𝐾-theory and topological cyclic homology of henselian pairs

Clausen, Dustin; Mathew, Akhil; Morrow, Matthew

doi:10.1090/jams/961

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-theory and topological cyclic homology of henselian pairs

Authors: Dustin Clausen, Akhil Mathew and Matthew Morrow
Journal: J. Amer. Math. Soc.
MSC (2020): Primary 19D55
DOI: https://doi.org/10.1090/jams/961
Published electronically: January 27, 2021
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Abstract: Given a henselian pair of commutative rings, we show that the relative -theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $K \to \mathrm {TC}$ . This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod coefficients, with invertible in ) and McCarthy's theorem on relative -theory (when is nilpotent).

We deduce that the cyclotomic trace is an equivalence in large degrees between -adic -theory and topological cyclic homology for a large class of -adic rings. In addition, we show that -theory with finite coefficients satisfies continuity for complete noetherian rings which are -finite modulo . Our main new ingredient is a basic finiteness property of $\mathrm {TC}$ with finite coefficients.

[1] Jiří Adámek and Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, Cambridge, 1994. MR 1294136
[2] Stephen T. Ahearn and Nicholas J. Kuhn, Product and other fine structure in polynomial resolutions of mapping spaces, Algebr. Geom. Topol. 2 (2002), 591–647. MR 1917068, https://doi.org/10.2140/agt.2002.2.591
[3] Benjamin Antieau, Akhil Mathew, and Thomas Nikolaus, On the Blumberg-Mandell Künneth theorem for TP, Selecta Math. (N.S.) 24 (2018), no. 5, 4555–4576. MR 3874698, https://doi.org/10.1007/s00029-018-0427-x
[4] Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354652
[5] D. Ayala, A. Mazel-Gee, and N. Rozenblyum,
The geometry of the cyclotomic trace, 1710.06409, 2017.
[6] C. Barwick and S. Glasman,
Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin, 1602.02163, 2016.
[7] Hyman Bass, Algebraic 𝐾-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
[8] B. Bhatt, J. Lurie, and A. Mathew,
The de Rham-Witt complex revisited,
Astérisque, to appear.
[9] Bhargav Bhatt, Matthew Morrow, and Peter Scholze, Topological Hochschild homology and integral 𝑝-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 129 (2019), 199–310. MR 3949030, https://doi.org/10.1007/s10240-019-00106-9
[10] Spencer Bloch, Hélène Esnault, and Moritz Kerz, Deformation of algebraic cycle classes in characteristic zero, Algebr. Geom. 1 (2014), no. 3, 290–310. MR 3238152, https://doi.org/10.14231/AG-2014-015
[11] Spencer Bloch, Hélène Esnault, and Moritz Kerz, 𝑝-adic deformation of algebraic cycle classes, Invent. Math. 195 (2014), no. 3, 673–722. MR 3166216, https://doi.org/10.1007/s00222-013-0461-4
[12] Andrew J. Blumberg and Michael A. Mandell, Localization theorems in topological Hochschild homology and topological cyclic homology, Geom. Topol. 16 (2012), no. 2, 1053–1120. MR 2928988, https://doi.org/10.2140/gt.2012.16.1053
[13] Andrew J. Blumberg and Michael A. Mandell, The homotopy theory of cyclotomic spectra, Geom. Topol. 19 (2015), no. 6, 3105–3147. MR 3447100, https://doi.org/10.2140/gt.2015.19.3105
[14] M. Bökstedt, W. C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic 𝐾-theory of spaces, Invent. Math. 111 (1993), no. 3, 465–539. MR 1202133, https://doi.org/10.1007/BF01231296
[15] Francis Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. Basic category theory. MR 1291599
[16] Francis Borceux, Handbook of categorical algebra. 2, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge University Press, Cambridge, 1994. Categories and structures. MR 1313497
[17] Pierre Cartier, Une nouvelle opération sur les formes différentielles, C. R. Acad. Sci. Paris 244 (1957), 426–428 (French). MR 84497
[18] D. Clausen and A. Mathew,
Hyperdescent and étale -theory, 1905.06611, 2019.
