Library of Mathexandria

For some reason, I have a hard time understanding the proof that in an intersection of two affine open subschemes, there exists an open set that is principal in both affine opens. Here is my attempt to make it slightly more clear. The precise statement that I want to prove is the following. Let $X$ be a scheme and $U=\mathrm{Spec}~A$ and $V=\mathrm{Spec}~B$ be two affine open subschemes of $X$. For all $x \in U \cap V$, there exists $W = D_U(f) = D_V(g)$, a neighborhood of $x$, such that $f \in A$, $g \in B$, and $W \subset U \cap V$. 1) We would like to reduce to a simpler case. $U \cap V$ is an open subset of $V$. Hence there exists a principal open set $D(g) \subset U \cap V$ with $g \in B$ containing $x$. Then we may identify $D(g) = \mathrm{Spec}~B_g$. Therefore, we have $x \in \mathrm{Spec}~B_g \subset \mathrm{Spec}~A$. Suppose we found an open set $W$ containing $x$ that is principal inside both $\mathrm{Spec}~B_g$ and $\mathrm{Spec}~A$, then there exists some $h/g^n \i...

수학자들 중에서 정수론을 전문적으로 공부하는 사람을 정수론자라고 한다. 워낙 추상적인 개념을 많이 다루는 직업이기에 매번 "그래서 정수론을 배워서 뭐에 써먹는데?" 라는 질문을 자주 받게 된다. 솔직히 말하면 정수론이 실생활에 응용되어 쓰여지는 일은 드물다. 하지만 컴퓨터의 등장 이후, 모듈러 산술을 이용해 암호론에 정수론에 적용되기 시작되었다. 그 중에서 RSA 암호에 대해서 설명해보려고 한다. 정수론에 대해서 이야기 하기 전에 먼저 암호에 대해서 이야기 해보자. 다음 용어를 숙지해두자. 용어정리 평문 : 보호해야 할 메세지 암호문 : 평문을 특정 대상 외에는 이해하지 못하게 변환한 메세지 암호화 : 평문을 암호문으로 바꾸는 과정 복호화 : 암호문을 평문으로 바꾸는 과정 기본적으로 우리는 다음을 하고 싶다. 김씨가 박씨에게 비밀 메세지를 보내고 싶다고 하자. 1. 김씨는 평문을 암호화 하여 암호문을 만든다. 2. 김씨는 암호문을 박씨에게 넘긴다. 3. 박씨는 김씨의 암호문을 받아 복호화를 하여 평문을 읽는다. 두 사람이 다른 사람에게 들키지 않고 비밀 메세지를 주고 받는 방법에 대해서 생각해보자. 일단 가장 간단한 방법은 두 사람이 같은 "키"를 가지고 있는 것이다. 대칭키 암호 (Symmetric-Key Cryptography) 예시 1 자물쇠가 달린 상자로 비유해보자. 김씨와 박씨는 다른 사람은 보지 못하게 비밀 메세지를 교환하고 싶다. 이를 위해서 그들은 자물쇠가 달린 상자를 구매했고 서로 같은 키를 구매했다. 1. 김씨는 비밀 메세지 (평문)을 상자에 넣어 자물쇠를 채운다 (암호화). 2. 김씨는 상자 (암호문)을 박씨에게 넘긴다. 3. 박씨는 김씨의가 보낸 상자를 키를 이용해서 자물쇠를 연다 (복호화). 이와 같이 김씨와 박씨가 같은 키를 쓰는 것을 대칭 키 암호 (Symmetric-Key Algorithm) 이라고 부른다. 만약 박씨가 ...

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...

We come back to the original paper. $A:=A_1$ is the representing algebra of the deformation functor $\mathcal{F}_1: \mathcal{C} \to (\mathrm{Set})$ that sends local $W$-algebras with additional properties to the set of isomorphism classes of deformations with level-$\pi$-structures. By the theorem of Drinfeld, we have the universal deformation and universal level $\pi$-structure \[ \widehat{\Sigma} :=\widetilde{\Sigma} \otimes_{A_0} A_1 \text{ and } \varphi:= \varphi_1 : (\pi^{-1} \mathcal{O}_K/\mathcal{O})^n \to \mathfrak{m}_{\widehat{\Sigma}} \]  Then $X_i:=\varphi(e_i)$ where $e_i$ are the standard basis of $(\pi^{-1} \mathcal{O}_K/\mathcal{O})^n$. This forms a system of local parameters of $A$. One of the conditions of level $\pi$-structure is that  \[ P_\varphi(T) { \Large| } [\pi](T) \] with $P_\varphi(T)U(T) = [\pi](T)$ (*) $U(T)$ has constant term $u_{\widehat{\Sigma}} \in 1 + \mathfrak{m}$. \[ P_\varphi(T) = \prod_{x \in (\pi^{-1} \mathcal{O}_K/\mathcal{O}...

Some remarks. Let $f:X \to Y$ be a morphism of (integral) schemes. Then we say that $f$ is a Galois covering  if $K(X)/K(Y)$ is a Galois extension. $A_0$ is clearly a domain and $A_m$ is a regular local ring, then by Matsumura, Theorem 14.3, we know that $A_m$ is a domain. Therefore $\mathrm{Spec}~A_m \to \mathrm{Spec}~A_0$ is a morphism of integral schemes. According to the post in mathSE, $Y$ becomes a quotient of $X$ by the group of deck transformations of $f$ which turns out to be the Galois group.  According to the Wiki article , discrete valuation ring is a regular ring with dimension 1. The converse is also true. Therefore $A_0:=W[[X_1, \ldots, X_{n-1}]]$ is a regular ring with dimension $n$. Review on representable functor A functor $\mathcal{F}: \mathcal{C} \to (\mathrm{Set})$ is called representable if there exists $A \in \mathcal{C}$ such that $\mathcal{F}$ is naturally isomorphic to $Hom(A,-)$ (or $Hom(-,A)$ for contravariant functor). A pair $(...

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. Excursion. Moduli Space 1. Moduli Space of Elliptic Curves  \( \require{AMScd} \) Consider the category \( (\mathrm{Ell}) \) where the objects are elliptic curves over an arbitrary base scheme \[ \begin{CD} E \\ @VV{\pi}V \\ S \end{CD} \] and the morphisms are the Cartesian squares \[ \begin{CD} E_1 @>{a}>> E \\ @V{\pi_1}VV @VV{\pi}V \\ S_1 @>{f}>> S \end{CD} \] The above Cartesian diagram induces an isomorphism \[ E_1 \xrightarrow{~(\alpha, \pi_1)~} E \times_S S_1 \] \( (\mathrm{Ell}) \) is called the modular stack of Deligne-Rapoport . A contravariant functor \( \mathcal{P} ...

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 f...

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, \ld...

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. 2. Review on the Moduli Spaces of Formal \( \mathcal{O}_K \)-modules 2.1 Formal \( \mathcal{O}_K \)-modules Definition of Formal \(\mathcal{O}_K \)-modules, homomorphism/endomorphism, base change Additive group is a formal \(\mathcal{O}_K \)-module For any formal \(\mathcal{O}_K \)-module (over \( \overline{\mathbb{F}}_q \) ) non-isomorphic to the additive group, there exists a unique height. We can "normalize" the formal \(\mathcal{O}_K \)-module to satisfy certain properties. 2.2  Deformation of formal \( \mathcal{O}_K \)-modules Category \( \mathcal{C} \) of local Noetherian \( W = \mat...