Geodesic
Prismatic Dieudonné Theory
Forum of Mathematics, Pi, Tome 11 (2023)

Voir la notice de l'article provenant de la source Cambridge University Press

We define, for each quasisyntomic ring R (in the sense of Bhatt et al., Publ. Math. IHES 129 (2019), 199–310), a category $\mathrm {DM}^{\mathrm {adm}}(R)$ of admissible prismatic Dieudonné crystals over R and a functor from p-divisible groups over R to $\mathrm {DM}^{\mathrm {adm}}(R)$. We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze. 
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     author = {Johannes Ansch\"utz and Arthur-C\'esar Le Bras},
     title = {Prismatic {Dieudonn\'e} {Theory}},
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Johannes Anschütz; Arthur-César Le Bras. Prismatic Dieudonné Theory. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2022.22

[1] Anschütz, J. and Le Bras, A.-C., ‘The -completed cyclotomic trace in degree 2’, Ann. K-theory 5(3) (2020), 539–580.Google Scholar | DOI

[2] Avramov, L. L., ‘Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology’, Ann. of Math. (2), 150(2) (1999), 455–487.Google Scholar | DOI

[3] Bauer, W., ‘On the conjecture of Birch and Swinnerton-Dyer for abelian varieties over function fields in characteristic ’, Inventiones mathematicae 108(1) (1992), 263–287.Google Scholar | DOI

[4] Berthelot, P., ‘Théorie de Dieudonné sur un anneau de valuation parfait’, Ann. Sci. École Norm. Sup. (4), 13(2) (1980), 225–268.Google Scholar | DOI

[5] Berthelot, P., ‘Cohomologie cristalline des schémas de caractéristique ’, in Lecture Notes in Mathematics, vol. 407 (Springer, Berlin, 2006).0$+’,+in+Lecture+Notes+in+Mathematics,+vol.+407+(Springer,+Berlin,+2006).>Google Scholar

[6] Berthelot, P., Breen, L. and Messing, W., ‘Théorie de Dieudonné cristalline. II’, in Lecture Notes in Mathematics, vol. 930 (Springer-Verlag, Berlin, 1982).Google Scholar

[7] Berthelot, P. and Messing, W., ‘Théorie de Dieudonné cristalline. III. Théorèmes d’équivalence et de pleine fidélité’, in A collection of articles written in honor of the 60th birthday of Alexander Grothendieck, modern Birkhäuser classics, The Grothendieck Festschrift, Vol. I (Birkhäuser Boston, Boston, MA, 1990), 173–247.Google Scholar

[8] Bhatt, B., ‘Lectures on prismatic cohomology’, (2018). URL: http://www-personal.umich.edu/bhattb/teaching/prismatic-columbia/.Google Scholar

[9] Bhatt, B., Iyengar, S. B. and Ma, L., ‘Regular rings and perfect (oid) algebras’, Preprint, 2018, .Google Scholar | arXiv | DOI

[10] Bhatt, B. and Lurie, J., ‘Absolute prismatic cohomology’, Preprint, 2022, .Google Scholar | arXiv | DOI

[11] Bhatt, B., Morrow, M. and Scholze, P., ‘Integral -adic Hodge theory’, Publ. Math. l’IHÉS 128(1) (2018), 219–397.Google Scholar | DOI

[12] Bhatt, B., Morrow, M. and Scholze, P., ‘Topological Hochschild homology and integral -adic Hodge theory’, Publ. Math. IHES 129 (2019), 199–310. URL: .Google Scholar | DOI

[13] Bhatt, B. and Scholze, P., ‘Prisms and prismatic cohomology’. URL: http://www.math.uni-bonn.de/people/scholze/Publikationen.html.Google Scholar

[14] Bhatt, B. and Scholze, P., ‘Prismatic -crystals and crystalline Galois representations’, Preprint, 2021, .Google Scholar | arXiv

[15] Breuil, C., ‘Schémas en groupes et corps de normes’, 13 pages, 1998. URL: https://www.imo.universite-paris-saclay.fr/~christophe.breuil/publications.html.Google Scholar

[16] Breuil, C., ‘Groupes -divisibles, groupes finis et modules filtrés’, Ann. Math.-Second Series 152(2) (2000), 489–550.Google Scholar | DOI

[17] Cais, B. and Lau, E., ‘Dieudonné crystals and Wach modules for -divisible groups’, Preprint, 2014, .Google Scholar | arXiv

[18] Cesnavic̆Ius, K. and Scholze, P., ‘Purity for flat cohomology’, In preparation.Google Scholar

[19] Clausen, D., Mathew, A. and Morrow, M., ‘K-theory and topological cyclic homology of Henselian pairs’, Preprint, 2018, .Google Scholar | arXiv

[20] De Jong, A. J., ‘Finite locally free group schemes in characteristic and Dieudonné modules’, Invent. Math. 114(1) (1993), 89–137.Google Scholar | DOI

[21] Elkik, R., ‘Solutions d’équations à coefficients dans un anneau hensélien’, Ann. Sci. École Norm. Sup. 6(4) (1973), 553–603.Google Scholar | DOI

[22] Fargues, L., ‘Quelques résultats et conjectures concernant la courbe’, Preprint, 2015, URL: https://webusers.imj-prg.fr/laurent.fargues/AuDela.pdf.Google Scholar

