemes

    Bhargav Bhatt and Jacob Lurie. “The prismatization of p-adic formal schemes”. In: (2022). eprint: 2201.06124. url: https://arxiv.org/abs/2201.06124

    work page arXiv 2022

  10. [10]

    Prismatic F -crystals and crystalline Galois representations

    Bhargav Bhatt and Peter Scholze. “Prismatic F -crystals and crystalline Galois representations”. In: Camb. J. Math. 11.2 (2023), pp. 507–562

    work page 2023

  11. [11]

    Prisms and prismatic cohom ology

    Bhargav Bhatt and Peter Scholze. “Prisms and prismatic cohom ology”. In: Ann. of Math. 196.3 (Nov. 2022), pp. 1135–1275

    work page 2022

  12. [12]

    N´ eron models

    Siegfried Bosch, Werner L¨ utkebohmert, and Michel Raynaud . N´ eron models. Vol. 21. Ergebnisse der Mathe- matik und ihrer Grenzgebiete. Berlin: Springer-Verlag, 1990

    work page 1990

  13. [13]

    Representability of cohomolog y of finite flat abelian group schemes

    Daniel Bragg and Martin Olsson. “Representability of cohomolog y of finite flat abelian group schemes”. In: (2021). eprint: 2107.11492

    work page arXiv 2021

  14. [14]

    ( G,µ )-Windows and Deformations of (G,µ )-Displays

    Oliver Bueltel and Mohammad Hadi Hedayatzadeh. ( G,µ )-Windows and Deformations of (G,µ )-Displays

    work page

  15. [15]

    url: https://arxiv.org/abs/2011.09163

    work page arXiv 2011

  16. [16]

    ( G,µ )-displays and Rapoport-Zink spaces

    O. B¨ ultel and G. Pappas. “( G,µ )-displays and Rapoport-Zink spaces”. In: J. Inst. Math. Jussieu 19.4 (2020), pp. 1211–1257

    work page 2020

  17. [17]

    R andom Dieudonn ˜A© modules, random p- divisible groups, and random curves over finite fields

    Bryden Cais, Jordan S. Ellenberg, and David Zureick-Brown. “R andom Dieudonn ˜A© modules, random p- divisible groups, and random curves over finite fields”. In: Journal of the Institute of Mathematics of Jussieu 12.3 (2013)

    work page 2013

  18. [18]

    Finite locally free group schemes in characterist ic p and Dieudonn´ e modules

    A. J. de Jong. “Finite locally free group schemes in characterist ic p and Dieudonn´ e modules”. In: Invent. Math. 114.1 (1993), pp. 89–137

    work page 1993

  19. [19]

    Homomorphisms of Barsotti-Tate groups and cry stals in positive characteristic

    A J de Jong. “Homomorphisms of Barsotti-Tate groups and cry stals in positive characteristic”. In: Invent. Math. 134.2 (Oct. 1998), pp. 301–333

    work page 1998

  20. [20]

    Crystalline Dieudonn´ e module theory via fo rmal and rigid geometry

    Aise Johan de Jong. “Crystalline Dieudonn´ e module theory via fo rmal and rigid geometry”. In: Inst. Hautes ´Etudes Sci. Publ. Math. 82.1 (1995), pp. 5–96

    work page 1995

  21. [21]

    Homologie nicht-additiver Funk toren. Anwendungen

    Albrecht Dold and Dieter Puppe. “Homologie nicht-additiver Funk toren. Anwendungen”. In: Ann. Inst. Fourier (Grenoble) 11 (1961), pp. 201–312. REFERENCES 135

    work page 1961

  22. [22]

    On algebraic spaces with an action of G_m

    Vladimir Drinfeld. “On algebraic spaces with an action of Gm”. In: (2015). url: https://arxiv.org/abs/1308.2604

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  23. [23]

    On Shimurian generalizations of the stack BT1⊗Fp

    Vladimir Drinfeld. “On Shimurian generalizations of the stack BT1⊗Fp”. In: (2023). arXiv: 2304.11709 [math.AG]

    work page arXiv 2023

  24. [24]

    On the Lau group scheme

    Vladimir Drinfeld. “On the Lau group scheme”. In: (2024). url: https://arxiv.org/abs/2307.06194

    work page arXiv 2024

  25. [25]

