Geometry & Topology

Localization theorems in topological Hochschild homology and topological cyclic homology

Andrew J Blumberg and Michael A Mandell

Full-text: Open access

Abstract

We construct localization cofibration sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of small spectral categories. Using a global construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofibration sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason–Trobaugh in K–theory. We also deduce versions of Thomason’s blow-up formula and the projective bundle formula for THH and TC.

Article information

Source
Geom. Topol., Volume 16, Number 2 (2012), 1053-1120.

Dates
Received: 18 November 2010
Revised: 7 February 2012
Accepted: 7 March 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732414

Digital Object Identifier
doi:10.2140/gt.2012.16.1053

Mathematical Reviews number (MathSciNet)
MR2928988

Zentralblatt MATH identifier
1282.19004

Subjects
Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]
Secondary: 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)

Keywords
topological Hochschild homology topological cyclic homology localization sequence Mayer–Vietoris sequence projective bundle theorem blow-up formula

Citation

Blumberg, Andrew J; Mandell, Michael A. Localization theorems in topological Hochschild homology and topological cyclic homology. Geom. Topol. 16 (2012), no. 2, 1053--1120. doi:10.2140/gt.2012.16.1053. https://projecteuclid.org/euclid.gt/1513732414


Export citation

References

  • S Bloch, S Lichtenbaum, A spectral sequence for motivic cohomology, preprint (1995) Available at \setbox0\makeatletter\@url http://www.math.uiuc.edu/K-theory/0062/ {\unhbox0
  • A J Blumberg, M A Mandell, Algebraic $K$–theory and abstract homotopy theory, Adv. Math. 226 (2011) 3760–3812
  • M B ökstedt, Topological Hochschild homology, Bielefeld preprint (1988)
  • M B ökstedt, W C Hsiang, I Madsen, The cyclotomic trace and algebraic $K$–theory of spaces, Invent. Math. 111 (1993) 465–539
  • A I Bondal, M Larsen, V A Lunts, Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. 2004 (2004) 1461–1495
  • A K Bousfield, The localization of spaces with respect to homology, Topology 14 (1975) 133–150
  • G Cortiñas, Infinitesimal $K$–theory, J. Reine Angew. Math. 503 (1998) 129–160
  • G Cortiñas, C Haesemeyer, M Schlichting, C Weibel, Cyclic homology, cdh-cohomology and negative $K$–theory, Ann. of Math. 167 (2008) 549–573
  • G Cortiñas, C Haesemeyer, C Weibel, $K$–regularity, $cdh$–fibrant Hochschild homology, and a conjecture of Vorst, J. Amer. Math. Soc. 21 (2008) 547–561
  • V Drinfeld, DG quotients of DG categories, J. Algebra 272 (2004) 643–691
  • D Dugger, Spectral enrichments of model categories, Homology, Homotopy Appl. 8 (2006) 1–30
  • D Dugger, B Shipley, Enriched model categories and an application to additive endomorphism spectra, Theory Appl. Categ. 18 (2007) 400–439
  • B I Dundas, R McCarthy, Topological Hochschild homology of ring functors and exact categories, J. Pure Appl. Algebra 109 (1996) 231–294
  • A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math.Surveys and Monogr. 47, Amer. Math. Soc. (1997) With an appendix by M Cole
  • E M Friedlander, A Suslin, The spectral sequence relating algebraic $K$–theory to motivic cohomology, Ann. Sci. École Norm. Sup. 35 (2002) 773–875
  • T Geisser, L Hesselholt, Topological cyclic homology of schemes, from: “Algebraic $K$–theory (Seattle, WA, 1997)”, (W Raskind, C Weibel, editors), Proc. Sympos. Pure Math. 67, Amer. Math. Soc., Providence, RI (1999) 41–87
  • T Geisser, L Hesselholt, On the vanishing of negative $K$–groups, Math. Ann. 348 (2010) 707–736
  • T G Goodwillie, Relative algebraic $K$–theory and cyclic homology, Ann. of Math. 124 (1986) 347–402
  • L Hesselholt, I Madsen, Cyclic polytopes and the $K$–theory of truncated polynomial algebras, Invent. Math. 130 (1997) 73–97
  • L Hesselholt, I Madsen, On the $K$–theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997) 29–101
  • L Hesselholt, I Madsen, On the $K$–theory of local fields, Ann. of Math. 158 (2003) 1–113
  • P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003)
  • M Hovey, J H Palmieri, N P Strickland, Axiomatic stable homotopy theory, 128, no. 610, Amer. Math. Soc. (1997)
  • M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
  • B Keller, On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999) 1–56
  • G M Kelly, Basic concepts of enriched category theory, London Math. Soc. Lecture Note Series 64, Cambridge Univ. Press (1982)
  • T A Kro, Model structure on operads in orthogonal spectra, Homology, Homotopy Appl. 9 (2007) 397–412
  • M Levine, The homotopy coniveau tower, J. Topol. 1 (2008) 217–267
  • L G Lewis, Jr, Is there a convenient category of spectra?, J. Pure Appl. Algebra 73 (1991) 233–246
  • L G Lewis, Jr, J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Math. 1213, Springer, Berlin (1986) With contributions by J E McClure
  • M A Mandell, J P May, Equivariant orthogonal spectra and $S$–modules, 159, no. 755, Amer. Math. Soc. (2002)
  • M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
  • J P May, The geometry of iterated loop spaces, Lectures Notes in Math. 271, Springer, Berlin (1972)
  • J P May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Math. 91, Conference Board of the Math. Sciences, Washington, DC (1996) With contributions by M Cole, G Comezaña, S Costenoble, A D Elmendorf, J P C Greenlees, L G Lewis, Jr., R J Piacenza, G Triantafillou and S Waner
  • R McCarthy, Relative algebraic $K$–theory and topological cyclic homology, Acta Math. 179 (1997) 197–222
  • A Neeman, The connection between the $K$–theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. 25 (1992) 547–566
  • S Schwede, B Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003) 287–334
  • S Schwede, B Shipley, Stable model categories are categories of modules, Topology 42 (2003) 103–153
  • B Shipley, Symmetric spectra and topological Hochschild homology, $K$–Theory 19 (2000) 155–183
  • B Shipley, $H\mathbb Z$–algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351–379
  • R W Thomason, Algebraic $K$–theory and étale cohomology, Ann. Sci. École Norm. Sup. 18 (1985) 437–552
  • R W Thomason, Les $K$–groupes d'un schéma éclaté et une formule d'intersection excédentaire, Invent. Math. 112 (1993) 195–215
  • R W Thomason, T Trobaugh, Higher algebraic $K$–theory of schemes and of derived categories, from: “The Grothendieck Festschrift, Vol. III”, (P Cartier, L Illusie, N M Katz, G Laumon, K A Ribet, editors), Progr. Math. 88, Birkhäuser, Boston, MA (1990) 247–435
  • B Toën, G Vezzosi, A remark on $K$–theory and $S$–categories, Topology 43 (2004) 765–791
  • F Waldhausen, Algebraic $K$–theory of topological spaces. II, from: “Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, 1978)”, (J L Dupont, I H Madsen, editors), Lecture Notes in Math. 763, Springer, Berlin (1979) 356–394
  • F Waldhausen, Algebraic $K$–theory of spaces, from: “Algebraic and geometric topology (New Brunswick, NJ, 1983)”, (A Ranicki, N Levitt, F Quinn, editors), Lecture Notes in Math. 1126, Springer, Berlin (1985) 318–419