Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A lift from group cohomology to spectra for trivial profinite actions   Let $G$ be a profinite group, $X$ a discrete $G$-spectrum with trivial
action, and $X^{hG}$ the continuous homotopy fixed points. For any $N
\trianglelefteq_o G$ ("$o$" for open), $X = X^N$ is a $G/N$-spectrum with
trivial action. We construct a zigzag $\text{colim}\,_N \,X^{hG/N}
\buildrel\Phi\over\longrightarrow \text{colim}\,_N \,(X^{hN})^{hG/N}
\buildrel\Psi\over\longleftarrow X^{hG}$, where $\Psi$ is a weak equivalence.
When $\Phi$ is a weak equivalence, this zigzag gives an interesting model for
$X^{hG}$ (for example, its Spanier-Whitehead dual is $\text{holim}\,_N
\,F(X^{hG/N}, S^0)$). We prove that this happens in the following cases: (1)
$|G| < \infty$; (2) $X$ is bounded above; (3) there exists $\{U\}$ cofinal in
$\{N\}$, such that for each $U$, $H^s_c(U, \pi_\ast(X)) = 0$, for $s > 0$.
Given (3), for each $U$, there is a weak equivalence $X
\buildrel\simeq\over\longrightarrow X^{hU}$ and $X^{hG} \simeq X^{hG/U}$. For
case (3), we give a series of corollaries and examples. As one instance of a
family of examples, if $p$ is a prime, $K(n_p,p)$ the $n_p$th Morava $K$-theory
$K(n_p)$ at $p$ for some $n_p \geq 1$, and $\mathbb{Z}_p$ the $p$-adic
integers, then for each $m \geq 2$, (3) is satisfied when $G \leqslant \prod_{p
\leq m} \mathbb{Z}_p$ is closed, $X = \bigvee_{p > m} (H\mathbb{Q} \vee
K(n_p,p))$, and $\{U\} := \{N_G \mid N_G \trianglelefteq_o G\}$.

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The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg