Geodesic
Syntomic complexes and p-adic étale Tate twists
Forum of Mathematics, Pi, Tome 11 (2023)

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The primary goal of this paper is to identify syntomic complexes with the p-adic étale Tate twists of Geisser–Sato–Schneider on regular p-torsion-free schemes. Our methods apply naturally to a broader class of schemes that we call ‘F-smooth’. The F-smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes. 
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Bhargav Bhatt; Akhil Mathew. Syntomic complexes and p-adic étale Tate twists. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2022.21

[AMMN22] Antieau, B., Mathew, A., Morrow, M. and Nikolaus, T., ‘On the Beilinson fiber square’, Duke Math. J. 171(18) (2022), 3707–3806. MR4516307.Google Scholar

[Avr99] 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. MR 1726700Google Scholar | DOI

[Bha12] Bhatt, B., ‘-adic derived de Rham cohomology’, Preprint, 2012, .Google Scholar | arXiv

[BK86] Bloch, S. and Kato, K., ‘-adic étale cohomology’, Inst. Hautes Études Sci. Publ. Math. (63) (1986), 107–152. MR 849653Google Scholar | DOI

[BL22a] Bhatt, B. and Lurie, J., ‘Absolute prismatic cohomology’, Preprint 2022, .Google Scholar | arXiv | DOI

[BL22b] Bhatt, B. and Lurie, J., ‘The prismatization of -adic formal schemes’, Preprint, 2022, .Google Scholar | arXiv

[BLM21] Bhatt, B., Lurie, J. and Mathew, A., ‘Revisiting the de Rham-Witt complex’, Astérisque (424) (2021), viii+165. MR 4275461Google Scholar

[BMS18] Bhatt, B., Morrow, M. and Scholze, P., ‘Integral -adic Hodge theory’, Publ. Math. Inst. Hautes Études Sci. 128(1) (2018), 1, 219–397.Google Scholar | DOI

[BMS19] Bhatt, B., Morrow, M. and Scholze, P., ‘Topological Hochschild homology and integral -adic Hodge theory’, Publ. Math. Inst. Hautes Études Sci. 129 (2019), 199–310. MR 3949030Google Scholar | DOI

[Bou22] Bouis, V., ‘Cartier smoothness in prismatic cohomology’, Preprint, .Google Scholar | arXiv

[BS22] Bhatt, B. and Scholze, P., ‘Prisms and prismatic cohomology’, Ann. of Math. (2) 196(3) (2022), 1135–1275. MR 4502597Google Scholar

[CDN21] Colmez, P., Dospinescu, G. and Nizioł, W., ‘Integral -adic étale cohomology of Drinfeld symmetric spaces’, Duke Math. J. 170(3) (2021), 575–613. MR 4255044Google Scholar | DOI

[CMM21] Clausen, D., Mathew, A. and Morrow, M., ‘ -theory and topological cyclic homology of Henselian pairs’, J. Amer. Math. Soc. 34(2) (2021), 411–473. MR 4280864Google Scholar | DOI

[CN17] Colmez, P. and Nizioł, W., ‘Syntomic complexes and -adic nearby cycles’, Invent. Math. 208(1) (2017), 1–108. MR 3621832Google Scholar | DOI

[DM17] Dundas, B. I. and Morrow, M., ‘Finite generation and continuity of topological Hochschild and cyclic homology’, Ann. Sci. Éc. Norm. Supér. (4) 50(1) (2017), 201–238. MR 3621430Google Scholar

[Dri20] Drinfeld, V., ‘Prismatization’, Preprint, 2020, .Google Scholar | arXiv

[EN19] Ertl, V. and Nizioł, W. A., ‘Syntomic cohomology and -adic motivic cohomology’, Algebr. Geom. 6(1) (2019), 100–131. MR 3904801Google Scholar

[FM87] Fontaine, J.-M. and Messing, W., ‘-adic periods and -adic étale cohomology’, Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67 (Amer. Math. Soc., Providence, RI, 1987), 179–207. MR 902593Google Scholar

[Gab92] Gabber, O., ‘-theory of Henselian local rings and Henselian pairs’, Algebraic -Theory, Commutative Algebra, and Algebraic Geometry (Santa Margherita Ligure, 1989), Contemp. Math., vol. 126 (Amer. Math. Soc., Providence, RI, 1992), 59–70. MR 1156502Google Scholar

[Gei04] Geisser, T., ‘Motivic cohomology over Dedekind rings’, Math. Z. 248(4) (2004), 773–794. MR 2103541Google Scholar

[GH99] Geisser, T. and Hesselholt, L., ‘Topological cyclic homology of schemes’, Algebraic -theory (Seattle, WA, 1997), Proc. Sympos. Pure Math., vol. 67 (Amer. Math. Soc., Providence, RI, 1999), 41–87. MR 1743237Google Scholar | DOI

[GH06] Geisser, T. and Hesselholt, L., ‘The de Rham–Witt complex and -adic vanishing cycles’, J. Amer. Math. Soc. 19(1) (2006), 1–36. MR 2169041Google Scholar | DOI

[Gro85] Gros, M., ‘Classes de Chern et classes de cycles en cohomologie de Hodge–Witt logarithmique’, Mém. Soc. Math. France (N.S.) (21) (1985), 87. MR 844488Google Scholar

[HJ21] Hochster, M. and Jeffries, J., ‘A Jacobian criterion for nonsingularity in mixed characteristic’, Preprint, 2021, .Google Scholar | arXiv

[HM03] Hesselholt, L. and Madsen, I., ‘On the -theory of local fields’, Ann. of Math. (2) 158(1) (2003), 1–113. MR 1998478Google Scholar | DOI