[19] Guillermo Cortiñas, Infinitesimal 𝐾-theory, J. Reine Angew. Math. 503 (1998), 129–160. MR 1650347, https://doi.org/10.1515/crll.1998.094
[20] Bjørn Ian Dundas, Relative 𝐾-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 223–242. MR 1607556, https://doi.org/10.1007/BF02392744
[21] Bjørn Ian Dundas, Continuity of 𝐾-theory: an example in equal characteristics, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1287–1291. MR 1452802, https://doi.org/10.1090/S0002-9939-98-04382-2
[22] Bjørn Ian Dundas, Thomas G. Goodwillie, and Randy McCarthy, The local structure of algebraic K-theory, Algebra and Applications, vol. 18, Springer-Verlag London, Ltd., London, 2013. MR 3013261
[23] Bjørn Ian Dundas and Harald Øyen Kittang, Integral excision for 𝐾-theory, Homology Homotopy Appl. 15 (2013), no. 1, 1–25. MR 3031812, https://doi.org/10.4310/HHA.2013.v15.n1.a1
[24] Bjørn Ian Dundas and Matthew Morrow, Finite generation and continuity of topological Hochschild and cyclic homology, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 1, 201–238 (English, with English and French summaries). MR 3621430, https://doi.org/10.24033/asens.2319
[25] Renée Elkik, Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup. (4) 6 (1973), 553–603 (1974) (French). MR 345966
[26] Ioannis Emmanouil, Mittag-Leffler condition and the vanishing of \varprojlim¹, Topology 35 (1996), no. 1, 267–271. MR 1367284, https://doi.org/10.1016/0040-9383(94)00056-5
[27] Ofer Gabber, 𝐾-theory of Henselian local rings and Henselian pairs, Algebraic 𝐾-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989) Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 59–70. MR 1156502, https://doi.org/10.1090/conm/126/00509
[28] Ofer Gabber, Affine analog of the proper base change theorem, Israel J. Math. 87 (1994), no. 1-3, 325–335. MR 1286833, https://doi.org/10.1007/BF02773001
[29] Thomas Geisser, Motivic cohomology, 𝐾-theory and topological cyclic homology, Handbook of 𝐾-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 193–234. MR 2181824, https://doi.org/10.1007/3-540-27855-9_6
[30] Thomas Geisser and Lars Hesselholt, Topological cyclic homology of schemes, Algebraic 𝐾-theory (Seattle, WA, 1997) Proc. Sympos. Pure Math., vol. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 41–87. MR 1743237, https://doi.org/10.1090/pspum/067/1743237
[31] Thomas Geisser and Lars Hesselholt, Bi-relative algebraic 𝐾-theory and topological cyclic homology, Invent. Math. 166 (2006), no. 2, 359–395. MR 2249803, https://doi.org/10.1007/s00222-006-0515-y
[32] Thomas Geisser and Lars Hesselholt, On the 𝐾-theory and topological cyclic homology of smooth schemes over a discrete valuation ring, Trans. Amer. Math. Soc. 358 (2006), no. 1, 131–145. MR 2171226, https://doi.org/10.1090/S0002-9947-04-03599-8
[33] Thomas Geisser and Lars Hesselholt, On the 𝐾-theory of complete regular local 𝔽_{𝕡}-algebras, Topology 45 (2006), no. 3, 475–493. MR 2218752, https://doi.org/10.1016/j.top.2005.09.002
[34] Thomas Geisser and Marc Levine, The 𝐾-theory of fields in characteristic 𝑝, Invent. Math. 139 (2000), no. 3, 459–493. MR 1738056, https://doi.org/10.1007/s002220050014
[35] Henri A. Gillet and Robert W. Thomason, The 𝐾-theory of strict Hensel local rings and a theorem of Suslin, Proceedings of the Luminy conference on algebraic 𝐾-theory (Luminy, 1983), 1984, pp. 241–254. MR 772059, https://doi.org/10.1016/0022-4049(84)90037-9
[36] Thomas G. Goodwillie, Relative algebraic 𝐾-theory and cyclic homology, Ann. of Math. (2) 124 (1986), no. 2, 347–402. MR 855300, https://doi.org/10.2307/1971283
[37] Thomas G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645–711. MR 2026544, https://doi.org/10.2140/gt.2003.7.645
[38] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 199181
