Abstract
Let 𝑋 be a quasi-compact quasi-separated 𝑝-adic formal scheme that is smooth either over a perfectoid
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 444845124
Funding statement: The second author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 444845124 – TRR 326.
Acknowledgements
We would like to thank Bhargav Bhatt, Hui Gao, Tongmu He, Juan Esteban Rodríguez Camargo, Peter Scholze, Yupeng Wang, Matti Würthen and Bogdan Zavyalov for helpful discussions. We would like to thank the referee for their close reading and very helpful and detailed comments.
References
[1] A. Abbes, M. Gros and T. Tsuji, The 𝑝-adic Simpson correspondence, Ann. of Math. Stud. 193, Princeton University, Princeton 2016. 10.23943/princeton/9780691170282.001.0001Suche in Google Scholar
[2] Y. André, La conjecture du facteur direct, Publ. Math. Inst. Hautes Études Sci. 127 (2018), 71–93. 10.1007/s10240-017-0097-9Suche in Google Scholar
[3] G. Andreychev, Pseudocoherent and perfect complexes and vector bundles on analytic adic spaces, preprint (2021), https://arxiv.org/abs/2105.12591. Suche in Google Scholar
[4] J. Anschütz, B. Heuer and A.-C. Le Bras, v-vector bundles on 𝑝-adic fields and Sen theory via the Hodge–Tate stack, preprint (2022), https://arxiv.org/abs/2211.08470. Suche in Google Scholar
[5] J. Anschütz, B. Heuer and A.-C. Le Bras, The small 𝑝-adic Simpson correspondence in terms of moduli spaces, preprint (2023), https://arxiv.org/abs/2312.07554; to appear in Math. Res. Lett. Suche in Google Scholar
[6] J. Anschütz and A.-C. Le Bras, A Fourier transform for Banach–Colmez spaces, preprint (2021), https://arxiv.org/abs/2111.11116; to appear in J. Eur. Math. Soc. Suche in Google Scholar
[7] P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University, Princeton 1978. Suche in Google Scholar
[8] B. Bhatt, Lectures on prismatic cohomology, Lecture notes (2018), http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/. Suche in Google Scholar
[9] B. Bhatt and J. Lurie, Absolute prismatic cohomology, preprint (2022), https://arxiv.org/abs/2201.06120. Suche in Google Scholar
[10] B. Bhatt and J. Lurie, Prismatic F-gauges, Lecture notes (2022), https://www.math.ias.edu/~bhatt/teaching/mat549f22/lectures.pdf. Suche in Google Scholar
[11] B. Bhatt and J. Lurie, The prismatization of 𝑝-adic formal schemes, preprint (2022), https://arxiv.org/abs/2201.06124. Suche in Google Scholar
[12] B. Bhatt, M. Morrow and P. Scholze, Integral 𝑝-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 219–397. 10.1007/s10240-019-00102-zSuche in Google Scholar
[13] B. Bhatt and P. Scholze, Prisms and prismatic cohomology, Ann. of Math. (2) 196 (2022), no. 3, 1135–1275. 10.4007/annals.2022.196.3.5Suche in Google Scholar
[14] O. Brinon, Une généralisation de la théorie de Sen, Math. Ann. 327 (2003), no. 4, 793–813. 10.1007/s00208-003-0472-3Suche in Google Scholar
[15] K. Česnavičius and P. Scholze, Purity for flat cohomology, Ann. of Math. (2) 199 (2024), no. 1, 51–180. 10.4007/annals.2024.199.1.2Suche in Google Scholar
[16] D. Clausen and P. Scholze, Lectures on complex geometry, Lecture notes (2019), https://people.mpim-bonn.mpg.de/scholze/Complex.pdf. Suche in Google Scholar
[17] V. Drinfeld, Prismatization, Selecta Math. (N. S.) 30 (2024), no. 3, Paper No. 49. 10.1007/s00029-024-00937-3Suche in Google Scholar
[18] G. Faltings, A 𝑝-adic Simpson correspondence, Adv. Math. 198 (2005), no. 2, 847–862. 10.1016/j.aim.2005.05.026Suche in Google Scholar
[19] H. Gao, Integral 𝑝-adic Hodge theory in the imperfect residue field case, preprint (2020), https://arxiv.org/abs/2007.06879. Suche in Google Scholar
[20] H. Gao, On 𝑝-adic Simpson and Riemann–Hilbert correspondences in the imperfect residue field case, Trans. Amer. Math. Soc. 378 (2025), no. 1, 279–315. 10.1090/tran/9311Suche in Google Scholar
[21] T. He, Sen operators and Lie algebras arising from Galois representations over 𝑝-adic varieties, preprint (2022), https://arxiv.org/abs/2208.07519. Suche in Google Scholar
[22] B. Heuer, 𝐺-torsors on perfectoid spaces, preprint (2022), https://arxiv.org/abs/2207.07623. Suche in Google Scholar
