Galois equivariance and stable motivic homotopy theory

Heller, J.; Ormsby, K.

doi:10.1090/tran6647

Galois equivariance and stable motivic homotopy theory
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by J. Heller and K. Ormsby PDF

Trans. Amer. Math. Soc. 368 (2016), 8047-8077 Request permission

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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