### 数学代考|代数几何代写algebraic geometry代考|Sheaves and Schemes

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|Sheaves Revisited

Recall that a sheaf of sets $\mathcal{F}$ can be defined on a topological space $X$. It assigns to every open set $U \subseteq X$ the set of sections $\mathcal{F}(U)$. The following properties are required:

1. Restriction: For $V \subseteq U$, we have a restriction map
$$\mathcal{F}(U) \rightarrow \mathcal{F}(V)$$

The restriction is required to be transitive (i.e. for $W \subseteq V \subseteq U$, restriction from $\mathcal{F}(U)$ to $\mathcal{F}(V)$ and then to $\mathcal{F}(W)$ is the same thing as restricting to $\mathcal{F}(W)$ directly). Also, the restriction from $\mathcal{F}(U)$ to itself is just the identity.

1. Gluing: If we have sections $s_{i} \in \mathcal{F}\left(U_{i}\right)$ where $U_{i}$ are open sets, such that $s_{i}$ and $s_{j}$ restrict to the same section in $\mathcal{F}\left(U_{i} \cap U_{j}\right)$, then there exists a unique section $s \in$ $\mathcal{F}\left(\bigcup U_{i}\right)$ which restricts to all the functions $s_{i}$.

The stalk $\mathcal{F}_{X}$ of a sheaf $\mathcal{F}$ at a point $x \in X$ is the set of equivalence classes of sections in $\mathcal{F}(U)$ with $x \in U$ where $U$ is any open set containing $x$, where two sections $s \in \mathcal{F}(U)$, $t \in \mathcal{F}(V)$, are equivalent if they restrict to the same section in $\mathcal{F}(U \cap V)$.

We can also have sheaves of algebraic structures such as groups, abelian groups or rings defined analogously except that $\mathcal{F}(U)$ are groups, abelian groups or rings, and restrictions are homomorphisms.

A morphism of sheaves $\phi: \mathcal{F} \rightarrow \mathcal{G}$ gives for an open set $U$ a map (resp. homomorphism of whatever algebraic structures we are considering)
$$\phi(U): \mathcal{F}(U) \rightarrow \mathcal{G}(U)$$
such that $\phi$ of a restriction of a section $s$ is the restriction of $\phi(s)$.
A morphism of sheaves $\phi: \mathcal{F} \rightarrow \mathcal{G}$ induces, for every $x \in X$, a map (or homomorphism of whatever algebraic structures we have) $\phi_{x}: \mathcal{F}{x} \rightarrow \mathcal{G}{x}$.

If $f: X \rightarrow Y$ is a continuous map and $\mathcal{F}$ is a sheaf on $X$, we have a sheaf $f_{*} \mathcal{F}$ (sometimes called the pushforward) on $Y$ where
$$f_{\circledast} \mathcal{F}(U)=\mathcal{F}\left(f^{-1}(U)\right)$$
for every open set $U \subseteq Y$.

## 数学代考|代数几何代写algebraic geometry代考|Ringed Spaces and Locally Ringed Spaces

Recall that, unless otherwise specified, by a ring, we mean a commutative ring. A ringed space is a topological space $X$ with a sheaf of rings $\mathcal{O}{X}$ (called the structure sheaf). A morphism of ringed spaces $f: X \rightarrow Y$ is a continuous map together with a morphism of sheaves of rings $$\phi: \mathcal{O}{Y} \rightarrow f_{*} \mathcal{O}{X}$$ A locally ringed space is a ringed space where every stalk $\mathcal{O}{X, x}=\left(\mathcal{O}{X}\right){X}$ is a local ring. (Recall that a local ring is a ring which has a unique maximal ideal; a maximal ideal of a ring $R$ is an ideal $m \neq R$ such that there exists no ideal $I$ with $m \subseteq I \subsetneq R$. Equivalently, an ideal $m$ is maximal if and only if $R / m$ is a field.)