    Prismatization

    Vladimir Drinfeld. “Prismatization”. In: (2022). eprint: 2005.04746. url: https://arxiv.org/abs/2005.04746

    work page arXiv 2022

  26. [26]

    Toward Shimurian analogs of Barsotti-Tate gr oups

    Vladimir Drinfeld. “Toward Shimurian analogs of Barsotti-Tate gr oups”. In: (2024). url: https://arxiv.org/abs/2309.023

    work page 2024

  27. [27]

    Group Schemes with Strict O-action

    G Faltings. “Group Schemes with Strict O-action”. In: Mosc. Math. J. 2.2 (2002), pp. 249–279

    work page 2002

  28. [28]

    Integral crystalline cohomology over very ram ified valuation rings

    Gerd Faltings. “Integral crystalline cohomology over very ram ified valuation rings”. In: J. Amer. Math. Soc. 12.1 (1999), pp. 117–144

    work page 1999

  29. [29]

    Groupesp-divisibles sur les corps locaux

    Jean-Marc Fontaine. Groupesp-divisibles sur les corps locaux. Vol. No. 47-48. Ast´ erisque. Soci´ et´ e Math´ ematique de France, Paris, 1977, pp. i+262

    work page 1977

  30. [30]

    Sch´ emas en groupes (SGA 3)

    Philippe Gille and Patrick Polo, eds. Sch´ emas en groupes (SGA 3). Tome I. Propri´ et´ es g´ en´ erales des sch´ emas en groupes. annotated. Vol. 7. Documents Math´ ematiques (Paris) [Mathematical Documents (Paris)]. S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois Marie 1962–64. [Algebraic Geometry Seminar of Bois Marie 1962–64], A seminar directed by ...

    work page 1962

  31. [31]

    Frobenius height of prismatic coh omology with coefficients

    Haoyang Guo and Shizhang Li. “Frobenius height of prismatic coh omology with coefficients”. In: (2023). url: https://arxiv.org/abs/2309.06663

    work page arXiv 2023

  32. [32]

    Mapping stacks and categorical notions of properness

    Daniel Halpern-Leistner and Anatoly Preygel. “Mapping stacks and categorical notions of properness”. In: Compos. Math. 159.3 (2023), pp. 530–589. issn: 0010-437X,1570-5846. doi: 10.1112/S0010437X22007667. url: https://doi.org/10.1112/S0010437X22007667

    work page doi:10.1112/s0010437x22007667 2023

  33. [33]

    Deformations of Prismatic Higher $(G,\mu)$-Displays over Quasi-Syntomic Rings

    S. Mohammad Hadi Hedayatzadeh and Ali Partofard. “Deform ations of Prismatic Higher (G,µ )-Displays over Quasi-Syntomic Rings”. In: (2024). url: https://arxiv.org/abs/2402.12879

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  34. [34]

    Derived δ-Rings and Relative Prismatic Cohomology

    Adam Holeman. “Derived δ-Rings and Relative Prismatic Cohomology”. In: (2023). eprint: 2303.17447. url: https://arxiv.org/abs/2303.17447

    work page arXiv 2023

  35. [35]

    D´ eformations de groupes de Barsotti-Tate (d’ap r` es A. Grothendieck)

    Luc Illusie. “D´ eformations de groupes de Barsotti-Tate (d’ap r` es A. Grothendieck)”. In: 127. Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84). 19 85, pp. 151–198

    work page 1983

  36. [36]

    The prismatic realizat ion functor for Shimura varieties of abelian type

    Naoki Imai, Hiroki Kato, and Alex Youcis. “The prismatic realizat ion functor for Shimura varieties of abelian type”. In: (2023). url: https://arxiv.org/abs/2310.08472

    work page arXiv 2023

  37. [37]

    Deformation theory for prismatic G-displays

    Kazuhiro Ito. “Deformation theory for prismatic G-displays”. In: (2023). eprint: 2306.05361. url: https://arxiv.org/abs

    work page arXiv 2023

  38. [38]

    Prismatic G-displays and descent theory

    Kazuhiro Ito. Prismatic G-displays and descent theory . 2024. url: https://arxiv.org/abs/2303.15814

    work page arXiv 2024

  39. [39]