[HM04] Hesselholt, L. and Madsen, I., ‘On the De Rham–Witt complex in mixed characteristic’, Ann. Sci. École Norm. Sup. (4) 37(1) (2004), 1–43. MR 2050204Google Scholar | DOI

[HN20] Hesselholt, L. and Nikolaus, T., ‘Topological cyclic homology’, Handbook of Homotopy Theory, CRC Press/Chapman Hall Handb. Math. Ser. (CRC Press, Boca Raton, FL, 2020), 619–656. MR 4197995Google Scholar | DOI

[HW22] Hahn, J. and Wilson, D., ‘Redshift and multiplication for truncated Brown–Peterson spectra’, Ann. of Math. (2) 196(3) (2022), 1277–1351. MR 4503327Google Scholar

[Hyo88] Hyodo, O., ‘A note on -adic étale cohomology in the semistable reduction case’, Invent. Math. 91(3) (1988), 543–557. MR 928497Google Scholar

[Ill71] Illusie, L., Complexe cotangent et déformations. I , Lecture Notes in Mathematics, vol. 239 (Springer-Verlag, Berlin-New York, 1971). MR 0491680Google Scholar

[Ill79] Illusie, L., Complexe de de Rham–Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4) 12(4) (1979), 501–661. MR 565469Google Scholar | DOI

[Iye07] Iyengar, S., ‘André–Quillen homology of commutative algebras’, Interactions between Homotopy Theory and Algebra, Contemp. Math., vol. 436 (Amer. Math. Soc., Providence, RI, 2007), 203–234. MR 2355775Google Scholar | DOI

[Kat87] Kato, K., ‘On -adic vanishing cycles (application of ideas of Fontaine–Messing)’, Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10 (North-Holland, Amsterdam, 1987), 207–251. MR 946241Google Scholar | DOI

[KM21] Kelly, S. and Morrow, M., ‘-theory of valuation rings’, Compos. Math. 157(6) (2021), 1121–1142. MR 4264079Google Scholar

[KST21] Kerz, M., Strunk, F. and Tamme, G., ‘Towards Vorst’s conjecture in positive characteristic’, Compos. Math. 157(6) (2021), 1143–1171. MR 4270122Google Scholar | DOI

[Kun76] Kunz, E., ‘On Noetherian rings of characteristic ’, Amer. J. Math. 98(4) (1976), 999–1013. MR 432625Google Scholar | DOI

[Kur87] Kurihara, M., ‘A note on -adic étale cohomology’, Proc. Japan Acad. Ser. A Math. Sci. 63(7) (1987), 275–278. MR 931263Google Scholar | DOI

[LM21] Lüders, M. and Morrow, M., ‘Milnor -theory of -adic rings’, Preprint, 2021, .Google Scholar | arXiv

[Lur18] Lurie, J., Elliptic Cohomology II: Orientations URL: https://www.math.ias.edu/~lurie/papers/Elliptic-II.pdf.Google Scholar

[LW22] Liu, R. and Wang, G., ‘Topological cyclic homology of local fields’, Invent. Math. 230(2) (2022), 851–932. MR 4493328Google Scholar | DOI

[Mat22] Mathew, A., ‘Some recent advances in topological Hochschild homology’, Bull. Lond. Math. Soc. 54(1) (2022), 1–44. MR 4396920Google Scholar

[Mil76] Milne, J. S., ‘Duality in the flat cohomology of a surface’, Ann. Sci. École Norm. Sup. (4) 9(2) (1976), 171–201. MR 460331Google Scholar

[Niz06] Nizioł, W., ‘-adic Motivic Cohomology in Arithmetic’, International Congress of Mathematicians. vol. II (Eur. Math. Soc., Zürich, 2006), 459–472. MR 2275605Google Scholar

[Niz12] Nizioł, W., ‘-theory of log-schemes II: Log-syntomic -theory’, Adv. Math. 230(4–6) (2012), 1646–1672. MR 2927351Google Scholar | DOI

[Sai22] Saito, T., ‘Frobenius–Witt differentials and regularity’, Algebra Number Theory 16(2) (2022), 369–391. MR 4412577Google Scholar

[Sat05] Sato, K., ‘Higher-dimensional arithmetic using -adic étale Tate twists’, Homology Homotopy Appl. 7(3) (2005), 173–187. MR 2205174Google Scholar | DOI

[Sat07] Sato, K., ‘-adic étale Tate twists and arithmetic duality’, Ann. Sci. École Norm. Sup. (4) 40(4) (2007), 519–588. With an appendix by Kei Hagihara. MR 2191526Google Scholar

[Sch94] Schneider, P., ‘-adic points of motives’, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55 (Amer. Math. Soc., Providence, RI, 1994), 225–249. MR 1265555Google Scholar

[SS20] Saito, S. and Sato, K., ‘On -adic vanishing cycles of log smooth families’, Tunis. J. Math. 2(2) (2020), 309–335. MR 3990821Google Scholar | DOI

[Sta19] The Stacks Project authors, The Stacks Project, https://stacks.math.columbia.edu, 2019.Google Scholar

[Sus83] Suslin, A., ‘On the -theory of algebraically closed fields’, Invent. Math. 73(2) (1983), 241–245. MR 714090Google Scholar

[Tat76] Tate, J., ‘Relations between and Galois cohomology’, Invent. Math. 36 (1976), 257–274. MR 429837Google Scholar | DOI

[Tsu99] Tsuji, T., ‘-adic étale cohomology and crystalline cohomology in the semi-stable reduction case’, Invent. Math. 137(2) (1999), 233–411. MR 1705837Google Scholar

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