Geodesic
THH and base-change for Galois extensions of ring spectra
Mathew, Akhil1
1Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, United States
Algebraic and Geometric Topology, Tome 17 (2017) no. 2, pp. 693-704
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We treat the question of base-change in THH for faithful Galois extensions of ring spectra in the sense of Rognes. Given a faithful Galois extension A → B of ring spectra, we consider whether the map THH(A) ⊗AB → THH(B) is an equivalence. We reprove and extend positive results of Weibel and Geller, and McCarthy and Minasian, and offer new examples of Galois extensions for which base-change holds. We also provide a counterexample where base-change fails.

DOI : 10.2140/agt.2017.17.693
Classification : 55P43, 13D03, 55P42
Keywords: topological Hochschild homology, Galois extensions, structured ring spectra

Mathew, Akhil  1

1 Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, United States 
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Mathew, Akhil. THH and base-change for Galois extensions of ring spectra. Algebraic and Geometric Topology, Tome 17 (2017) no. 2, pp. 693-704. doi: 10.2140/agt.2017.17.693

[1] R Banerjee, Galois descent for real spectra, J. Homotopy Relat. Struct. (2016) | DOI

[2] A J Blumberg, D Gepner, G Tabuada, A universal characterization of higher algebraic K–theory, Geom. Topol. 17 (2013) 733 | DOI

[3] A J Blumberg, M A Mandell, Localization theorems in topological Hochschild homology and topological cyclic homology, Geom. Topol. 16 (2012) 1053 | DOI

[4] D Clausen, A Mathew, N Naumann, J Noel, Descent in algebraic K–theory and a conjecture of Ausoni–Rognes, preprint (2016)

[5] J Lurie, Higher topos theory, 170, Princeton University Press (2009) | DOI

[6] J Lurie, Derived algebraic geometry, VII: Spectral schemes, preprint (2011)

[7] J Lurie, Derived algebraic geometry, XIII : Rational and p–adic homotopy theory, preprint (2011)

[8] J Lurie, Higher algebra, unpublished (2014)

[9] M A Mandell, E∞ algebras and p–adic homotopy theory, Topology 40 (2001) 43 | DOI

[10] A Mathew, The Galois group of a stable homotopy theory, Adv. Math. 291 (2016) 403 | DOI

[11] R Mccarthy, V Minasian, HKR theorem for smooth S–algebras, J. Pure Appl. Algebra 185 (2003) 239 | DOI

[12] J Mcclure, R Schwänzl, R Vogt, THH(R)≅R ⊗ S1 for E∞ ring spectra, J. Pure Appl. Algebra 121 (1997) 137 | DOI

[13] L Meier, United elliptic homology, PhD thesis, Universät Bonn (2012)

[14] J Rognes, Galois extensions of structured ring spectra: stably dualizable groups, 898, Amer. Math. Soc. (2008) | DOI

[15] C A Weibel, S C Geller, Étale descent for Hochschild and cyclic homology, Comment. Math. Helv. 66 (1991) 368 | DOI

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