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We establish a formal framework for Rogness homotopical Galois theory and adapt it to the context of motivic spaces and spectra. We discuss examples of Galois extensions between Eilenberg-MacLane motivic spectra and between the Hermitian and algebraic K-theory spectra.
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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 Serres 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.
We show a number of Toda brackets in the homotopy of the motivic bordism spectrum $MGL$ and of the Real bordism spectrum $MU_{mathbb R}$. These brackets are red-shifting in the sense that while the terms in the bracket will be of some chromatic height $n$, the bracket itself will be of chromatic height $(n+1)$. Using these, we deduce a family of exotic multiplications in the $pi_{(ast,ast)}MGL$-module structure of the motivic Morava $K$-theories, including non-trivial multiplications by $2$. These in turn imply the analogous family of exotic multiplications in the $pi_{star}MU_mathbb R$-module structure on the Real Morava $K$-theories.
We explain how to reconstruct the category of Artin-Tate $mathbb{R}$-motivic spectra as a deformation of the purely topological $C_2$-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of $C_2$-equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of $tau$ philosophy that has revolutionized classical stable homotopy theory. A key observation is that the Artin-Tate subcategory of $mathbb{R}$-motivic spectra is easier to understand than the previously studied cellular subcategory. In particular, the Artin-Tate category contains a variant of the $tau$ map, which is a feature conspicuously absent from the cellular category.
Let E be a k-local profinite G-Galois extension of an E_infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rogness Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points.
We generalize Quillens $F$-isomorphism theorem, Quillens stratification theorem, the stable transfer, and the finite generation of cohomology rings from finite groups to homotopical groups. As a consequence, we show that the category of module spectra over $C^*(Bmathcal{G},mathbb{F}_p)$ is stratified and costratified for a large class of $p$-local compact groups $mathcal{G}$ including compact Lie groups, connected $p$-compact groups, and $p$-local finite groups, thereby giving a support-theoretic classification of all localizing and colocalizing subcategories of this category. Moreover, we prove that $p$-compact groups admit a homotopical form of Gorenstein duality.
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