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Register a new userProfinite and discrete G-spectra and iterated homotopy fixed points
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For a profinite group $G$, let $(text{-})^{hG}$, $(text{-})^{h_dG}$, and $(text{-})^{hG}$ denote continuous homotopy fixed points for profinite $G$-spectra, discrete $G$-spectra, and continuous $G$-spectra (coming from towers of discrete $G$-spectra), respectively. We establish some connections between the first two notions, and by using Postnikov towers, for $K vartriangleleft_c G$ (a closed normal subgroup), give various conditions for when the iterated homotopy fixed points $(X^{hK})^{hG/K}$ exist and are $X^{hG}$. For the Lubin-Tate spectrum $E_n$ and $G <_c G_n$, the extended Morava stabilizer group, our results show that $E_n^{hK}$ is a profinite $G/K$-spectrum with $(E_n^{hK})^{hG/K} simeq E_n^{hG}$, by an argument that possesses a certain technical simplicity not enjoyed by either the proof that $(E_n^{hK})^{hG/K} simeq E_n^{hG}$ or the Devinatz-Hopkins proof (which requires $|G/K| < infty$) of $(E_n^{dhK})^{h_dG/K} simeq E_n^{dhG}$, where $E_n^{dhK}$ is a construction that behaves like continuous homotopy fixed points. Also, we prove that (in general) the $G/K$-homotopy fixed point spectral sequence for $pi_ast((E_n^{hK})^{hG/K})$, with $E_2^{s,t} = H^s_c(G/K; pi_t(E_n^{hK}))$ (continuous cohomology), is isomorphic to both the strongly convergent Lyndon-Hochschild-Serre spectral sequence of Devinatz for $pi_ast(E_n^{dhG})$, with $E_2^{s,t} = H^s_c(G/K; pi_t(E_n^{dhK}))$, and the descent spectral sequence for $pi_ast((E_n^{hK})^{hG/K})$.
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If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X^{hG} is often given by (H_{G,X})^G, where H_{G,X} is a certain explicit construction given by a homotopy limit in the category of discrete G-spectra. Thus, in each of two common equivariant settings, the homotopy fixed point spectrum is equal to the fixed points of an explicit object in the ambient equivariant category. We enrich this pattern by proving in a precise sense that the discrete G-spectrum H_{G,X} is just a profinite version of Map_*(EK_+, Z): at each stage of its construction, H_{G,X} replicates in the setting of discrete G-spectra the corresponding stage in the formation of Map_*(EK_+, Z) (up to a certain natural identification).
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.
In this thesis we will investigate rational G-spectra for a profinite group G. We will provide an algebraic model for this model category whose injective dimension can be calculated in terms of the Cantor-Bendixson rank of the space of closed subgroups of G, denoted SG. The algebraic model we consider is chain complexes of Weyl-G-sheaves of rational vector spaces over the spaces. The key step in proving that this is an algebraic model for G-spectra is in proving that the category of rational G-Mackey functors is equivalent to Weyl-G-sheaves. In addition to the fact that this sheaf description utilises the topology of G and the closed subgroups of G in a more explicit way than Mackey functors do, we can also calculate the injective dimension. In the final part of the thesis we will see that the injective dimension of the category of Weyl-G-sheaves can be calculated in terms of the Cantor-Bendixson rank of SG, hence giving the injective dimension of the category of Mackey functors via the earlier equivalence.
We give a new description of Rosenthals generalized homotopy fixed point spaces as homotopy limits over the orbit category. This is achieved using a simple categorical model for classifying spaces with respect to families of subgroups.
For a finite p-group G and a bounded below G-spectrum X of finite type mod p, the G-equivariant Segal conjecture for X asserts that the canonical map X^G --> X^{hG} is a p-adic equivalence. Let C_{p^n} be the cyclic group of order p^n. We show that if the C_p Segal conjecture holds for a C_{p^n} spectrum X, as well as for each of its C_{p^e} geometric fixed points for 0 < e < n, then then C_{p^n} Segal conjecture holds for X. Similar results hold for weaker forms of the Segal conjecture, asking only that the canonical map induces an equivalence in sufficiently high degrees, on homotopy groups with suitable finite coefficients.
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