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Descent in algebraic \(K\)-theory and a conjecture of Ausoni-Rognes. (English) Zbl 1453.18011

Summary: Let \(A \to B\) be a \(G\)-Galois extension of rings, or more generally of \(\mathbb{E}_\infty \)-ring spectra in the sense of J. Rognes [in: Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. II: Invited lectures. Seoul: KM Kyung Moon Sa. 1259–1283 (2014; Zbl 1373.19002)]. A basic question in algebraic \(K\)-theory asks how close the map \(K(A) \to K(B)^{hG}\) is to being an equivalence, i.e., how close algebraic \(K\)-theory is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of R. W. Thomason [Ann. Sci. Éc. Norm. Supér. (4) 18, 437–552 (1985; Zbl 0596.14012)], one also expects such a result after periodic localization. We formulate and prove a general result which enables one to promote rational descent statements as above into descent statements after periodic localization. This reduces the localized descent problem to establishing an elementary condition on \(K_0(-)\otimes \mathbb{Q} \). As applications, we prove various descent results in the periodically localized \(K\)-theory, \(TC, THH\), etc. of structured ring spectra, and verify several cases of a conjecture of C. Ausoni and J. Rognes [Geom. Topol. 16, No. 4, 2037–2065 (2012; Zbl 1260.19004)].

MSC:

18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
55P42 Stable homotopy theory, spectra
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
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[1] Adams, J. F.: Stable Homotopy and Generalised Homology. Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL (1995)Zbl 0309.55016 MR 0402720 · Zbl 0309.55016
[2] Angeltveit, V.: Topological Hochschild homology and cohomology ofA∞ring spectra. Geom. Topol.12, 987-1032 (2008)Zbl 1149.55006 MR 2403804 · Zbl 1149.55006
[3] Antieau, B., Gepner, D.: Brauer groups and ´etale cohomology in derived algebraic geometry. Geom. Topol.18, 1149-1244 (2014)Zbl 1308.14021 MR 3190610 · Zbl 1308.14021
[4] Ausoni, C., Rognes, J.: The chromatic red-shift in algebraic K-theory. Enseign. Math. 54, 9-11 (2008)
[5] Antol´ın-Camarena, O., Barthel, T.: A simple universal property of Thom ring spectra. J. Topol.12, 56-78 (2019)Zbl 1417.55007 MR 3875978 · Zbl 1417.55007
[6] Ausoni, C., Rognes, J.: Rational algebraicK-theory of topologicalK-theory. Geom. Topol.16, 2037-2065 (2012)Zbl 1260.19004 MR 2975299 · Zbl 1260.19004
[7] Barwick, C.: On the algebraicK-theory of higher categories. J. Topol.9, 245-347 (2016)Zbl 1364.19001 MR 3465850 · Zbl 1364.19001
[8] Barwick, C.: Spectral Mackey functors and equivariant algebraicK-theory (I). Adv. Math.304, 646-727 (2017)Zbl 1348.18020 MR 3558219 · Zbl 1348.18020
[9] Barwick, C.: Glasman, S., Shah, J.: Spectral Mackey functors and equivariant algebraicK-theory II. Tunis. J. Math.2, 97-146 (2020)Zbl 07074072 MR 3933393 · Zbl 1461.18009
[10] Barwick, C., Lawson, T.: Regularity of structured ring spectra and localization inKtheory.arXiv:1402.6038(2014)
[11] Behrens, M., Lawson, T.: Topological automorphic forms. Mem. Amer. Math. Soc. 204, no. 958, xxiv+141 pp. (2010)Zbl 1210.55005 MR 2640996 · Zbl 1210.55005
[12] Ben-Zvi, D., Francis, J., Nadler, D.: Integral transforms and Drinfeld centers in derived algebraic geometry. J. Amer. Math. Soc.23, 909-966 (2010)Zbl 1202.14015 MR 2669705 · Zbl 1202.14015
[13] Bhatt, B.: Algebraization and Tannaka duality. Cambridge J. Math.4, 403-461 (2016) Zbl 1356.14006 MR 3572635 · Zbl 1356.14006
[14] Blumberg, A. J., Gepner, D., Tabuada, G.: A universal characterization of higher algebraicK-theory. Geom. Topol.17, 733-838 (2013)Zbl 1267.19001 MR 3070515 · Zbl 1267.19001
