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 $pi_ast((E_n^{hK})^{hG/K})$.
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 gi
ven 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 n be any positive integer and p any prime. Also, let X be any spectrum and let K(n) denote the nth Morava K-theory spectrum. Then we construct a descent spectral sequence with abutment pi_*(L_{K(n)}(X)) and E_2-term equal to the continuous cohomo
logy of G_n, the extended Morava stabilizer group, with coefficients in a certain discrete G_n-module that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)-local E_n-Adams spectral sequence for pi_*(L_{K(n)}(X)), whose E_2-term is not known to always be equal to a continuous cohomology group.
We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of M. J. Hop
kins that certain towers of generalized Moore spectra, closely related to the K(n)-local sphere, are E-infinity algebras in the category of pro-spectra. In addition, we show that Adams resolutions automatically satisfy the above rigidity criterion. In order to carry this out we develop the concept of an operadic model category, whose objects have homotopically tractable endomorphism operads.
Let G be a profinite group, {X_alpha}_alpha a cofiltered diagram of discrete G-spectra, and Z a spectrum with trivial G-action. We show how to define the homotopy fixed point spectrum F(Z, holim_alpha X_alpha)^{hG} and that when G has finite virtual
cohomological dimension (vcd), it is equivalent to F(Z, holim_alpha (X_alpha)^{hG}). With these tools, we show that the K(n)-local Spanier-Whitehead dual is always a homotopy fixed point spectrum, a well-known Adams-type spectral sequence is actually a descent spectral sequence, and, for a sufficiently nice k-local profinite G-Galois extension E, with K a closed normal subgroup of G, the equivalence (E^{h_kK})^{h_kG/K} simeq E^{h_kG} (due to Behrens and the author), where (-)^{h_k(-)} denotes k-local homotopy fixed points, can be upgraded to an equivalence that just uses ordinary (non-local) homotopy fixed points, when G/K has finite vcd.
Let n geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author
of this note, show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_n wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n wedge X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to pi_ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts to pi_ast((L_{K(n)}(E_n wedge X))^{hG_n}).
If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a
profinite group, the fibrant objects in the model category of discrete G-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete G-spectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G.
