Home Cartier smoothness in prismatic cohomology
Article
Licensed
Unlicensed Requires Authentication

Cartier smoothness in prismatic cohomology

  • Tess Vincent Bouis EMAIL logo
Published/Copyright: November 21, 2023

Abstract

We introduce the notion of a 𝑝-Cartier smooth algebra. It generalises that of a smooth algebra and includes valuation rings over a perfectoid base. We give several characterisations of 𝑝-Cartier smoothness in terms of prismatic cohomology and deduce a comparison theorem between syntomic and étale cohomologies under this hypothesis.

Award Identifier / Grant number: 101001474

Funding statement: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101001474).

Acknowledgements

I am very grateful to Matthew Morrow for suggesting this project to me, sharing many insights and for careful readings of this manuscript. I would also like to thank Elden Elmanto and Arnab Kundu for helpful discussions, Javier Fresán, Akhil Mathew and Mohamed Moakher for comments on a draft of this paper, and the referee for many helpful comments and corrections.

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] B. Antieau, A. Mathew, M. Morrow and T. Nikolaus, On the Beilinson fiber square, Duke Math. J. 171 (2022), no. 18, 3707–3806. 10.1215/00127094-2022-0037Search in Google Scholar

[3] L. L. Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Ann. of Math. (2) 150 (1999), no. 2, 455–487. 10.2307/121087Search in Google Scholar

[4] B. Bhatt, 𝑝-adic derived de Rham cohomology, preprint (2012), https://arxiv.org/abs/1204.6560. Search in Google Scholar

[5] B. Bhatt, An imperfect ring with trivial cotangent complex, preprint (2013), https://www.math.ias.edu/~bhatt/math/trivial-cc.pdf. Search in Google Scholar

[6] B. Bhatt, Specializing varieties and their cohomology from characteristic 0 to characteristic 𝑝, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math. 97, American Mathematical Society, Providence (2018), 43–88. 10.1090/pspum/097.2/02Search in Google Scholar

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

[8] B. Bhatt and A. Mathew, The arc-topology, Duke Math. J. 170 (2021), no. 9, 1899–1988. 10.1215/00127094-2020-0088Search in Google Scholar

[9] B. Bhatt and A. Mathew, Syntomic complexes and 𝑝-adic étale Tate twists, Forum Math. Pi 11 (2023), Paper No. e1. 10.1017/fmp.2022.21Search in Google Scholar

[10] 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

[11] B. Bhatt, M. Morrow and P. Scholze, Topological Hochschild homology and integral 𝑝-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 129 (2019), 199–310. 10.1007/s10240-019-00106-9Search in Google Scholar

[12] B. Bhatt and P. Scholze, Projectivity of the Witt vector affine Grassmannian, Invent. Math. 209 (2017), no. 2, 329–423. 10.1007/s00222-016-0710-4Search 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] P. Deligne and L. Illusie, Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270. 10.1007/BF01389078Search in Google Scholar

[15] O. Gabber, 𝐾-theory of Henselian local rings and Henselian pairs, Algebraic 𝐾-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure 1989), Contemp. Math. 126, American Mathematical Society, Providence (1992), 59–70. 10.1090/conm/126/00509Search in Google Scholar

[16] O. Gabber and L. Ramero, Almost ring theory, Lecture Notes in Math. 1800, Springer, Berlin 2003. 10.1007/b10047Search in Google Scholar

[17] T. Geisser and M. Levine, The 𝐾-theory of fields in characteristic 𝑝, Invent. Math. 139 (2000), no. 3, 459–493. 10.1007/s002220050014Search in Google Scholar

[18] R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Math. E30, Friedrich Vieweg & Sohn, Braunschweig 1996. 10.1007/978-3-663-09991-8Search in Google Scholar

[19] A. Huber-Klawitter and S. Kelly, Differential forms in positive characteristic, II: Cdh-descent via functorial Riemann–Zariski spaces, Algebra Number Theory 12 (2018), no. 3, 649–692. 10.2140/ant.2018.12.649Search in Google Scholar

[20] L. Illusie, Complexe cotangent et déformations. I, Lecture Notes in Math. 239, Springer, Berlin 1971. 10.1007/BFb0059052Search in Google Scholar

[21] S. Kelly and M. Morrow, 𝐾-theory of valuation rings, Compos. Math. 157 (2021), no. 6, 1121–1142. 10.1112/S0010437X21007119Search in Google Scholar

[22] M. Kerz, F. Strunk and G. Tamme, Towards Vorst’s conjecture in positive characteristic, Compos. Math. 157 (2021), no. 6, 1143–1171. 10.1112/S0010437X21007120Search in Google Scholar

[23] M. Lüders and M. Morrow, Milnor 𝐾-theory of 𝑝-adic rings, J. reine angew. Math. 796 (2023), 69–116. 10.1515/crelle-2022-0079Search in Google Scholar

[24] P. Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 245–313. 10.1007/s10240-012-0042-xSearch in Google Scholar

[25] A. Suslin, On the 𝐾-theory of algebraically closed fields, Invent. Math. 73 (1983), no. 2, 241–245. 10.1007/BF01394024Search in Google Scholar

[26] A. Suslin and V. Voevodsky, Bloch–Kato conjecture and motivic cohomology with finite coefficients, The arithmetic and geometry of algebraic cycles (Banff 1998), NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer Academic, Dordrecht (2000), 117–189. 10.1007/978-94-011-4098-0_5Search in Google Scholar

[27] The Stacks Project authors, Stacks Project, https://stacks.math.columbia.edu, 2019. Search in Google Scholar

Received: 2022-12-15
Revised: 2023-07-19
Published Online: 2023-11-21
Published in Print: 2023-12-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 9.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2023-0074/html
Scroll to top button