形式A-模在A变换下的模
Moduli of formal A-modules under change of A
摘要：
我们开发了从高度$dn$形式群律的自同构群的上同调计算限制映射的方法（即，高度$dn$ Morava稳定子群）与$A$ -高度$n$形式$A$ -模的自同构群的上同调，其中$A$是$\mathbb{Q}_p$的度$d$域扩展中的整数环。然后，我们计算这个映射的二次扩张的$\mathbb{Q}_p$和高度$2$摩拉瓦稳定剂组在素数$p>3$。证明了形式模的这些自同构群是Morava稳定群的闭子群，并利用局部类域理论证明了一个$A$ -高度$1$ -形式$A$ -模的自同构群与$K$的绝对Galois群的阿贝尔化的分歧部分，对于每个二次扩展$K/\mathbb{Q}_p$，在Lubin-Tate/Morava $E$ -理论谱$E_2$上产生$Gal(K^{ab}/K^{nr})$的作用。最后，我们运行相关的下降谱序列来计算该作用的同伦不动点的$V(1)$ -同伦群;一个结果是，对于史密斯-户田复形$V(1)$的$K(2)$ -局部同伦群中的每个元素，该元素或其对偶元素在$\mathbb{Q}_p$的某个二次扩张的阿贝尔闭包的伽罗瓦上同调中被检测到。
英文摘要：
We develop methods for computing the restriction map from the cohomology of the automorphism group of a height $dn$ formal group law (i.e., the height $dn$ Morava stabilizer group) to the cohomology of the automorphism group of an $A$-height $n$ formal $A$-module, where $A$ is the ring of integers in a degree $d$ field extension of $\mathbb{Q}_p$. We then compute this map for the quadratic extensions of $\mathbb{Q}_p$ and the height $2$ Morava stabilizer group at primes $p>3$. We show that the these automorphism groups of formal modules are closed subgroups of the Morava stabilizer groups, and we use local class field theory to identify the automorphism group of an $A$-height $1$-formal $A$-module with the ramified part of the abelianization of the absolute Galois group of $K$, yielding an action of $Gal(K^{ab}/K^{nr})$ on the Lubin-Tate/Morava $E$-theory spectrum $E_2$ for each quadratic extension $K/\mathbb{Q}_p$. Finally, we run the associated descent spectral sequence to compute the $V(1)$-homotopy groups of the homotopy fixed-points of this action; one consequence is that, for each element in the $K(2)$-local homotopy groups of the Smith-Toda complex $V(1)$, either that element or its dual is detected in the Galois cohomology of the abelian closure of some quadratic extension of $\mathbb{Q}_p$.
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