Galois equivariance and stable motivic homotopy theory
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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.References
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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
- MR Author ID: 901183
- Email: jeremiahheller.math@gmail.com
- K. Ormsby
- Affiliation: Department of Mathematics, Reed College, 3203 SE Woodstock Boulevard, Portland, Oregon 97202-8199
- MR Author ID: 928471
- Email: ormsbyk@reed.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8047-8077
- MSC (2010): Primary 14F42, 55P91; Secondary 11E81, 19E15
- DOI: https://doi.org/10.1090/tran6647
- MathSciNet review: 3546793