    The classification of p-divisible groups over 2-adic discrete valuation rings

    Wansu Kim. “The classification of p-divisible groups over 2-adic discrete valuation rings”. In: Math. Res. Lett. 19.1 (2012), pp. 121–141

    work page 2012

  40. [40]

    Crystalline representations and F-crystals

    Mark Kisin. “Crystalline representations and F-crystals”. In: Algebraic Geometry and Number Theory. Vol. 253. Progress in Mathematics. Birkh¨ auser Boston, 2006, pp. 459–49 6

    work page 2006

  41. [41]

    Integral models for Shimura varieties of abelian typ e

    Mark Kisin. “Integral models for Shimura varieties of abelian typ e”. In: J. Amer. Math. Soc. 23.4 (2010), pp. 967–1012

    work page 2010

  42. [42]

    Dieudonn´ e theory over semiperfect rings and perf ectoid rings

    Eike Lau. “Dieudonn´ e theory over semiperfect rings and perf ectoid rings”. In: Compos. Math. 154.9 (2018), pp. 1974–2004

    work page 2018

  43. [43]

    Divided Dieudonn\'e crystals

    Eike Lau. “Divided Dieudonn´ e crystals”. In: (2018). url: https://arxiv.org/abs/1811.09439

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  44. [44]

    Higher frames and G-displays

    Eike Lau. “Higher frames and G-displays”. In: Algebra Number Theory 15.9 (2021), pp. 2315–2355

    work page 2021

  45. [45]

    Relations between Dieudonn´ e displays and crystalline Dieudonn´ e theory

    Eike Lau. “Relations between Dieudonn´ e displays and crystalline Dieudonn´ e theory”. In: Algebra Number Theory 8.9 (2014), pp. 2201–2262

    work page 2014

  46. [46]

    Smoothness of the truncated display functor

    Eike Lau. “Smoothness of the truncated display functor”. In : J. Amer. Math. Soc. 26.1 (2013), pp. 129–165

    work page 2013

  47. [47]

    p-isogenies with G-structure and their applications

    Si Ying Lee and Keerthi Madapusi. “ p-isogenies with G-structure and their applications”. In: (2025). in preparation

    work page 2025

  48. [48]

    Derived algebraic geometry

    Jacob Lurie. Derived algebraic geometry . Thesis (Ph.D.)–Massachusetts Institute of Technology. ProQue st LLC, Ann Arbor, MI, 2004, (no paging)

    work page 2004

  49. [49]

    Higher Algebra

    Jacob Lurie. Higher Algebra. 2017. url: https://www.math.ias.edu/~lurie/papers/HA.pdf

    work page 2017

  50. [50]

    Higher Topos Theory

    Jacob Lurie. Higher Topos Theory. en. Princeton University Press, July 2009. url: https://www.degruyter.com/document 136 REFERENCES

    work page 2009

  51. [51]

    Spectral algebraic geometry

    Jacob Lurie. “Spectral algebraic geometry”. In: preprint (2018). url: https://www.math.ias.edu/~lurie/papers/SAG-ro

    work page 2018

  52. [52]

    Ambidexterity in K(n)-local stable homotopy theory

    Jacob Lurie and Mike Hopkins. “Ambidexterity in K(n)-local stable homotopy theory”. In: (2013). url: https://people.math.harvard.edu/~lurie/papers/Ambidexterity.pdf

    work page 2013

  53. [53]

    Derived special cycles on Shimura varieties

    Keerthi Madapusi. “Derived special cycles on Shimura varieties ”. In: (Dec. 2022). arXiv: 2212.12849 [math.NT] . url: http://arxiv.org/abs/2212.12849

    work page arXiv 2022

  54. [54]

    Ordinary loci: a stack theoretic approach

    Keerthi Madapusi. “Ordinary loci: a stack theoretic approach ”. In: (2025). in preparation

    work page 2025

  55. [55]

    Finite flat group sch emes and F -gauges

    Keerthi Madapusi and Shubhodip Mondal. “Finite flat group sch emes and F -gauges”. In: (2024)

    work page 2024

  56. [56]