A morphism of locally ringed spaces $f: X \rightarrow Y$ is a morphism of ringed spaces such that for every point $x \in X$,
$$\mathcal{O}{Y, f(x)} \stackrel{\phi{f(x)}}{\longrightarrow}\left(f_{*} \mathcal{O}{X}\right){f(x)} \longrightarrow \mathcal{O}{X, x}$$ is a morphism of local rings (where the second map is defined in the obvious way). Here by a morphism of local rings $\phi: R \rightarrow S$ where the maximal ideal of $R$ is $m$ and the maximal ideal of $S$ is $n$, we mean a homomorphism of rings such that $\phi^{-1}(n)=m$ or, equivalently, $\phi(m) \subseteq n$ (see Exercise 2); an example of a homomorphism between local rings which is not a morphism of local rings is the inclusion $\mathbb{Z}{(p)} \subset \mathbb{Q}$, where $\mathbb{Z}_{(p)}$ is $\mathbb{Z}$ localized at the prime ideal ( $p$ ) for $p$ prime, or, in other words, the set of rational numbers whose denominators are not divisible by $p$.)

Note that if $X$ is a locally ringed space and $U \subseteq X$ is an open set, then $U$ with $\mathcal{O}{U}$ equal to the restriction $\left.\mathcal{O}{X}\right|{U}$ of the sheaf $\mathcal{O}{X}$ to $U$ (given by $\left.\mathcal{O}{X}\right|{U}(V)=\mathcal{O}_{X}(V)$ for $V \subseteq U$ open) is a locally ringed space. Let us call it the restriction of the locally ringed space $X$ to $U$.

## 数学代考|代数几何代写algebraic geometry代考|Schemes

An affine scheme is a locally ringed space of the form
$$\operatorname{Spec}(R)={p \mid p \text { is a prime ideal in } R}$$
The topology is the Zariski topology where closed sets are of the form
$$Z_{l}={p \in \operatorname{Spec}(R) \mid I \subseteq p}$$
for an ideal $I$.
As in Chap. 1, we have $Z_{I \cdot J}=Z_{I} \cup Z_{J}$ and
$$Z_{\sum I_{i}}=\bigcap_{i} Z_{I_{i}}$$
thus showing that we have indeed defined a topology. Denote by
$$U_{I}=\operatorname{Spec}(R) \backslash Z_{I}$$
the complementary open set.
A distinguished open set is a set of the form $U_{(r)}$ for $r \in R$ (i.e. $U_{I}$ where $I$ is a principal ideal). Every open set is a union of distinguished open sets. The structure sheaf $\mathcal{O}_{S p e c(R)}$

is uniquely determined by its sections on distinguished open sets by gluing. We have
$$\mathcal{O}{S p e c(R)}\left(U{(r)}\right)=r^{-1} R$$
(recall that $r^{-1} R$ is the set of equivalence classes of fractions $s / r^{n}, s \in R$ by the equivalence relation $s / r^{n} \sim t / r^{m}$ when $r^{n+k} t=r^{m+k} s$ for some $k=0,1,2, \ldots$ ). It is possible to use (1.3.1) as a definition. Some consistency checks are needed. We prefer, however, a definition using actual functions; using our definition, we will prove (1.3.1) in Sect. $2.2$ below (see Lemma 2.2.2).

More concretely, recall that for a commutative ring $R$ and a prime ideal $p$, the localization $R_{p}$ of $R$ at $p$ is the set of equivalence classes
$${r / s \mid r, s \in R, s \notin p} / \sim$$
where
$$\frac{r_{1}}{s_{1}} \sim \frac{r_{2}}{s_{2}}$$
when
$r_{1} s_{2} u=r_{2} s_{1} u$ for some $u \notin p .$

## 数学代考|代数几何代写algebraic geometry代考|Sheaves Revisited

1. 限制：对于在⊆在，我们有一个限制图
F(在)→F(在)

1. 胶合：如果我们有部分s一世∈F(在一世)在哪里在一世是开集，这样s一世和sj限制在同一部分F(在一世∩在j), 那么存在一个唯一的部分s∈ F(⋃在一世)仅限于所有功能s一世.

φ(在):F(在)→G(在)

F⊛F(在)=F(F−1(在))

## 数学代考|代数几何代写algebraic geometry代考|Schemes

（回想起那个r−1R是分数的等价类的集合s/rn,s∈R由等价关系s/rn∼吨/r米什么时候rn+ķ吨=r米+ķs对于一些ķ=0,1,2,…）。可以使用 (1.3.1) 作为定义。需要进行一些一致性检查。然而，我们更喜欢使用实际函数的定义；使用我们的定义，我们将在 Sect. 中证明 (1.3.1)。2.2下面（见引理 2.2.2）。

r/s∣r,s∈R,s∉p/∼

r1s1∼r2s2

r1s2在=r2s1在对于一些在∉p.

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