[23] Fontaine, J.-M., ‘Groupes -divisibles sur les corps locaux’, Société Mathématique de France, Paris, Astérisque, nos. 47–48 1977.Google Scholar

[24] Illusie, L., ‘Complexe cotangent et déformations. I ,’ in Lecture Notes in Mathematics, vol. 239 (Springer-Verlag, Berlin-New York, 1971).Google Scholar

[25] Illusie, L., ‘Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck),’ Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84). Astérisque 127 (1985), 151–198.Google Scholar

[26] Joyal, A., ‘-anneaux et vecteurs de Witt’, C. R. Math. Rep. Acad. Sci. Canada 7(3) (1985), 177–182.Google Scholar

[27] Katz, N. M., ‘-adic properties of modular schemes and modular forms’, in Lecture Notes in Mathematics, vol. 350 (Springer, Berlin, Heidelberg, 1973), 69–190.Google Scholar | DOI

[28] Kedlaya, K. S. and Liu, R., ‘Relative -adic Hodge theory: Foundations’, Preprint, 2013, .Google Scholar | arXiv

[29] Kim, W., ‘The classification of -divisible groups over 2-adic discrete valuation rings’, Preprint, 2010, .Google Scholar | arXiv

[30] Kisin, M., ‘Crystalline representations and -crystals’, in Algebraic geometry and number theory, Progress in Mathematics, vol. 253 (Birkhäuser Boston, Boston, MA, 2006), 459–496.Google Scholar | DOI

[31] Kunz, E., ‘Characterizations of regular local rings of characteristic ’, Am. J. Math. 91(3) (1969), 772–784.Google Scholar | DOI

[32] Lau, E., ‘Displays and formal -divisible groups’, Invent. Math. 171(3), (2008), 617–628.Google Scholar | DOI

[33] Lau, E., ‘Frames and finite group schemes over complete regular local rings’, Preprint, 2009, .Google Scholar | arXiv

[34] Lau, E., ‘Smoothness of the truncated display functor’, J. Amer. Math. Soc. 26(1) (2013), 129–165.Google Scholar | DOI

[35] Lau, E., ‘Relations between Dieudonné displays and crystalline Dieudonné theory’, Algebra number theory (Online) 8(9) (2014), 2201–2262.Google Scholar | DOI

[36] Lau, E., ‘Dieudonné theory over semiperfect rings and perfectoid rings’, Preprint, 2016, .Google Scholar | arXiv

[37] Lau, E., ‘Divided Dieudonné crystals’, Preprint, 2018, .Google Scholar | arXiv

[38] Liu, T., ‘The correspondence between Barsotti-Tate groups and Kisin modules when ’, J. Théor. Nr. Bordx. 25(3) (2013), 661–676.Google Scholar | DOI

[39] Lurie, J., ‘Higher algebra’. 2017, URL: http://www.math.harvard.edu/lurie/.Google Scholar

[40] Lurie, J., Higher Topos Theory (AM-170), vol. 189 (Princeton University Press, 2009).Google Scholar | DOI

[41] Lurie, J. A.. Elliptic cohomology I: Spectral abelian varieties. Preprint, 2017.Google Scholar

[42] Matsumura, H., Commutative Ring Theory, vol. 8 (Cambridge University Press, Cambridge, 1989).Google Scholar

[43] Mazur, B. and Messing, W., ‘Universal extensions and one dimensional crystalline cohomology’, in Lecture Notes in Mathematics, vol. 370 (Springer-Verlag, Berlin-New York, 1974).Google Scholar

[44] Messing, W., ‘The crystals associated to Barsotti-Tate groups’, in The crystals associated to Barsotti-Tate groups: With applications to abelian schemes (Springer, 1972), 112–149.Google Scholar | DOI

[45] Mondal, S., ‘Dieudonné theory via cohomology of classifying stacks’, in Forum of Mathematics, Sigma, vol. 9 (Cambridge University Press, Cambridge, 2021), 1–25.Google Scholar

[46] Mondal, S., ‘A computation of prismatic Dieudonné module’, Preprint, 2022, URL: http://www-personal.umich.edu/~smondal/.Google Scholar

[47] Oort, F., ‘Finite group schemes, local moduli for abelian varieties, and lifting problems’, Compositio Math. 23 (1971), 265–296.Google Scholar

[48] Scholze, P., ‘Lectures on condensed mathematics’, 2019, URL: https://www.math.uni-bonn.de/people/scholze/Condensed.pdf.Google Scholar

[49] Scholze, P., ‘Étale cohomology of diamonds’, Preprint, 2017, .Google Scholar | arXiv | DOI

[50] Scholze, P. and Weinstein, J., ‘Moduli of -divisible groups’, Camb. J. Math. 1(2) (2013), 145–237.Google Scholar | DOI

[51] Scholze, P. and Weinstein, J., Berkeley Lectures on -adic Geometry, vol. 207 (Princeton University Press, Princeton, 2020).Google Scholar

[52] The Stacks Project Authors, Project, Stacks. URL: http://stacks.math.columbia.edu, 2017.Google Scholar

[53] Waterhouse, W. C., ‘Principal homogeneous spaces and group scheme extensions’, Trans. Am. Math. Soc. 153 1971, 181–189.Google Scholar | DOI

[54] Zink, T., ‘The display of a formal -divisible group’, Astérisque 278 (2002), 127–248. Cohomologies -adiques et applications arithmétiques, I.Google Scholar

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