    Theory of commutative formal groups over field s of finite characteristic

    Ju. I. Manin. “Theory of commutative formal groups over field s of finite characteristic”. In: Uspehi Mat. Nauk 18.6(114) (1963), pp. 3–90

    work page 1963

  57. [57]

    Revisiting derived crystalline cohomology

    Zhouhang Mao. “Revisiting derived crystalline cohomology”. In: (July 2021). arXiv: 2107.02921 [math.AG] . url: http://arxiv.org/abs/2107.02921

    work page arXiv 2021

  58. [58]

    Prismatic F -crystals and Lubin-Tate (ϕq, Γ)-modules

    Samuel Marks. “Prismatic F -crystals and Lubin-Tate (ϕq, Γ)-modules”. In: (2023). eprint: 2303.07620. url: https://arxiv.org/abs/2303.07620

    work page arXiv 2023

  59. [59]

    Dieudonn´ e theory via cohomology of classif ying stacks II

    Shubhodip Mondal. “Dieudonn´ e theory via cohomology of classif ying stacks II”. In: (2024). url: https://personal.math.ub

    work page 2024

  60. [60]

    Group schemes with additional structures and W eyl group cosets

    Ben Moonen. “Group schemes with additional structures and W eyl group cosets”. In: Moduli of Abelian Vari- eties. Basel: Birkh ˜A¤user Basel, 2001, pp. 255–298. url: http://dx.doi.org/10.1007/978-3-0348-8303-0_10

    work page doi:10.1007/978-3-0348-8303-0_10 2001

  61. [61]

    Serre-Tate theory for moduli spaces of PEL ty pe

    Ben Moonen. “Serre-Tate theory for moduli spaces of PEL ty pe”. In: Ann. Sci. Ec. Norm. Super. 37.2 (Mar. 2004), pp. 223–269

    work page 2004

  62. [62]

    The geometry of filtrations

    Tasos Moulinos. “The geometry of filtrations”. In: Bull. Lond. Math. Soc. 53.5 (2021), pp. 1486–1499

    work page 2021

  63. [63]

    Newton polygons and p-divisible groups: a conjecture by Grothendieck

    Frans Oort. “Newton polygons and p-divisible groups: a conjecture by Grothendieck”. In: 298. Autom orphic forms. I. 2005, pp. 255–269

    work page 2005

  64. [64]

    F-zips with a dditional structure

    Richard Pink, Torsten Wedhorn, and Paul Ziegler. “F-zips with a dditional structure”. In: Pacific J. Math. 274.1 (Mar. 2015), pp. 183–236. url: http://msp.org/pjm/2015/274-1/p09.xhtml

    work page 2015

  65. [65]

    The Stacks project

    The Stacks project authors. The Stacks project . https://stacks.math.columbia.edu. 2023

    work page 2023

  66. [66]

    Moduli of objects in dg- categories

    Bertrand To¨ en and Michel Vaqui´ e. “Moduli of objects in dg- categories”. In: Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure40.3 (2007), pp. 387–444. issn: 0012-9593

    work page 2007

  67. [67]

    Homotopical algebraic geometry

    Bertrand To¨ en and Gabriele Vezzosi. Homotopical algebraic geometry. II. Geometric stacks and a pplications. Vol. 193. 2008, pp. 0–0. url: http://dx.doi.org/10.1090/memo/0902

    work page doi:10.1090/memo/0902 2008

  68. [68]

    Derived F -zips

    Can Yaylali. “Derived F -zips”. In: ´Epijournal de G´ eom´ etrie Alg´ ebriqueVolume 8, 5 (2024). issn: 2491-6765. doi: 10.46298/epiga.2024.10375. url: https://epiga.episciences.org/10375

    work page doi:10.46298/epiga.2024.10375 2024

  69. [69]

    Windows for displays of p-divisible groups

    Thomas Zink. “Windows for displays of p-divisible groups”. In: Moduli of abelian varieties (Texel Island, 1999). Vol. 195. Progr. Math. Birkh¨ auser, Basel, 2001, pp. 491–518. Zachary Gardner, Department of Mathematics, Maloney Hall, B oston College, Chestnut Hill, MA 02467, USA Email address : gardneza@bc.edu Keerthi Madapusi, Department of Mathematics, ...

    work page 1999