Chromatic Vanishing Conjecture for the Lubin–Tate ring

Let $R_n$ be the Lubin–Tate ring associated to the Honda formal group law $\mathbb{H}_n$ of height $n$ over $\mathbb{F}_{p^n}$, with $R_n\cong \mathbb{W}[[u_1,\ldots,u_{n-1}]]$, where $\mathbb{W}$ denotes the Witt vectors on $\mathbb{F}_{p^n}$. Let $\mathbb{S}_n$ be the group of automorphisms of $\mathbb{H}_n$ over $\mathbb{F}_{p^n}$, let $\mathbb{G}_n$ be its extension by $\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$, and consider the natural maps $\mathbb{W}\to R_n$ and $\mathbb{F}_{p^n}\to R_n/p$. Chromatic Vanishing Conjecture. The following statements hold: (1) the continuous cohomology and homology of $R_n/\mathbb{W}$ vanish in all degrees, so that

$$H^*(\mathbb{G}_n,R_n)\cong H^*(\mathbb{G}_n,\mathbb{W}),\qquad H_*(\mathbb{G}_n,R_n)\cong H_*(\mathbb{G}_n,\mathbb{W});$$

(2) the continuous cohomology and homology of $(R_n/p)/\mathbb{F}_{p^n}$ vanish in all degrees, so that

$$H^*(\mathbb{G}_n,R_n/p)\cong H^*(\mathbb{G}_n,\mathbb{F}_{p^n}),\qquad H_*(\mathbb{G}_n,R_n/p)\cong H_*(\mathbb{G}_n,\mathbb{F}_{p^n}).$$

These vanishing assertions are intended to support analyses of the Chromatic Splitting Conjecture at height $2$, especially at the primes $2$ and $3$; the supplied source does not establish the conjecture in general.