𝐾-theory and topological cyclic homology of henselian pairs

Clausen, Dustin; Mathew, Akhil; Morrow, Matthew

doi:10.1090/jams/961

$K$-theory and topological cyclic homology of henselian pairs
HTML articles powered by AMS MathViewer

by Dustin Clausen, Akhil Mathew and Matthew Morrow

J. Amer. Math. Soc. 34 (2021), 411-473

DOI: https://doi.org/10.1090/jams/961

Published electronically: January 27, 2021

HTML | PDF | Request permission

Abstract:

Given a henselian pair $(R, I)$ of commutative rings, we show that the relative $K$-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 $n$ coefficients, with $n$ invertible in $R$) and McCarthy’s theorem on relative $K$-theory (when $I$ is nilpotent).

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

References

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, DOI 10.1017/CBO9780511600579
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, DOI 10.2140/agt.2002.2.591
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, DOI 10.1007/s00029-018-0427-x
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
D. Ayala, A. Mazel-Gee, and N. Rozenblyum, The geometry of the cyclotomic trace, arXiv:1710.06409, 2017.
C. Barwick and S. Glasman, Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin, arXiv:1602.02163, 2016.
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
B. Bhatt, J. Lurie, and A. Mathew, The de Rham-Witt complex revisited, Astérisque, to appear.
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, DOI 10.1007/s10240-019-00106-9
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, DOI 10.14231/AG-2014-015
Spencer Bloch, Hélène Esnault, and Moritz Kerz, $p$-adic deformation of algebraic cycle classes, Invent. Math. 195 (2014), no. 3, 673–722. MR 3166216, DOI 10.1007/s00222-013-0461-4
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, DOI 10.2140/gt.2012.16.1053
Andrew J. Blumberg and Michael A. Mandell, The homotopy theory of cyclotomic spectra, Geom. Topol. 19 (2015), no. 6, 3105–3147. MR 3447100, DOI 10.2140/gt.2015.19.3105
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, DOI 10.1007/BF01231296
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
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
Pierre Cartier, Une nouvelle opération sur les formes différentielles, C. R. Acad. Sci. Paris 244 (1957), 426–428 (French). MR 84497
D. Clausen and A. Mathew, Hyperdescent and étale $K$-theory, arXiv:1905.06611, 2019.
Guillermo Cortiñas, Infinitesimal $K$-theory, J. Reine Angew. Math. 503 (1998), 129–160. MR 1650347, DOI 10.1515/crll.1998.094
Bjørn Ian Dundas, Relative $K$-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 223–242. MR 1607556, DOI 10.1007/BF02392744
Bjørn Ian Dundas, Continuity of $K$-theory: an example in equal characteristics, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1287–1291. MR 1452802, DOI 10.1090/S0002-9939-98-04382-2
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
Bjørn Ian Dundas and Harald Øyen Kittang, Integral excision for $K$-theory, Homology Homotopy Appl. 15 (2013), no. 1, 1–25. MR 3031812, DOI 10.4310/HHA.2013.v15.n1.a1
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, DOI 10.24033/asens.2319
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, DOI 10.24033/asens.1258
Ioannis Emmanouil, Mittag-Leffler condition and the vanishing of $\underleftarrow {\mmlToken {mi}{lim}}^1$, Topology 35 (1996), no. 1, 267–271. MR 1367284, DOI 10.1016/0040-9383(94)00056-5
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, DOI 10.1090/conm/126/00509
Ofer Gabber, Affine analog of the proper base change theorem, Israel J. Math. 87 (1994), no. 1-3, 325–335. MR 1286833, DOI 10.1007/BF02773001
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, DOI 10.1007/3-540-27855-9_{6}
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, DOI 10.1090/pspum/067/1743237
Thomas Geisser and Lars Hesselholt, Bi-relative algebraic $K$-theory and topological cyclic homology, Invent. Math. 166 (2006), no. 2, 359–395. MR 2249803, DOI 10.1007/s00222-006-0515-y
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, DOI 10.1090/S0002-9947-04-03599-8
Thomas Geisser and Lars Hesselholt, On the $K$-theory of complete regular local $\Bbb F_p$-algebras, Topology 45 (2006), no. 3, 475–493. MR 2218752, DOI 10.1016/j.top.2005.09.002
Thomas Geisser and Marc Levine, The $K$-theory of fields in characteristic $p$, Invent. Math. 139 (2000), no. 3, 459–493. MR 1738056, DOI 10.1007/s002220050014
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, DOI 10.1016/0022-4049(84)90037-9
Thomas G. Goodwillie, Relative algebraic $K$-theory and cyclic homology, Ann. of Math. (2) 124 (1986), no. 2, 347–402. MR 855300, DOI 10.2307/1971283
Thomas G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645–711. MR 2026544, DOI 10.2140/gt.2003.7.645
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
Lars Hesselholt, On the $p$-typical curves in Quillen’s $K$-theory, Acta Math. 177 (1996), no. 1, 1–53. MR 1417085, DOI 10.1007/BF02392597
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, DOI 10.1090/conm/399/07517
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] ©2018, pp. 157–180. MR 3807755, DOI 10.1090/conm/708/14264
