Yoshida - Non-abelian Lubin-Tate 7

• Now we would like to study the special fiber of $X:=\mathrm{A}$ over $S:=\mathrm{Spec}~W$. We denote the special fiber by $X_s:= X \times_S \mathrm{Spec}~\overline{k}$, i.e. the fiber of the special point (the maximal ideal). Sine this is simply just reducing mod $\pi$, combining with the result from the previous post where we computed the explicit formula for $A$, we obtain that
  
  $$X_s = \mathrm{Spec}~ \overline{k}[[\widetilde{X}_1, \ldots, \widetilde{X}_n]]/ \left ( \prod_{\underline{a} \in k^n \backslash \{ 0 \}} (P_{\underline{a}} ~\mathrm{mod}~ \pi ) \right )$$

• Let $Y_{\underline{a}}$ to be the closed subscheme of $X_s$ defined by $P_{\underline{a}} ~\mathrm{mod}~\pi = 0$. Equivalently, this is simply the closed subscheme of $X$ defined by $P_{\underline{a}} = 0$.
• It can be shown that $Y_{\underline{a}} = Y_{\underline{a'}}$ if and only if $a$ is a scalar multiple of $a'$ by some element in $k^\times$. Therefore, we can label the irreducible component of $X_s$ by $\underline{a} \in k^n \backslash \{ 0 \}$. As one can guess, this looks a lot like the projective space over $k$, so we may establish the following bijection.
• Let $\mathbb{P}$ denote the projective space over $\overline{k}$. Then for each $\underline{a} \in k^n \backslash \{ 0 \}$, we can define a $k$-rational hyperplane
  $$M_{\underline{a}} : a_1 X_1^* + \cdots + a_n X_n^* = 0$$
  where $X_i^*$ is the projective coordinate for $\mathbb{P}$. Define $Y_M = Y_{\underline{a}}$ for $M=M_{\underline{a}}$. This establishes a bijection between $k$-rational hyperplanes and the irreducible components of $X_s$. In particular, there exists $\frac{q^{n}-1}{q-1}$ many irreducible components.
• We will identify $\mathbb{P}$ as the exceptional divisor of the first blow up.

Blow-ups

Reference: Gortz pg 409 - 411.

• (Gortz) $X$ be a scheme and $Z$ be a closed subscheme of $X$. We define the blow-up of X along Z to be a morphism $p: X' \to X$ such that $p^{-1}(X)$ is a effective Cartier divisor. $p$ is universal in the following sense: if $p': X'' \to X$ is a morphism such that $p'^{-1}(X)$ is an effective Cartier divisor, then there exists a map $f: X'' \to X'$ such that $p' = p \circ f$.
• $p^{-1}(Z)$ is called the exceptional divisor of the blow-up. 
• Denote $Bl_Z(X)$ to be the blow-up of $X$ along $Z$ and $f: X' \to X$ a morphism of schemes. Then there exists a unique morphism $Bl_Z(f) : Bl_{f^{-1}(Z)}(X') \to Bl_Z(X)$ such that the following diagram commutes.
  $\require{AMScd}$
  $$\begin{CD} Bl_{f^{-1}(Z)}(X') @>{Bl_Z(f)}>>  Bl_Z(X) \\ @VVV @VVV \\ X' @>{f}>> X \end{CD}$$
• The above diagram becomes Cartesian if $f$ is flat.
• [Explicit Construction] Let $Z$ correspond to some quasi-coherent  sheaf of ideals $\mathcal{I}$. We denote $\mathcal{I}^n$ to be the $n$th power of $\mathcal{I}$ with $\mathcal{I}^0:=\mathcal{O}_X$. Then consider $X' = \mathrm{Proj} \bigoplus_{n \geq 0} \mathcal{I}^n$. As $\bigoplus \mathcal{I}^n$ is a $\mathcal{O}_X$-algebra, we obtain a structure morphism $p: X' \to X$. 
• The exceptional divisor also has an explicit form. It is
  $$\mathrm{Proj} \bigoplus_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n+1}$$
• Further assume that $\mathcal{I}$ is of finite type. This happens when $X$ is locally Noetherian. Then we have that $Bl_Z(X)$ is projective over $X$.
  

