I am trying to compare the coherent cohomology of the generic fiber of Lubin-Tate space to the coherent cohomology of Lubin-Tate space considered rationally, and I am going in circles! I would be very appreciative of any insight or direction on the topic.
Let's establish a bit of background and notation. Fix a $h \in \mathbb{N}$. Let $k = \mathbb{F}_{p^h}$. Let $\mathrm{Art}_k$ be the category of Artinian local rings $R$ and their maximal ideals, such that $R/\mathfrak{m}$ is a $k$-algebra. Let $\mathrm{FGL}_{R}$ be the category of formal group laws over a ring $R \in \mathrm{Art}_k$. Fix $H \in \mathrm{FGL}_k$ to be a height $h$ formal group law. The groupoid $\mathrm{Def_H(R)}$ has as objects: $$\{G \in \mathrm{FGL}_{R}, \iota: H \xrightarrow{\simeq} G \otimes_R k\},$$ and as morphisms: isomorphisms $f$ of formal group laws over $R$ which reduces to the identity modulo $\mathfrak{m}$.
We define another groupoid $\mathrm{DefU_H(R)}$ which has as objects: $$\{G \in \mathrm{FGL}_{R}, a \in R^\times, \iota: H \xrightarrow{\simeq} G \otimes_R k\},$$ and as morphisms: $f \in \mathrm{Hom}((G, a, \iota), (F, b, \delta))$ are isomorphisms $f$ of formal group laws $G, F$ over $R$, such that $a = f'(0)b$.
There is an action of the group $J := \mathrm{Aut}_k(H)$ on $\mathrm{Def}_H(R)$ and $\mathrm{DefU}_H(R)$, which acts by each $j \in J$ taking $(G, \iota)$ to $(G, \iota \circ j),$ and $(G, a, \iota)$ to $(G, a, \iota \circ j)$ respectively. The functor $\mathrm{Def}_H(-)$ is represented by a formal scheme, $LT := \mathrm{Spf} \text{ } W(k)[[u_1, ..., u_{h-1}]],$ where $W(k)$ denotes the Witt vectors of $k$. The functor $\mathrm{DefU}_H(-)$ is represented by the formal scheme $LTU :=\mathrm{Spf} \text{ }W(k)[[u_1, ..., u_{h-1}]][u, u^{-1}]$.
We denote the adic space associated to $LT$ by $LT^{ad}$. The adic space $LT^{ad}$ lives over $\mathrm{Spa}(\mathbb{Z}_p, \mathbb{Z}_p)$. We may take the rigid generic fiber of the formal scheme $LT$, $$LT^\eta := X^{ad} \times_{\mathrm{Spa}(\mathbb{Z}_p, \mathbb{Z}_p)} \mathrm{Spa}(\mathbb{Q}_p, \mathbb{Z}_p).$$$$LT^\eta := LT^{ad} \times_{\mathrm{Spa}(\mathbb{Z}_p, \mathbb{Z}_p)} \mathrm{Spa}(\mathbb{Q}_p, \mathbb{Z}_p).$$
Here is my question: What is the relationship between these two continuous group cohomologies? Are they isomorphic?
$$H_{cts}^*(J,\Gamma(\mathcal{O}_{LT}, LT)) \otimes_{\mathbb{Z}} \mathbb{Q} \stackrel{?}{\simeq} H_{cts}^*(J, \Gamma(\mathcal{O}_{LT^\eta}, LT^{\eta})).$$
I have the same question for the case with units thrown in: how are these two related? $$H_{cts}^*(J,\Gamma(\mathcal{O}_{LTU}, LTU)) \otimes_{\mathbb{Z}} \mathbb{Q} \stackrel{?}{\simeq} H_{cts}^*(J, \Gamma( \mathcal{O}_{LTU^\eta}, LTU^{\eta})).$$
Here is my sub-question: Is $\Gamma(\mathcal{O}_{LT^\eta}, LT^{\eta}) \simeq \Gamma(\mathcal{O}_{LT}, LT) \hat{\otimes}_{\mathbb{Z}_p} \mathbb{Q}_p$?
By the following argument, it seems that this is true. However, I have a strange feeling in my gut about it. I am almost sure I have missed something, but combing through it I can't figure out what.
Let us denote the ring $L := W(k)[[u_1, ..., u_{h-1}]] \simeq \Gamma(\mathcal{O}_{LT}, LT)$. One might expect $LT^{\eta}$ to be $\mathrm{Spa}(L \hat{\otimes}_{\mathbb{Z}_p} \mathbb{Q}_p, L)$, but $L$ isn't open in $L \hat{\otimes}_{\mathbb{Z}_p} \mathbb{Q}_p$. This is because, for example, $p^{-1}u_1^{\text{ }m} \to 0$, but this sequence never enters $L$. Thus, $(L \hat{\otimes}_{\mathbb{Z}_p} \mathbb{Q}_p, L)$ isn't a Huber pair. Instead, if I understand correctly, $$LT^{\eta} := \mathrm{colim}_m \text{ }\mathrm{Spa} \left(L \left\langle \frac{u_1^{\text{ }m}}{p}, ...,\frac{u_{h-1}^{\text{ }m}}{p} \right\rangle \otimes \mathbb{Q}_p, L\left\langle \frac{u_1^{\text{ }m}}{p}, ...,\frac{u_{h-1}^{\text{ }m}}{p} \right\rangle \right),$$
where $U^{\underline{s}}$ denotes $u_1^{s_1}\cdots u_{h-1}^{s_{h-1}}$, and
$$ L \left\langle \frac{u_1^{\text{ }m}}{p}, ...,\frac{u_{h-1}^{\text{ }m}}{p} \right\rangle := \left\{ \sum_{\underline{s} \in \mathbb{N}^{h-1}} c_{\underline{s}}U^{\underline{s}}; c_{\underline{s}} \in L^{h-1} \text{ } \Big| \text{ } \lim_{s_i \to \infty} |c_{s_i}p^{s_i/m}| = 0 \text{ },\forall i \in \{1, ..., h-1\}\right \}.$$
This $(LT^{\eta})_m$ defines an increasing family of neighborhoods of the origin. Since a power of $u_i$ is divisible by $p$, the $(p, u_1, ..., u_{h-1})$-adic topology agrees with the $p$-adic topology on each of these rings. It seems that the global sections $\Gamma(\mathcal{O}_{LT^\eta}, LT^{\eta})$ are the completion of the intersection of these rings as $m \to \infty$, which is $L \hat{\otimes} \mathbb{Q}_p$. Thus, $\Gamma(\mathcal{O}_{LT^\eta}, LT^{\eta}) \simeq L \hat{\otimes} _{\mathbb{Z}_p} \mathbb{Q}_p$ as claimed.