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, DOI 10.1016/0040-9383(96)00003-1
Lars Hesselholt and Ib Madsen, On the $K$-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1–113. MR 1998478, DOI 10.4007/annals.2003.158.1
Howard L. Hiller, $\lambda$-rings and algebraic $K$-theory, J. Pure Appl. Algebra 20 (1981), no. 3, 241–266. MR 604319, DOI 10.1016/0022-4049(81)90062-1
Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
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, DOI 10.24033/asens.1374
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, DOI 10.1007/BF02684688
Moritz Kerz, Milnor $K$-theory of local rings with finite residue fields, J. Algebraic Geom. 19 (2010), no. 1, 173–191. MR 2551760, DOI 10.1090/S1056-3911-09-00514-1
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
Ernst Kunz, On Noetherian rings of characteristic $p$, Amer. J. Math. 98 (1976), no. 4, 999–1013. MR 432625, DOI 10.2307/2374038
Markus Land and Georg Tamme, On the $K$-theory of pullbacks, Ann. of Math. (2) 190 (2019), no. 3, 877–930. MR 4024564, DOI 10.4007/annals.2019.190.3.4
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, DOI 10.1017/S1474748004000088
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, DOI 10.1007/978-3-662-21739-9
J. Lurie, Spectral algebraic geometry. Available at https://www.math.ias.edu/~lurie/.
Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659, DOI 10.1515/9781400830558
J. Lurie, Higher algebra, 2014. Available at https://www.math.ias.edu/~lurie/.
Akhil Mathew, A thick subcategory theorem for modules over certain ring spectra, Geom. Topol. 19 (2015), no. 4, 2359–2392. MR 3375530, DOI 10.2140/gt.2015.19.2359
Akhil Mathew, Niko Naumann, and Justin Noel, Nilpotence and descent in equivariant stable homotopy theory, Adv. Math. 305 (2017), 994–1084. MR 3570153, DOI 10.1016/j.aim.2016.09.027
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
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
J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin-New York, 1972. MR 0420610, DOI 10.1007/BFb0067491
Randy McCarthy, Relative algebraic $K$-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 197–222. MR 1607555, DOI 10.1007/BF02392743
Matthew Morrow, Pro unitality and pro excision in algebraic $K$-theory and cyclic homology, J. Reine Angew. Math. 736 (2018), 95–139. MR 3769987, DOI 10.1515/crelle-2015-0007
M. Morrow, $K$-theory and logarithmic Hodge–Witt sheaves of formal scheme in characteristic $p$, Ann. Sci. École Norm. Sup. (4) 52 (2019), no. 6, 1537–1601.
Thomas Nikolaus and Peter Scholze, On topological cyclic homology, Acta Math. 221 (2018), no. 2, 203–409. MR 3904731, DOI 10.4310/ACTA.2018.v221.n2.a1
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, DOI 10.1016/S0012-9593(98)80003-7
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
Dorin Popescu, General Néron desingularization, Nagoya Math. J. 100 (1985), 97–126. MR 818160, DOI 10.1017/S0027763000000246
Dorin Popescu, General Néron desingularization and approximation, Nagoya Math. J. 104 (1986), 85–115. MR 868439, DOI 10.1017/S0027763000022698
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, DOI 10.2307/1970825
Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432, DOI 10.1007/BFb0097438
Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics, Vol. 169, Springer-Verlag, Berlin-New York, 1970 (French). MR 0277519
Les Reid, $N$-dimensional rings with an isolated singular point having nonzero $K_{-N}$, $K$-Theory 1 (1987), no. 2, 197–205. MR 899922, DOI 10.1007/BF00533419
Peter Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 245–313. MR 3090258, DOI 10.1007/s10240-012-0042-x
P. Scholze, Lectures on condensed mathematics, 2019. Available at https://www.math.uni-bonn.de/people/scholze/Condensed.pdf.
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
Atsushi Shiho, On logarithmic Hodge-Witt cohomology of regular schemes, J. Math. Sci. Univ. Tokyo 14 (2007), no. 4, 567–635. MR 2396000
Victor P. Snaith, Stable homotopy around the Arf-Kervaire invariant, Progress in Mathematics, vol. 273, Birkhäuser Verlag, Basel, 2009. MR 2498881, DOI 10.1007/978-3-7643-9904-7
T. Stacks Project Authors, Stacks project, 2017. Available at http://stacks.math.columbia.edu.
A. Suslin, On the $K$-theory of algebraically closed fields, Invent. Math. 73 (1983), no. 2, 241–245. MR 714090, DOI 10.1007/BF01394024
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, DOI 10.1016/0022-4049(84)90043-4
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
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, DOI 10.1007/978-0-8176-4576-2_{1}0
Wilberd van der Kallen, The $K_{2}$ of rings with many units, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 473–515. MR 506170, DOI 10.24033/asens.1334
Wilberd van der Kallen, Homology stability for linear groups, Invent. Math. 60 (1980), no. 3, 269–295. MR 586429, DOI 10.1007/BF01390018
Wilberd van der Kallen, Descent for the $K$-theory of polynomial rings, Math. Z. 191 (1986), no. 3, 405–415. MR 824442, DOI 10.1007/BF01162716
Charles A. Weibel, Pic is a contracted functor, Invent. Math. 103 (1991), no. 2, 351–377. MR 1085112, DOI 10.1007/BF01239518
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, DOI 10.1090/gsm/145