I am trying to understand example I.2.3 of the article "On Topological Cyclic Homology", i.e. im trying to see the necesity of the bounded below assumption in the Tate Orbit Lemma (Lemma I.2.1) by seeing that the Anderson dual of the sphere spectrum $I_Z$ equipped with the trivial $C_{p^2}$-action does not satisfy the Tate Orbit lemma directly. This should follow analogously to the previous example which was for the sphere spectrum.

If we consider the fiber sequence induced from the tate construction, and apply $(-)^{t(C_{p^2}/C_p)}$,

\begin{align*} ((I_Z)_{hC_p})^{t(C_{p^2}/C_p)} \to ((I_Z)^{hC_p})^{t(C_{p^2}/C_p)} \to ((I_Z)^{tC_p})^{t(C_{p^2}/C_p)} \end{align*}

Because the Anderson dual is bounded above the middle term vanishes by the Tate fixpoint theorem (Lemma I.2.2), and hence from a long exact sequence we have

\begin{align*} \Sigma((I_Z)_{hC_p})^{t(C_{p^2}/C_p)} \simeq ((I_Z)^{tC_p})^{t(C_{p^2}/C_p)} \simeq (I_Z)^{tC_p}. \end{align*}

At this point in the example for the sphere spectrum the Segal conjecture is employed, to see that $(\mathbb{S}^{hC_p})^{t(C_{p^2}/C_p)} \simeq \mathbb{S}^{\wedge}_p$, and hence not trivial. As far as i can tell the analog is not available for $I_Z$?

I have tried to use the Tate spectral sequence, but to no avail. I have also tried to use the defining fiber sequence of $I_Z$ together with the above fiber sequence, again with no payoff.

At this point i would be happy with a hint as to why $(I_Z)^{tC_p}$ is non-vanishing.


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