Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T11:45:03.970Z Has data issue: false hasContentIssue false
  • English
  • Français

Towards an understanding of ramified extensions of structured ring spectra

Published online by Cambridge University Press:  25 March 2018

BJØRN IAN DUNDAS ,
AYELET LINDENSTRAUSS  and
BIRGIT RICHTER
BJØRN IAN DUNDAS
Affiliation:
Department of Mathematics, University of Bergen, Postboks 7800, 5020 Bergen, Norway. e-mail: dundas@math.uib.no
AYELET LINDENSTRAUSS
Affiliation:
Mathematics Department, Indiana University, 831 East Third Street Bloomington, IN 47405, U.S.A., e-mail: alindens@indiana.edu
BIRGIT RICHTER
Affiliation:
Fachbereich Mathematik der Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany. e-mail: birgit.richter@uni-hamburg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose topological Hochschild homology as a tool for measuring ramification of maps of structured ring spectra. We determine second order topological Hochschild homology of the p-local integers. For the tamely ramified extension of the map from the connective Adams summand to p-local complex topological K-theory we determine the relative topological Hochschild homology and show that it detects the tame ramification of this extension. We show that the complexification map from connective topological real to complex K-theory shows features of a wildly ramified extension. We also determine relative topological Hochschild homology for some quotient maps with commutative quotients.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 3 , May 2020 , pp. 435 - 454
Copyright
Copyright © Cambridge Philosophical Society 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Angeltveit, Vigleik. Topological Hochschild homology and cohomology of A ∞ ring spectra. Geom. Topol. 12 (2008), 987–1032.10.2140/gt.2008.12.987CrossRefGoogle Scholar
Ausoni, Christian. Topological Hochschild homology of connective complex K-theory. Amer. J. Math. 127 (2005), 1261–1313.10.1353/ajm.2005.0036CrossRefGoogle Scholar
Blumberg, Andrew, Cohen, Ralph L. and Schlichtkrull, Christian. Topological Hochschild homology of Thom spectra and the free loop space. Geom. Topol. 14 (2010), 1165–1242.10.2140/gt.2010.14.1165CrossRefGoogle Scholar
Bobkova, Irina, Lindenstrauss, Ayelet, Poirier, Kate, Richter, Birgit and Zakharevich, Inna. On the higher topological Hochschild homology of 𝔽p and commutative 𝔽p-group algebras, Women in Topology: Collaborations in Homotopy Theory. Contemp. Math. 641 AMS, (2015), 97–122.10.1090/conm/641/12856CrossRefGoogle Scholar
Bobkova, Irina, Höning, Eva, Lindenstrauss, Ayelet, Poirier, Kate, Richter, Birgit and Zakharevich, Inna. Higher THH and higher Shukla homology of ℤ/pmℤ and of truncated polynomial algebras over 𝔽p, in preparation.Google Scholar
Bökstedt, Marcel. Topological Hochschild homology, preprint.Google Scholar
Bökstedt, Marcel. The topological Hochschild homology of ℤ and of ℤ/pℤ, preprint.Google Scholar
Robert, R. Bruner, May, J. Peter, McClure, James E. and Steinberger, Mark. H ∞ ring spectra and their applications. Lecture Notes in Math., 1176 (Springer-Verlag, Berlin, 1986), viii+388 pp.Google Scholar
Dundas, Bjørn Ian, Lindenstrauss, Ayelet and Richter, Birgit. On higher topological Hochschild homology of rings of integers, to appear in Math. Res. Lett. arXiv:1502.02504.Google Scholar
Anthony, D. Elmendorf, Kriz, Igor, Mandell, Michael A. and May, J. Peter. Rings, modules, and algebras in stable homotopy theory, With an appendix by M. Cole. Math. Surv. Monogr. 47 (American Mathematical Society, Providence, RI, 1997), xii+249 pp.Google Scholar
Franjou, Vincent, Lannes, Jean and Schwartz, Lionel. Autour de la cohomologie de Mac Lane des corps finis. Invent. Math. 115 (1994), 513–538.10.1007/BF01231771CrossRefGoogle Scholar
Franjou, Vincent and Pirashvili, Teimuraz. On the MacLane cohomology for the ring of integers. Topology 37 (1998), 109–114.10.1016/S0040-9383(97)00005-0CrossRefGoogle Scholar
John, P. C. Greenlees. Ausoni-Bökstedt duality for topological Hochschild homology, J. Pure Appl. Alg. 220 (2016), 1382–1402.Google Scholar
Greither, Cornelius. Cyclic Galois extensions of commutative rings. Lecture Notes in Math. 1534 (Springer-Verlag, Berlin, 1992), x+145 pp.Google Scholar
Hill, Michael and Lawson, Tyler. Automorphic forms and cohomology theories on Shimura curves of small discriminant. Adv. Math. 225 (2010), 1013–1045.10.1016/j.aim.2010.03.009CrossRefGoogle Scholar
Klang, Inbar. The factorisation homology of Thom spectra and twisted non-abelian Poincaré duality, preprint, arXiv:1606.03805.Google Scholar
Larsen, Michael and Lindenstrauss, Ayelet. Cyclic homology of Dedekind domains. K-Theory 6 (1992), 301–334.10.1007/BF00966115CrossRefGoogle Scholar
Lindenstrauss, Ayelet and Madsen, Ib. Topological Hochschild homology of number rings. Trans. Amer. Math. Soc. 352 (2000), 2179–2204.10.1090/S0002-9947-00-02611-8CrossRefGoogle Scholar
Mathew, Akhil. The Galois group of a stable homotopy theory. Adv. Math. 291 (2016), 403–541.10.1016/j.aim.2015.12.017CrossRefGoogle Scholar
Lawson, Tyler and Naumann, Niko. Commutativity conditions for truncated Brown-Peterson spectra of height 2. J. Topol. 5 (2012), 137–168.10.1112/jtopol/jtr030CrossRefGoogle Scholar
Lawson, Tyler and Naumann, Niko. Strictly commutative realisations of diagrams over the Steenrod algebra and topological modular forms at the prime 2. Int. Math. Res. Not. 10 (2014), 2773–2813.10.1093/imrn/rnt010CrossRefGoogle Scholar
Robinson, Alan. Gamma homology, Lie representations and E ∞ multiplications. Invent. Math. 152 (2003), 331–348.10.1007/s00222-002-0272-5CrossRefGoogle Scholar
Rognes, John. Galois extensions of structured ring spectra. Stably dualizable groups. Mem. Amer. Math. Soc. 898 (2008), viii+137 pp.Google Scholar
Sagave, Steffen. Logarithmic structures on topological K-theory spectra. Geom. Topol. 18 (2014), 447–490.10.2140/gt.2014.18.447CrossRefGoogle Scholar
Schlichtkrull, Christian. Higher topological Hochschild homology of Thom spectra. J. Topol. 4 (2011), 161–189.10.1112/jtopol/jtq037CrossRefGoogle Scholar