Yoshida - Non-abelian Lubin-Tate 2

This will be a series of personal notes on the paper of Yoshida, "On Non-abelian Lubin-Tate Theory via Vanishing Cycles". The general format would be to have outlines in the first part and (stupid) questions in the second part. Also, in the middle, I will put background materials. Hopefully, most of the questions will be answered by the end of the month.

3. The Level $\pi$ Deformation Space

3.1 The equation of the space

• We would like to study $X = \mathrm{Spec}~A_1$ where $A_1$ is the deformation ring defined from the previous post. Then $X$ is a regular flat scheme over $S = \mathrm{Spec}~W$ of relative dimension $n-1$ with a (formally) smooth generic fiber.
• Notation-wise, we shorten $\widetilde{\Sigma}_n$ as simply $\widetilde{\Sigma}$. Similarly, we shorten $A_1$ as simply $A$.
• We denote the generators of $\mathfrak{m}$, the maximal ideal of $A$, by $X_1, \ldots, X_n$, i.e., $\mathfrak{m} = (X_1, \ldots, X_n )$
• (Assuming that $\widetilde{X}_i$ are variables, but I am not sure), we know that $X_1, \ldots, X_n$ is a set regular local parameters, hence the natural evaluation map
  $$W[[ \widetilde{X}_1, \ldots, \widetilde{X}_n ]] \to A$$
  defined by $\widetilde{X}_i \mapsto X_i$ is a surjective local $W$-algebra homomorphism. Hence if $I$ is the kernel, we obtain that $A= W[[ \widetilde{X}_1, \ldots, \widetilde{X}_n ]]/I$.
• $W[[ \widetilde{X}_1, \ldots, \widetilde{X}_n ]]$ is a $n+1$-dimensional regular local ring with the maximal ideal $\widetilde{\mathfrak{m}} = (\pi, \widetilde{X}_1, \ldots, \widetilde{X}_n )$
• $A$ regular $\Rightarrow I$ has height $1$. Therefore $I$ is principal. Let $t$ be its generator. $t$ is a part of the system of regular parameters of  $W[[ \widetilde{X}_1, \ldots, \widetilde{X}_n ]]$. This implies that $t \in I \backslash \widetilde{\mathfrak{m}}^2$.
• For any $t' \in I \backslash \widetilde{\mathfrak{m}}^2$,
• $\widetilde{\mathfrak{m}}(t') = \widetilde{\mathfrak{m}}^2 \cap (t')$
• $(t')/\widetilde{\mathfrak{m}}(t') \to I/\widetilde{\mathfrak{m}}I$ is an injection.
• Injection between 1-dimensional $\bar{k}$-vector spaces is an isomorphism
• Nakayama's Lemma implies that (t') = I.
• Then we can express $\pi$ as follows
  $$\pi = u_{\widetilde{\Sigma}} \cdot \prod_{\underline{a} \in k^n \backslash \{ \underline{0} \}} ( [a_1](X_1) +_{\widetilde{\Sigma}} \cdots +_{\widetilde{\Sigma}} [a_n](X_n))$$
  where $+_{\widetilde{\Sigma}}, [\cdot]$ denotes the operation of $\mathfrak{m}_{\widetilde{\Sigma} \otimes A}$. 
• $\widetilde{\Sigma}$ is defined over $A_0$. 
• $\widetilde{\Sigma}$ is a formal $\mathcal{O}_K$-module over $A_0$. We base change to $A$ by tensor. This means that $A:=A_1$ is an $A_0$-algebra. This is because of the fact that there exists a representing morphism (of the forgetting morphism) $A_0 \to A_m$ for all $m \geq 1$.

Questions:

• Why is $X$ defined above a regular flat scheme over $S = \mathrm{Spec}~W$ of relative dimension $n - 1$ with a (formally) smooth generic fiber?
• Is $\widetilde{X}_i$ simply a variable?
• So far, nothing went wrong when I assumed that $\widetilde{X}_i$ is just a variable.
• What do you mean by $\widetilde{\mathfrak{m}}(t')$? Why does it become an injection?
• Ideal product. The question boils down to saying that $\widetilde{\mathfrak{m}}I \cap (t') = \widetilde{\mathfrak{m}}(t')$ which is highly likely.