Hodge–Tate stacks and non-abelian 𝑝-adic Hodge theory of v-perfect complexes on rigid spaces

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/9780691170282.001.0001Search in Google Scholar

[2] Y. André, La conjecture du facteur direct, Publ. Math. Inst. Hautes Études Sci. 127 (2018), 71–93. 10.1007/s10240-017-0097-9Search 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. Search 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. Search 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. Search 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. Search in Google Scholar

[7] P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University, Princeton 1978. Search in Google Scholar

[8] B. Bhatt, Lectures on prismatic cohomology, Lecture notes (2018), http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/. Search in Google Scholar

[9] B. Bhatt and J. Lurie, Absolute prismatic cohomology, preprint (2022), https://arxiv.org/abs/2201.06120. Search 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. Search in Google Scholar

[11] B. Bhatt and J. Lurie, The prismatization of 𝑝-adic formal schemes, preprint (2022), https://arxiv.org/abs/2201.06124. Search 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-zSearch 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.2022.196.3.5Search in Google Scholar

[14] O. Brinon, Une généralisation de la théorie de Sen, Math. Ann. 327 (2003), no. 4, 793–813. 10.1007/s00208-003-0472-3Search 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.2024.199.1.2Search 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. Search in Google Scholar

[17] V. Drinfeld, Prismatization, Selecta Math. (N. S.) 30 (2024), no. 3, Paper No. 49. 10.1007/s00029-024-00937-3Search in Google Scholar

[18] G. Faltings, A 𝑝-adic Simpson correspondence, Adv. Math. 198 (2005), no. 2, 847–862. 10.1016/j.aim.2005.05.026Search 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. Search 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/9311Search 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. Search in Google Scholar

[22] B. Heuer, 𝐺-torsors on perfectoid spaces, preprint (2022), https://arxiv.org/abs/2207.07623. Search in Google Scholar

[23] B. Heuer, Moduli spaces in 𝑝-adic non-abelian Hodge theory, preprint (2022), https://arxiv.org/abs/2207.13819. Search 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. Search in Google Scholar

[25] J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math. 9, Springer, New York 1978. Search 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.1986.365.97Search in Google Scholar

[27] G. Laumon, Transformation de Fourier généralisée, preprint (1996), https://arxiv.org/abs/alg-geom/9603004. Search 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/s00222-016-0671-7Search in Google Scholar

[29] J. Lurie, Spectral algebraic geometry, (2018), http://www.math.harvard.edu/~lurie/. Search in Google Scholar

[30] Y. Min and Y. Wang, On the Hodge–Tate crystals over O K , preprint (2021), https://arxiv.org/abs/2112.10140. Search in Google Scholar

[31] Y. Min and Y. Wang, 𝑝-adic Simpson correpondence via prismatic crystals, preprint (2022), https://arxiv.org/abs/2201.08030. Search 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-zSearch 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/s00209-010-0726-1Search in Google Scholar

[34] J. E. Rodríguez Camargo, Locally analytic completed cohomology of Shimura varieties and overconvergent B ⁢ G ⁢ G maps, preprint (2022), https://arxiv.org/abs/2205.02016. Search in Google Scholar

[35] P. Scholze, 𝑝-adic Hodge theory for rigid-analytic varieties, Forum Math. Pi 1 (2013), Paper No. e1. 10.1017/fmp.2013.1Search 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.a4Search 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_10Search 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-2023-0032Search 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/s00208-018-1655-2Search in Google Scholar

[40] T. Yamauchi, A generalization of Sen–Brinon’s theory, Manuscripta Math. 133 (2010), no. 3–4, 327–346. 10.1007/s00229-010-0372-2Search 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.2023.17.1453Search in Google Scholar

[42] T. Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2024. Search in Google Scholar