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

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

Algebraic geometry builds fundamental concepts of geometry out of pure algebra (rings and polynomials). A very basic concept of geometry is topology. A topology on a set $X$ is specified by open (and/or closed) sets. An open set containing a point $x \in X$ is also called an open neighborhood of $x$. A set $X$ with a topology is called a topological space. An open set is the same thing as a complement of a closed set, and vice versa, so it suffices to specify either open sets or closed sets. Open sets in a topology are required to satisfy the following properties (or axioms):

1. $\emptyset, X$ are open.
2. A union of arbitrarily (possibly infinitely) many open sets is open.
3. An intersection of two (hence finitely many) open sets is open.
One can equivalently formulate the axioms for closed sets by swapping union and intersection.

For any set $S \subseteq X$, we then have a smallest closed set (with respect to inclusion) $\bar{S}$ containing $S$ (namely, the intersection of all closed sets containing $S$ ). It is called the closure of $S$. Symmetrically, the interior $S^{\circ}$ is the largest open set (i.e. the union of all open sets) contained in $S$.

## 数学代考|代数几何代写algebraic geometry代考|Zariski and Analytic Topology

In algebraic geometry, the set of all $n$-tuples of complex numbers is called the affine space $\mathbb{A}{\mathrm{C}}^{n}$. For the purposes of algebraic geometry, we consider the Zariski topology on $\mathrm{A}{\mathrm{C}}^{n}$, in which closed sets are affine algebraic sets (see Sect. 1.1). Similarly, in the Zariski topology on any affine algebraic set $X$, the closed sets are affine algebraic sets in $A_{C}^{n}$ which are subsets of $X$. To verify the axioms of topology, one notes that for sets of $n$-variable polynomials $S_{i}$, we have
$$Z\left(\bigcup_{i} S_{i}\right)=\bigcap_{i} Z\left(S_{i}\right)$$
and for sets of $n$-variable polynomials $S, T$, we have
$$Z({p \cdot q \mid p \in S, q \in T})=Z(S) \cup Z(T)$$
The Zariski topology is not the most typical kind of topology one considers outside of algebraic geometry. In analysis, the key example of a topology is the analytic topology. In the analytic topology on $\mathbb{A}{C}^{n}=\mathbb{C}^{n}$ (or on $\mathbb{R}^{n}$ ), a set $U$ is open when with any point $x \in U$, the set $U$ also contains all points of distance $<\epsilon$ for some $\epsilon>0$ (where $\epsilon$ can depend on $x$ ). As the name suggests, the analytic topology is very important in mathematical analysis. The Zariski topology has “far fewer” closed (and open) sets than the analytic topology. For example, in $\mathbb{R}^{n}$ (or $\mathbb{C}^{n}$ ), any open ball is open and any closed ball is closed in the analytic topology. On the other hand, the only Zariski closed sets in $A{C}^{1}$ are itself and finite subsets.

Still, we can use the analytic topology for intuition about the Zariski topology on algebraic sets. For example, a single point is closed (in both analytic and Zariski topology), and is not open, unless we are in $\mathrm{A}_{\mathrm{C}^{*}}^{0}$.

## 数学代考|代数几何代写algebraic geometry代考|Affine and Quasi-Affine Varieties

In a topological space $X$, a non-empty closed set $Z$ is called irreducible if there do not exist closed subsets $Z_{1} \neq Z, Z_{2} \neq Z$ of $Z$ such that $Z=Z_{1} \cup Z_{2}$ (i.e. $Z$ is not a union of two closed subsets other than itself). $Z$ is called connected if it is not a union of two disjoint closed subsets other than itself.

An affine variety is an affine algebraic set which is irreducible in the Zariski topology. A quasi-affine variety is a Zariski open subset $U$ of an affine variety $X$. (Caution: $U$ is open a topology on $S$ where, by definition, open (resp. closed) sets in $S$ are of the form $V \cap S$ where $V$ is an open (resp. closed) set in $X$. This topology is called the induced topology.

The Zariski topology on an affine algebraic set is induced from the Zariski topology on $\mathrm{A}^{n} \mathrm{C}$.

Recall that an ideal $I \subseteq R$ in a ring $R$ is called prime if $I \neq R$ and for $x, y \in R$, $x y \in I$ implies $x \in I$ or $y \in I$. The ideal $I$ is called maximal if $I \neq R$ and for every ideal $J \subseteq R$ with $I \subseteq J$, we have $J=I$ or $J=R$. An ideal $I \subseteq R$ is maximal if and only if the quotient ring $R / I$ (consisting of all cosets $x+I, x \in R$ ) is a field. Similarly, $I \subseteq R$ is prime if and only if $R / I$ is an integral domain which means that it satisfies $0 \neq 1$ and has no zero divisors (i.e. non-zero elements $x, y$ such that $x y=0$ ).

Any ideal $I \neq R$ is contained in a maximal ideal by a principle called Zorn’s lemma, which states that any partially ordered set $P$ (such as the set of ideals in a ring $R$ ordered with respect to inclusion) contains a maximal element (i.e. an element $m \in P$ such that $a \in P$ and $m \leq a$ implies $m=a$ ), provided that for any subset $L$ which is totally ordered (i.e. $a, b \in L$ implies $a \leq b$ or $b \leq a$ ) there exists an element $\ell \in P$ greater or equal than all elements of $L$.

Now it is easy to see that an affine algebraic set $X$ is irreducible (i.e. is an affine variety) if and only if the ideal $I(X)$ is prime. Indeed, if $I(X)$ is not prime, then there exists $f, g \notin I(X)$ such that $f g \in I(X)$, so $X$ is a union of the two closed subsets $Z(f) \cap X$, $Z(g) \cap X$ neither of which is equal to $X$. On the other hand, if $X=X_{1} \cup X_{2}$ where $X_{i} \neq X$ are closed, then by definition, there are $f_{i} \in I\left(X_{i}\right) \backslash I(X)$, while $f_{1} f_{2} \in I(X)$.

In particular, since polynomials over a field obviously form an integral domain, the 0 ideal is prime, and thus, the affine space $\mathrm{A}_{C}^{n}$ is irreducible (and hence, an affine variety).

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

1. ∅,X是开放的。
2. 任意（可能无限）许多开集的并集是开集。
3. 两个（因此是有限多个）开集的交集是开集。
可以通过交换并集和交集来等效地制定封闭集的公理。

## 数学代考|代数几何代写algebraic geometry代考|Zariski and Analytic Topology

Zariski 拓扑不是代数几何之外最典型的拓扑。在分析中，拓扑的关键示例是解析拓扑。在解析拓扑上一种Cn=Cn（或在Rn）， 一套在任何时候都打开X∈在, 集合在还包含所有距离点<ε对于一些ε>0（在哪里ε可以依赖X）。顾名思义，解析拓扑在数学分析中非常重要。Zariski 拓扑的封闭（和开放）集比解析拓扑“少得多”。例如，在Rn（或者Cn)，在解析拓扑中，任何开球都是开球，任何闭球都是闭球。另一方面，Zariski 唯一的封闭式一种C1是自身和有限子集。

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