[15] Blumberg, A. J., Gepner, D., Tabuada, G.: Uniqueness of the multiplicative cyclotomic trace. Adv. Math.260, 191-232 (2014)Zbl 1297.19002 MR 3209352 · Zbl 1297.19002
[16] Blumberg, A. J., Hill, M. A.: Operadic multiplications in equivariant spectra, norms, and transfers. Adv. Math.285, 658-708 (2015)Zbl 1329.55012 MR 3406512 · Zbl 1329.55012
[17] Blumberg, A. J., Mandell, M.: The localization sequence for the algebraicKtheory of topologicalK-theory. Acta Math.200, 155-179 (2008)Zbl 1149.18008 MR 2413133 · Zbl 1149.18008
[18] Bondal, A., van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Moscow Math. J.3, 1-36 (258, 2003) Zbl 1135.18302 MR 1996800 · Zbl 1135.18302
[19] Bousfield, A. K.: The localization of spectra with respect to homology. Topology18, 257-281 (1979)Zbl 0417.55007 MR 0551009 · Zbl 0417.55007
[20] Brun, M.: Witt vectors and equivariant ring spectra applied to cobordism. Proc. London Math. Soc. (3)94, 351-385 (2007)Zbl 1121.55007 MR 2308231 · Zbl 1121.55007
[21] Chouinard, L. G.: Projectivity and relative projectivity over group rings. J. Pure Appl. Algebra7, 287-302 (1976)Zbl 0327.20020 MR 0401943 · Zbl 0327.20020
[22] Conrad, B.: Arithmetic moduli of generalized elliptic curves. J. Inst. Math. Jussieu6, 209-278 (2007)Zbl 1140.14018 MR 2311664 · Zbl 1140.14018
[23] Deligne, P., Rapoport, M.: Les sch´emas de modules de courbes elliptiques. In: Modular Functions of One Variable, II (Antwerp, 1972), Lecture Notes in Math. 349, Springer, 143-316 (1973)Zbl 0281.14010 MR 0337993 · Zbl 0281.14010
[24] Devinatz, E. S., Hopkins, M. J., Smith, J. H.: Nilpotence and stable homotopy theory. I. Ann. of Math. (2)128, 207-241 (1988)Zbl 0673.55008 MR 0960945 · Zbl 0673.55008
[25] Douglas, C. L., Francis, J., Henriques, A. G., Hill, M. A. (eds.): Topological Modular Forms. Math. Surveys Monogr. 201, Amer. Math. Soc., Providence, RI (2014) Zbl 1304.55002
[26] Elmendorf, A. D., Kriz, I., Mandell, M., May, J.: Rings, Modules, and Algebras in Stable Homotopy Theory. Math. Surveys Monogr. 47, Amer. Math. Soc., Providence, RI (1997)Zbl 0894.55001 MR 1417719 · Zbl 0894.55001
[27] Gaitsgory, D.: Generalities on DG-categories.http://www.math.harvard.edu/∼ gaitsgde/GL/textDG.pdf(2012)
[28] Gepner, D., Groth, M., Nikolaus, T.: Universality of multiplicative infinite loop space machines. Algebr. Geom. Topol.15, 3107-3153 (2015)Zbl 1336.55006 MR 3450758 · Zbl 1336.55006
[29] Glasman, S.: Day convolution for∞-categories. Math. Res. Lett.23, 1369-1385 (2016)Zbl 1375.18041 MR 3601070 · Zbl 1375.18041
[30] Goerss,P.,Hopkins,M.:Moduliproblemsforstructuredringspectra. http://www.math.northwestern.edu/∼pgoerss/spectra/obstruct.pdf(2005)
[31] G¨ortz, U., Wedhorn, T.: Algebraic Geometry I. Vieweg + Teubner, Wiesbaden (2010) Zbl 1213.14001 MR 2675155
[32] Grothendieck, A.: ´El´ements de g´eom´etrie alg´ebrique. IV. ´Etude locale des sch´emas et des morphismes de sch´emas. III. Inst. Hautes ´Etudes Sci. Publ. Math.28, 255 pp. (1966)Zbl 0144.19904 MR 0217086
[33] Hazrat, H.: Graded Rings and Graded Grothendieck Groups. London Math. Soc. Lecture Note Ser. 435, Cambridge Univ. Press (2016)Zbl 1390.13001 · Zbl 1390.13001
[34] Heard, D., Mathew, A., Stojanoska, V.: Picard groups of higher realK-theory spectra at heightp−1. Compos. Math.153, 1820-1854 (2017)Zbl 1374.14006 MR 3705278 · Zbl 1374.14006
[35] Hewett, T.: Finite subgroups of division algebras over local fields. J. Algebra173, 518-548 (1995)Zbl 0829.16023 MR 1327867 · Zbl 0829.16023
[36] Hill, M., Lawson, T.: Topological modular forms with level structure. Invent. Math. 203, 359-416 (2016)Zbl 1338.55006 MR 3455154 · Zbl 1338.55006