Regular Immersion

Reference: Tag 01AL, Tag 01B1 (Locally generated), Section 30.20-21

• Let $(X, \mathcal{O}_X)$ be a ringed space. $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. A (global section) $s \in \Gamma(X, \mathcal{F})$ induces a map $\mathcal{O}_X \to \mathcal{F}$ defined by $f \mapsto fs$. We say that $\mathcal{F}$ is generated by sections $(s_i)$ if there exists $s_i \in \Gamma(X, \mathcal{F})$ such that
  $$\bigoplus_{i \in I} \mathcal{O}_X \to \mathcal{F}$$
  is surjective. 
• The support of $\mathcal{F}$ is the set of $x \in X$ such that $\mathcal{F}_x \neq 0$. It is denoted by $\mathrm{Supp}(\mathcal{F})$.
• Let $(X, \mathcal{O}_X)$ be a ringed space. We say that $\mathcal{F}$, a sheaf of $\mathcal{O}_X$-modules, is locally generated by sections if  for every $x \in X$, there exists an open neighborhood $U$ of $x$ such that $\mathcal{F}|_U$ is (globally) generated generated as $\mathcal{O}_U$-module.
• Recall that if $R$ is a ring, then $f_1, \ldots, f_r \in R$ is called a regular sequence if $f_i$ is not a zero divisor in $R/(f_1, \ldots, f_{i-1})$ and $R/(f_1, \ldots, f_r) \neq 0$.
• Let $\mathcal{I} \subset \mathcal{O}_X$ be a sheaf of ideals. We say that $\mathcal{I}$ is regular if for all $x \in \mathrm{Supp}(\mathcal{O}_X/\mathcal{I})$, there exists an open neighborhood of $x$, denoted $U \subset X$ and a regular sequence $f_1, \ldots, f_r \in \mathcal{O}_X(U)$ such that $\mathcal{I}|_U$ is generated by $f_1, \ldots, f_r$.
• Given an immersion $f: Y \to X$. We don't have a unique corresponding quasi-coherent sheaf as we did for closed immersion. Let $Y$ be a closed subscheme of $U$ for some open subscheme $U$ of $X$. There exists $\mathcal{I}$ a quasi-coherent sheaf of ideals corresponding to $Y$ as a closed subscheme of $U$. If $U'$ is another open subscheme of $X$ such that $Y$ is a closed subscheme of, let $\mathcal{I}'$ the corresponding quasi-coherent sheaf of ideals. Then we have the following
• $\mathcal{I}|_{U \cap U'} = \mathcal{I}'|_{U \cap U'}$,
• $\mathcal{I}$ is regular if and only if $\mathcal{I}'$ is regular.
• We say that an immersion $Y \to X$ is regular if the corresponding quasi-coherent sheaf $\mathcal{I}$ is regular.

Codimension

Reference: Gortz, pg 128

• [Irreducible, closed] Let $Z \subset X$ be a closed irreducible subset of $X$. Then codimension of $Z$ in $X$ is the supremum of the length of chains of irreducible closed subsets $Z_0 \supsetneq \cdots \supsetneq Z_l = Z$.
• [Closed] Let $Z \subset X$ be a closed set. Then we say that $Z$ has codimension $r$ in $X$ if for all irreducible component of $Z$ has codimension $r$ in $X$.  
• [Arbitrary] Let $Y \subset X$ be an arbitrary subset. Then we define codimension of $Y$ in $X$ to be the infimum of $\mathrm{dim}~\mathcal{O}_{X,y}$ where infimum runs through $y \in Y$.
• (Yoshida) Let $Y \subset X$ be a regular immersion of codimension $r$, then this is equivalent to saying that $\mathcal{I}$ is locally generated by a regular sequence of sections of $\mathcal{O}_X$ of length $r$.
• Here why and how do we say that it is "locally generated"?

Commutativity

• Let $f: Z \to X$ be a flat morphism. Let $Y \subset X$ be a closed subscheme with sheaf of ideals $\mathcal{I}$ and $X' = Bl_Y(X)$. Then if $\mathcal{I}' = \mathrm{Im}(f^* \mathcal{I} \to \mathcal{O}_Z)$ where $f^* \mathcal{I} := f^{-1} \mathcal{I} \otimes_{f^{-1} \mathcal{O}_X} \mathcal{O}_Z$ and $Z':= Bl_\mathcal{I}'(Z)$ with abuse of notation. Then the map $Z' \to Z \times_X X'$ given by universality is an isomorphism.
• Let $\widehat{X}_x = \mathrm{Spec} \widehat{\mathcal{O}}_{X,x}$, the spectrum of a completion (a completion of strict henselianization) of the local ring of $X$ at $x$. Let $X_x'$ be the blow-up of $\widehat{X}_x$ along $\mathcal{I} \mathcal{O}_{\widehat{X}_x}$. Then we have an isomorphism $X_x' \to \widehat{X}_x \times_X X'$.
• Need to check how we get a flat morphism $f: \widehat{X}_x' \to X$ and why $\mathrm{Im}(f^* \mathcal{I} \to \mathcal{O}_{\widehat{X}_x}) = \mathcal{I} \mathcal{O}_{\widehat{X}_x}$.