[39] Lars Hesselholt, On the 𝑝-typical curves in Quillen’s 𝐾-theory, Acta Math. 177 (1996), no. 1, 1–53. MR 1417085, https://doi.org/10.1007/BF02392597
[40] Lars Hesselholt, On the topological cyclic homology of the algebraic closure of a local field, An alpine anthology of homotopy theory, Contemp. Math., vol. 399, Amer. Math. Soc., Providence, RI, 2006, pp. 133–162. MR 2222509, https://doi.org/10.1090/conm/399/07517
[41] Lars Hesselholt, Topological Hochschild homology and the Hasse-Weil zeta function, An alpine bouquet of algebraic topology, Contemp. Math., vol. 708, Amer. Math. Soc., Providence, RI, 2018, pp. 157–180. MR 3807755, https://doi.org/10.1090/conm/708/14264
[42] Lars Hesselholt and Ib Madsen, On the 𝐾-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), no. 1, 29–101. MR 1410465, https://doi.org/10.1016/0040-9383(96)00003-1
[43] Lars Hesselholt and Ib Madsen, On the 𝐾-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1–113. MR 1998478, https://doi.org/10.4007/annals.2003.158.1
[44] Howard L. Hiller, 𝜆-rings and algebraic 𝐾-theory, J. Pure Appl. Algebra 20 (1981), no. 3, 241–266. MR 604319, https://doi.org/10.1016/0022-4049(81)90062-1
[45] Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
[46] Luc Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 501–661 (French). MR 565469
[47] Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232. MR 291177
[48] Moritz Kerz, Milnor 𝐾-theory of local rings with finite residue fields, J. Algebraic Geom. 19 (2010), no. 1, 173–191. MR 2551760, https://doi.org/10.1090/S1056-3911-09-00514-1
[49] Charles Kratzer, Opérations d’Adams en 𝐾-théorie algébrique, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 5, A297–A298 (French, with English summary). MR 506639
[50] Ernst Kunz, On Noetherian rings of characteristic 𝑝, Amer. J. Math. 98 (1976), no. 4, 999–1013. MR 432625, https://doi.org/10.2307/2374038
[51] Markus Land and Georg Tamme, On the 𝐾-theory of pullbacks, Ann. of Math. (2) 190 (2019), no. 3, 877–930. MR 4024564, https://doi.org/10.4007/annals.2019.190.3.4
[52] Andreas Langer and Thomas Zink, De Rham-Witt cohomology for a proper and smooth morphism, J. Inst. Math. Jussieu 3 (2004), no. 2, 231–314. MR 2055710, https://doi.org/10.1017/S1474748004000088
[53] Jean-Louis Loday, Cyclic homology, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1992. Appendix E by María O. Ronco. MR 1217970
[54] J. Lurie,
Spectral algebraic geometry.
Available at https://www.math.ias.edu/~lurie/.
[55] Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659
[56] J. Lurie,
Higher algebra, 2014.
Available at https://www.math.ias.edu/~lurie/.
[57] Akhil Mathew, A thick subcategory theorem for modules over certain ring spectra, Geom. Topol. 19 (2015), no. 4, 2359–2392. MR 3375530, https://doi.org/10.2140/gt.2015.19.2359
[58] Akhil Mathew, Niko Naumann, and Justin Noel, Nilpotence and descent in equivariant stable homotopy theory, Adv. Math. 305 (2017), 994–1084. MR 3570153, https://doi.org/10.1016/j.aim.2016.09.027
[59] Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
[60] Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
[61] J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin-New York, 1972. Lectures Notes in Mathematics, Vol. 271. MR 0420610
[62] Randy McCarthy, Relative algebraic 𝐾-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 197–222. MR 1607555, https://doi.org/10.1007/BF02392743
[63] Matthew Morrow, Pro unitality and pro excision in algebraic 𝐾-theory and cyclic homology, J. Reine Angew. Math. 736 (2018), 95–139. MR 3769987, https://doi.org/10.1515/crelle-2015-0007
[64] M. Morrow,
-theory and logarithmic Hodge-Witt sheaves of formal scheme in characteristic ,
Ann. Sci. École Norm. Sup. (4) 52 (2019), no. 6, 1537-1601.