[37] Hopkins, M. J., Palmieri, J. H., Smith, J. H.: Vanishing lines in generalized Adams spectral sequences are generic. Geom. Topol.3, 155-165 (1999)Zbl 0920.55020 MR 1697180 · Zbl 0920.55020
[38] Hopkins, M. J., Smith, J.: Nilpotence and stable homotopy theory. II. Ann. of Math. (2)148, 1-49 (1998)Zbl 0924.55010 MR 1652975 · Zbl 0927.55015
[39] Hoyois, M.: A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula. Algebr. Geom. Topol.14, 3603-3658 (2014)Zbl 1351.14013 MR 3302973 · Zbl 1351.14013
[40] Hoyois, M., Scherotzke, S., Sibilla, N.: Higher traces, noncommutative motives, and the categorified Chern character. Adv. Math.309, 97-154 (2017)Zbl 1361.14014 MR 3607274 · Zbl 1361.14014
[41] Katz, N. M., Mazur, B.: Arithmetic Moduli of Elliptic Curves. Ann. of Math. Stud. 108, Princeton Univ. Press, Princeton, NJ (1985)Zbl 0576.14026 MR 0772569 · Zbl 0576.14026
[42] Keller, B.: Deriving DG categories. Ann. Sci. ´Ecole Norm. Sup. (4)27, 63-102 (1994) Zbl 0799.18007 MR 1258406 · Zbl 0799.18007
[43] Kolster, M.: The norm residue theorem and the Quillen-Lichtenbaum conjecture. In: The Bloch-Kato Conjecture for the Riemann Zeta Function, London Math. Soc. Lecture Note Ser. 418, Cambridge Univ. Press, 97-120 (2015)Zbl 1353.14030 MR 3497675 · Zbl 1353.14030
[44] Lam, T. Y.: A First Course in Noncommutative Rings. Grad. Texts in Math. 131, Springer, New York (1991)Zbl 0728.16001 MR 1838439 · Zbl 0728.16001
[45] Lichtenbaum, S.: Values of zeta-functions, ´etale cohomology, and algebraicK-theory. In: AlgebraicK-theory, II: “Classical” AlgebraicK-theory and Connections with Arithmetic (Seattle, 1972), Lecture Notes in Math. 342, Springer, Berlin, 489-501 (1973)Zbl 0284.12005 MR 0406981 · Zbl 0284.12005
[46] Lurie,J.:CoursenotesonalgebraicK-theoryandmanifoldtopology. http://www.math.harvard.edu/∼lurie/281.html
[47] Lurie, J.: Spectral algebraic geometry.http://www.math.harvard.edu/∼lurie/papers/ SAG-rootfile.pdf(2017)
[48] Lurie, J.: Higher Topos Theory. Ann. of Math. Stud. 170, Princeton Univ. Press, Princeton, NJ (2009)Zbl 1175.18001 MR 2522659 · Zbl 1175.18001
[49] Lurie, J.: On the classification of topological field theories. In: Current Developments in Mathematics, 2008, Int. Press, Somerville, MA, 129-280 (2009)Zbl 1180.81122 MR 2555928 · Zbl 1180.81122
[50] Lurie, J.: Derived algebraic geometry XI: Descent theorems.http://www.math. harvard.edu/∼lurie/papers/DAG-XI.pdf(2011)
[51] Lurie, J.: Rotation invariance in algebraicK-theory.(2015)
[52] Lurie, J.: Higher algebra.http://www.math.harvard.edu/∼lurie/papers/HA.pdf(2016)
[53] Mathew, A.: A thick subcategory theorem for modules over certain ring spectra. Geom. Topol.19, 2359-2392 (2015)Zbl 1405.55009 MR 3375530 · Zbl 1405.55009
[54] Mathew, A.: Torus actions on stable module categories, Picard groups, and localizing subcategories.arXiv:1512.01716(2015)
[55] Mathew, A.: The Galois group of a stable homotopy theory. Adv. Math.291, 403-541 (2016)Zbl 1338.55009 MR 3459022 · Zbl 1338.55009
[56] Mathew, A.: The homology of tmf. Homology Homotopy Appl.18, 1-29 (2016) Zbl 1357.55002 MR 3515195 · Zbl 1357.55002
[57] Mathew, A., Meier, L.: Affineness and chromatic homotopy theory. J. Topol.8, 476- 528 (2015)Zbl 1325.55004 MR 3356769 · Zbl 1325.55004
[58] Mathew, A., Naumann, N., Noel, J.: On a nilpotence conjecture of J. P. May. J. Topol. 8, 917-932 (2015)Zbl 1335.55009 MR 3431664 · Zbl 1335.55009
[59] Mathew, A., Naumann, N., Noel, J.: Nilpotence and descent in equivariant stable homotopy theory. Adv. Math.305, 994-1084 (2017)Zbl 1420.55024 MR 3570153 · Zbl 1420.55024
[60] Merling, M.: Equivariant algebraic K-theory ofG-rings. Math. Z.285, 1205-1248 (2017)Zbl 1365.19007 MR 3623747 · Zbl 1365.19007