[23] B. Heuer, Moduli spaces in 𝑝-adic non-abelian Hodge theory, preprint (2022), https://arxiv.org/abs/2207.13819. Suche in Google Scholar
[24] B. Heuer, A. Werner and M. Zhang, 𝑝-adic Simpson correspondences for principal bundles in abelian settings, preprint (2023), https://arxiv.org/abs/2308.13456. Suche in Google Scholar
[25] J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math. 9, Springer, New York 1978. Suche in Google Scholar
[26] O. Hyodo, On the Hodge–Tate decomposition in the imperfect residue field case, J. reine angew. Math. 365 (1986), 97–113. 10.1515/crll.1986.365.97Suche in Google Scholar
[27] G. Laumon, Transformation de Fourier généralisée, preprint (1996), https://arxiv.org/abs/alg-geom/9603004. Suche in Google Scholar
[28] R. Liu and X. Zhu, Rigidity and a Riemann–Hilbert correspondence for 𝑝-adic local systems, Invent. Math. 207 (2017), no. 1, 291–343. 10.1007/s00222-016-0671-7Suche in Google Scholar
[29] J. Lurie, Spectral algebraic geometry, (2018), http://www.math.harvard.edu/~lurie/. Suche in Google Scholar
[30]
Y. Min and Y. Wang,
On the Hodge–Tate crystals over
[31] Y. Min and Y. Wang, 𝑝-adic Simpson correpondence via prismatic crystals, preprint (2022), https://arxiv.org/abs/2201.08030. Suche in Google Scholar
[32] A. Ogus and V. Vologodsky, Nonabelian Hodge theory in characteristic 𝑝, Publ. Math. Inst. Hautes Études Sci. 106 (2007), 1–138. 10.1007/s10240-007-0010-zSuche in Google Scholar
[33] S. Ohkubo, A note on Sen’s theory in the imperfect residue field case, Math. Z. 269 (2011), no. 1–2, 261–280. 10.1007/s00209-010-0726-1Suche in Google Scholar
[34]
J. E. Rodríguez Camargo,
Locally analytic completed cohomology of Shimura varieties and overconvergent
[35] P. Scholze, 𝑝-adic Hodge theory for rigid-analytic varieties, Forum Math. Pi 1 (2013), Paper No. e1. 10.1017/fmp.2013.1Suche in Google Scholar
[36] P. Scholze, Perfectoid spaces: A survey, Current developments in mathematics 2012, International Press, Somerville (2013), 193–227. 10.4310/CDM.2012.v2012.n1.a4Suche in Google Scholar
[37] R. W. Thomason and T. Trobaugh, Higher algebraic 𝐾-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math. 88, Birkhäuser, Boston (1990), 247–435. 10.1007/978-0-8176-4576-2_10Suche in Google Scholar
[38] Y. Tian, Finiteness and duality for the cohomology of prismatic crystals, J. reine angew. Math. 800 (2023), 217–257. 10.1515/crelle-2023-0032Suche in Google Scholar
[39] T. Tsuji, Notes on the local 𝑝-adic Simpson correspondence, Math. Ann. 371 (2018), no. 1–2, 795–881. 10.1007/s00208-018-1655-2Suche in Google Scholar
[40] T. Yamauchi, A generalization of Sen–Brinon’s theory, Manuscripta Math. 133 (2010), no. 3–4, 327–346. 10.1007/s00229-010-0372-2Suche in Google Scholar
[41] Y. Wang, A p-adic Simpson correspondence for rigid analytic varieties, Algebra Number Theory 17 (2023), no. 8, 1453–1499. 10.2140/ant.2023.17.1453Suche in Google Scholar
[42] T. Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2024. Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Covariant projective representations of Hilbert–Lie groups
- Solutions of the minimal surface equation and of the Monge–Ampère equation near infinity
- A Lindemann–Weierstrass theorem for 𝐸-functions
- Topology and dynamics of compact plane waves
- Diameter of Kähler currents
- CscK metrics near the canonical class
- On 3-nondegenerate CR manifolds in dimension 7 (I): The transitive case
- The rigidity of eigenvalues on Kähler manifolds with positive Ricci lower bound
- Hodge–Tate stacks and non-abelian 𝑝-adic Hodge theory of v-perfect complexes on rigid spaces
Artikel in diesem Heft
- Frontmatter
- Covariant projective representations of Hilbert–Lie groups
- Solutions of the minimal surface equation and of the Monge–Ampère equation near infinity
- A Lindemann–Weierstrass theorem for 𝐸-functions
- Topology and dynamics of compact plane waves
- Diameter of Kähler currents
- CscK metrics near the canonical class
- On 3-nondegenerate CR manifolds in dimension 7 (I): The transitive case
- The rigidity of eigenvalues on Kähler manifolds with positive Ricci lower bound
- Hodge–Tate stacks and non-abelian 𝑝-adic Hodge theory of v-perfect complexes on rigid spaces