Yoshida - Non-abelian Lubin-Tate 3

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.

Current Reference: Drinfeld, "Elliptic Modules".

1. Formal Modules

• The definition of a formal group and a homomorphism coincide with the one with Yoshida. 
• There is a canonical homomorphism $D: \mathrm{End}(F) \to B$ where $F$ is a formal group over a ring $B$. If $\phi$ is an endomorphism of $F$, then $D(\phi) = \phi'(0)$.
• This is a homomorphism as $D(\phi \circ \psi) = \phi'(\psi(0)) \cdot \psi'(0)$, but as $\psi \in (X) \subset B[[X]]$, we conclude that $D(\phi \circ \psi) = \phi'(0) \psi'(0)$.
• (Example) $F(X,Y) = X+Y$ is a formal group over any ring. This group is called additive. 
• We consider an endomorphism $\phi$ of $F$ over a ring $B$ with characteristic $p$. This means that
  $$\phi(F(X,Y)) = F(\phi(X), \phi(Y)) \Rightarrow \phi(X+Y) = \phi(X)+\phi(Y)$$  
• Write $\phi(X) = \sum a_i X^i$, then $\phi(X+Y) = \sum a_i (X+Y)^i$ and $\phi(X) + \phi(Y) = \sum a_i (X^i + Y^i)$. If $p|i$ or $i=1$, then $(X+Y)^i = X^i + Y^i$, so $a_i$ survives, but when $p\not| i$, ''intermediate coefficient" in $(X+Y)^i$ is nonzero, hence creates an addition term $a_i X^k Y^l$ with $k+l= i$. Therefore, $a_i$ dies.
  
  As a conclusion, $\phi(X) = \sum_{i=0} a_i X^{p^i}$. 
• Elements in $B$ is identified with the endomorphism $bX$. We also have the Frobenius endomorphism $X^p$ denoted. Hence the additive group consists of the ''series'' $\sum b_i \tau^p$ with the rule that
  $$\tau b = b^p \tau$$
  We denote the ring as $B\{ \{ \tau \} \}$.
• Let $B$ be an $\mathcal{O}$-algebra. A Formal $\mathcal{O}$-module over $B$ is the pair $(F, f)$ where $F$ is a formal group over $B$ and $f: \mathcal{O} \to \mathrm{End}(F)$ such that $D \circ f = \gamma$ where $\gamma$ is the structure map of $B$ as a $\mathcal{O}$-module. 
• As $D$ is exactly looking at its linear term and $\mathcal{O}$ acts on $B$ exactly by $\gamma$, $D(f(a)) = \gamma(a)$ is equivalent to saying that $f(a) \equiv aX ~(\mathrm{mod}~X^2)$, i.e., coincide with the definition given in Yoshida's paper.