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Galois equivariance and stable motivic homotopy theory


Authors: J. Heller and K. Ormsby
Journal: Trans. Amer. Math. Soc. 368 (2016), 8047-8077
MSC (2010): Primary 14F42, 55P91; Secondary 11E81, 19E15
DOI: https://doi.org/10.1090/tran6647
Published electronically: February 12, 2016
MathSciNet review: 3546793
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Abstract: For a finite Galois extension of fields $ L/k$ with Galois group $ G$, we study a functor from the $ G$-equivariant stable homotopy category to the stable motivic homotopy category over $ k$ induced by the classical Galois correspondence. We show that after completing at a prime and $ \eta $ (the motivic Hopf map) this results in a full and faithful embedding whenever $ k$ is real closed and $ L=k[i]$. It is a full and faithful embedding after $ \eta $-completion if a motivic version of Serre's finiteness theorem is valid. We produce strong necessary conditions on the field extension $ L/k$ for this functor to be full and faithful. Along the way, we produce several results on the stable $ C_2$-equivariant Betti realization functor and prove convergence theorems for the $ p$-primary $ C_2$-equivariant Adams spectral sequence.


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Additional Information

J. Heller
Affiliation: Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Address at time of publication: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
Email: jeremiahheller.math@gmail.com

K. Ormsby
Affiliation: Department of Mathematics, Reed College, 3203 SE Woodstock Boulevard, Portland, Oregon 97202-8199
Email: ormsbyk@reed.edu

DOI: https://doi.org/10.1090/tran6647
Keywords: Equivariant and motivic stable homotopy theory, equivariant Betti realization
Received by editor(s): June 14, 2014
Received by editor(s) in revised form: December 16, 2014, and March 22, 2015
Published electronically: February 12, 2016
Article copyright: © Copyright 2016 American Mathematical Society