[65] Thomas Nikolaus and Peter Scholze, On topological cyclic homology, Acta Math. 221 (2018), no. 2, 203–409. MR 3904731, https://doi.org/10.4310/ACTA.2018.v221.n2.a1
[66] Wiesława Nizioł, Crystalline conjecture via 𝐾-theory, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 5, 659–681 (English, with English and French summaries). MR 1643962, https://doi.org/10.1016/S0012-9593(98)80003-7
[67] I. A. Panin, The Hurewicz theorem and 𝐾-theory of complete discrete valuation rings, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 763–775, 878 (Russian). MR 864175
[68] Dorin Popescu, General Néron desingularization, Nagoya Math. J. 100 (1985), 97–126. MR 818160, https://doi.org/10.1017/S0027763000000246
[69] Dorin Popescu, General Néron desingularization and approximation, Nagoya Math. J. 104 (1986), 85–115. MR 868439, https://doi.org/10.1017/S0027763000022698
[70] Daniel Quillen, On the cohomology and 𝐾-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972), 552–586. MR 315016, https://doi.org/10.2307/1970825
[71] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
[72] Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics, Vol. 169, Springer-Verlag, Berlin-New York, 1970 (French). MR 0277519
[73] Les Reid, 𝑁-dimensional rings with an isolated singular point having nonzero 𝐾_{-𝑁}, 𝐾-Theory 1 (1987), no. 2, 197–205. MR 899922, https://doi.org/10.1007/BF00533419
[74] Peter Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 245–313. MR 3090258, https://doi.org/10.1007/s10240-012-0042-x
[75] P. Scholze,
Lectures on condensed mathematics,
2019. Available at https://www.math.uni-bonn.de/people/scholze/Condensed.pdf.
[76] J.-P. Serre, Arithmetic groups, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 105–136. MR 564421
[77] Atsushi Shiho, On logarithmic Hodge-Witt cohomology of regular schemes, J. Math. Sci. Univ. Tokyo 14 (2007), no. 4, 567–635. MR 2396000
[78] Victor P. Snaith, Stable homotopy around the Arf-Kervaire invariant, Progress in Mathematics, vol. 273, Birkhäuser Verlag, Basel, 2009. MR 2498881
[79] T. Stacks Project Authors,
Stacks project, 2017.
Available at http://stacks.math.columbia.edu.
[80] A. Suslin, On the 𝐾-theory of algebraically closed fields, Invent. Math. 73 (1983), no. 2, 241–245. MR 714090, https://doi.org/10.1007/BF01394024
[81] Andrei A. Suslin, On the 𝐾-theory of local fields, Proceedings of the Luminy conference on algebraic 𝐾-theory (Luminy, 1983), 1984, pp. 301–318. MR 772065, https://doi.org/10.1016/0022-4049(84)90043-4
[82] Robert W. Thomason, The local to global principle in algebraic 𝐾-theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 381–394. MR 1159226
[83] R. W. Thomason and Thomas Trobaugh, Higher algebraic 𝐾-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, https://doi.org/10.1007/978-0-8176-4576-2_10
[84] Wilberd van der Kallen, The 𝐾₂ of rings with many units, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 473–515. MR 506170
[85] Wilberd van der Kallen, Homology stability for linear groups, Invent. Math. 60 (1980), no. 3, 269–295. MR 586429, https://doi.org/10.1007/BF01390018
[86] Wilberd van der Kallen, Descent for the 𝐾-theory of polynomial rings, Math. Z. 191 (1986), no. 3, 405–415. MR 824442, https://doi.org/10.1007/BF01162716
[87] Charles A. Weibel, Pic is a contracted functor, Invent. Math. 103 (1991), no. 2, 351–377. MR 1085112, https://doi.org/10.1007/BF01239518
[88] Charles A. Weibel, The 𝐾-book, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. An introduction to algebraic 𝐾-theory. MR 3076731