[61] Miller, H.: Finite localizations. Bol. Soc. Mat. Mexicana (2)37, 383-389 (1992) Zbl 0852.55015 MR 1317588 · Zbl 0852.55015
[62] Mitchell, S. A.: Finite complexes withA(n)-free cohomology. Topology24, 227-246 (1985)Zbl 0568.55021 MR 0793186 · Zbl 0568.55021
[63] Mitchell, S. A.: The MoravaK-theory of algebraicK-theory spectra.K-Theory3, 607-626 (1990)Zbl 0709.55011 MR 1071898 · Zbl 0709.55011
[64] Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Amer. Math. Soc.9, 205-236 (1996)Zbl 0864.14008 MR 1308405 · Zbl 0864.14008
[65] Nisnevich, Ye. A.: The completely decomposed topology on schemes and associated descent spectral sequences in algebraicK-theory. In: AlgebraicK-theory: Connections with Geometry and Topology (Lake Louise, AB, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 279, Kluwer, Dordrecht, 241-342 (1989)Zbl 0715.14009 MR 1045853 · Zbl 0715.14009
[66] Quillen, D.: Higher algebraicK-theory. In: Proc. International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 171-176 (1975)Zbl 0359.18014 MR 0422392 · Zbl 0359.18014
[67] Ravenel, D. C.: Complex Cobordism and Stable Homotopy Groups of Spheres. Pure Appl. Math. 121, Academic Press, Orlando, FL (1986)Zbl 0608.55001 MR 0860042 · Zbl 0608.55001
[68] Ravenel, D. C.: Nilpotence and Periodicity in Stable Homotopy Theory. Ann. of Math. Stud. 128, Princeton Univ. Press, Princeton, NJ (1992)Zbl 0774.55001 MR 1192553 · Zbl 0774.55001
[69] Rezk, C.: Notes on the Hopkins-Miller theorem. In: Homotopy Theory via Algebraic Geometry and Group Representations (Evanston, IL, 1997), Contemp. Math. 220, Amer. Math. Soc., 313-366 (1997)Zbl 0910.55004 MR 1642902 · Zbl 0910.55004
[70] Rognes, J.: Galois extensions of structured ring spectra. Stably dualizable groups. Mem. Amer. Math. Soc.192, no. 898, viii+137 pp. (2008)Zbl 1166.55001 MR 2387923 · Zbl 1166.55001
[71] Rognes, J.: AlgebraicK-theory of strict ring spectra. In: Proc. ICM (Seoul, 2014), Vol. II, 1259-1283 (2014)Zbl 1373.19002 MR 3728661 · Zbl 1373.19002
[72] Rouquier, R.: Dimensions of triangulated categories. J. K-Theory1, 193-256 (2008) Zbl 1165.18008 MR 2434186 · Zbl 1165.18008
[73] S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1963-1964 (SGA 4). Th´eorie des topos et cohomologie ´etale des sch´emas. Tome 1: Th´eorie des topos. Lecture Notes in Math. 269, Springer, Berlin (1972)Zbl 0234.00007 MR 0354652
[74] Smith,J.H.:Finitecomplexeswithvanishinglinesofsmallslope. http://hopf.math.purdue.edu/SmithJH/idempotents.pdf
[75] The Stacks Project Authors. Stacks Project.http://stacks.math.columbia.edu(2017)
[76] Stojanoska, V.: Duality for topological modular forms. Documenta Math.17, 271-311 (2012)Zbl 1366.55005 MR 2946825 · Zbl 1366.55005
[77] Swan, R. G.: AlgebraicK-theory. Lecture Notes in Math. 76, Springer, Berlin (1968) Zbl 0193.34601 MR 0245634 · Zbl 0193.34601
[78] Tabuada, G.:A1-homotopy theory of noncommutative motives. J. Noncommut. Geom.9, 851-875 (2015)Zbl 1345.14008 MR 3420534 · Zbl 1345.14008
[79] Tambara, D.: On multiplicative transfer. Comm. Algebra21, 1393-1420 (1993) Zbl 0797.19001 MR 1209937 · Zbl 0797.19001
[80] Thomason, R. W.: AlgebraicK-theory and ´etale cohomology. Ann. Sci. ´Ecole Norm. Sup. (4)18, 437-552 (1985)Zbl 0596.14012 MR 1026753 · Zbl 0596.14012
[81] Thomason, R. W., Trobaugh, T.: Higher algebraicK-theory of schemes and of derived categories. In: The Grothendieck Festschrift, Vol. III, Progr. Math. 88, Birkh¨auser Boston, Boston, MA, 247-435 (1990)Zbl 0731.14001 MR 1106918
[82] Van den Bergh, M.: A note on gradedK-theory. Comm. Algebra14, 1561-1564 (1986)Zbl 0602.16021 MR 0859452 · Zbl 0602.16021
[83] Waldhausen, F.
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