In this section, we will introduce the notion of valuation, which is an important tool in algebraic number theory and algebraic geometry. It provides a measurement of elements of a field or a ring. We will first review some abstract algebra. Some materials can be found in [4], [52], [7], [18], and [35].

Recall that an abelian totally ordered group or an ordered abelian group is an abelian group (written multiplicatively) $G$ endowed with a total order, such that $x\leq y$ implies $xz \leq yz$ for all $z\in G$. Since $1<x$ implies $1 < x < x^{2} < x^{3} < \cdot\cdot\cdot< x^{n}< \cdot\cdot\cdot$ (note that $1<x$ would imply $x^{-1}<1$), the abelian ordered group is torsion-free, i.e. no elements in it except the identity have finite order.

And recall that a chain of subsets of a set $S$ is a family $(C_{j})_{j\in J}$ of subsets of $S$, such that for any pair $C_{i},C_{j}$ in it, we have $C_{i}\subseteq C_{j}$ or $C_{i}\supseteq C_{j}$. The length of a chain is the number of the inequalities. For example, a singleton is a chain of length zero. And in the sequel, the maximal chain of prime ideals in the valuation ring of the non-archimedean absolute value has length one.

Definition 2.1. An order-preserving map $\varphi : (G,\leqq) \rightarrow (H,\leqq)$ of partially ordered sets is a map $G\rightarrow H$ such that for $x \leqq y$ in $G$, we have $\varphi(x)\leqq \varphi(y)$ in $H$. A morphism of ordered abelian groups is an order-preserving homomorphism of abelian groups. An isomorphism of ordered abelian group is an order-preserving isomorphism of abelian groups.

Example 2.2. The logarithmic functions with base $a>1$ from $\mathbb{R}_{>0}$ to $\mathbb{R}$ is an isomorphism of ordered abelian groups, i.e. $\log_{a}: (\mathbb{R}_{>0},\cdot,\leq)\xrightarrow{\sim}(\mathbb{R},+,\leq)$ for some $a>1$.

Example 2.3.

1. $(\mathbb{R},+,\leq)$, $(\mathbb{R}_{>0},\cdot,\leq)$, and $(\mathbb{Q},+,\leq)$ are ordered abelian groups with the usual order. Note that $(\mathbb{R},+,\leq)\cong (\mathbb{R}_{>0},\cdot,\leq)$.
2. $(\mathbb{R}^{n}_{>0},\cdot)$ with the lexicographic order ($(x_{1},...,x_{n})<(y_{1},...,y_{n})$ if and only if $x_{1}< y_{1}$ or $x_{k}< y_{k}$ when there exists $k\leq n$ such that $x_{i}=y_{i}$ for all $i<k$) is an ordered abelian group.

Definition 2.4. Let $(\Gamma,\cdot,\leq)$ be an ordered abelian group. A subgroup $H$ of $\Gamma$ is a convex subgroup or an isolated subgroup if for any $x\in\Gamma,x'\in H$ such that $1\geq x\geq x'$, we have $x\in H$. The spectrum of $\Gamma$ is the set of all convex subgroups of $\Gamma$, denoted by Spec $\Gamma$.

Proposition 2.5. The set Spec $\Gamma$ forms a well-ordered set under inclusion. The cardinality of non-trivial convex subgroups in Spec $\Gamma$ is called the height of $\Gamma$ which is denoted by ht$(\Gamma)$.

Proof. Let $H,H'$ be two convex subgroups of $\Gamma$. Assume that $H\nsubseteq H',H'\nsubseteq H$, and $1\geq x\geq x'$ for $x\in H,x'\in H'$. Then since $H'$ is a convex subgroup, $x\in H'$, which is a contradiction! So $H'\subseteq H$ or $H\subseteq H'$.

Definition 2.6. The convex rank of $\Gamma$, denoted by $$ c.rk(\Gamma)\in\mathbb{N}\cup\{\infty\}, $$ is the supremum over the lengths of the chains $1\subsetneq H_{1}\subsetneq \cdot\cdot\cdot\subsetneq H_{r}:=\Gamma$ of convex subgroups of $\Gamma$.

Remark 2.7. Note that the convex rank of $\Gamma$ is equal to the height of $\Gamma$.

Example 2.8.

1. If $\textrm{c.rk}(\Gamma)=0$, then the only convex subgroup is the trivial group.
2. If $\textrm{c.rk}(\Gamma)=1$, then the convex subgroups of $\Gamma$ are the trivial group $1$ and $\Gamma$. If $\Gamma$ is a non-trivial subgroup of $\mathbb{R}_{>0}$, then $\textrm{c.rk}(\Gamma)=1$.
3. If $\Gamma=\mathbb{R}_{>0}^{n}$ (with lexicographic order), then $\textrm{c.rk}(\Gamma)=n$.

Proposition 2.9. Let $(\Gamma,\cdot,\leq)$ be an ordered abelian group. Then the following are equivalent:

1. ${\rm{c.rk}}(\Gamma)=1$;
2. for all $a,b\in\Gamma$ with $b<1,a\leq1$, there exists $n\geq0$ such that $b^{n}\leq a$;
3. there exists an injective morphism of ordered abelian group $(\Gamma,\cdot,\leq)\rightarrow(\mathbb{R}_{>0},\cdot,\leq)$.

Remark 2.10. Note that condition (2) is the multiplicative version of Archimedean property.

Definition 2.11. The rational rank of $\Gamma$, denoted by $$rk(\Gamma)\in\mathbb{N}\cup\{\infty\},$$ is the dimension of the $\mathbb{Q}$-vector space $\Gamma\otimes_{\mathbb{Z}}\mathbb{Q}$.

Remark 2.12. We define the $\mathbb{Q}$-vector space structure on $\Gamma\otimes_{\mathbb{Z}}\mathbb{Q}$ by $\lambda(a\otimes b)=a\otimes\lambda b$ for $a\in\Gamma;b,\lambda\in\mathbb{Q}$.

Lemma 2.13. Let $K$ be a field and let $\left|\cdot\right|$ be an absolute value on $K$. Then the topology induced from $\left|\cdot\right|$ makes $K$ a topological field.

Proof. Let $\mathbb{B}_{1}=\{x\mid\left|x-a\right|<r_{1}\}$ and $\mathbb{B}_{2}=\{y\mid\left|y-b\right|<r_{2}\}$ be two open balls in $K$. We first check the continuity of the addition. Since $\left|(x-y)-(a-b)\right|=\left|(x-a)+(-y+b)\right|$, we have $$\left|(x-y)-(a-b)\right|=\left|(x-a)+(-y+b)\right|\leq\left|x-a\right|+\left|y-b\right|<r_{1}+r_{2},$$ which implies that $$ \mathbb{B}_{1}-\mathbb{B}_{2}=\{x-y\mid\left|(x-y)-(a-b)\right|<r_{1}+r_{2}\}. $$

Next, for multiplication, dividing both sides by $\left|a\right|$ or $\left|b\right|$, we have $\displaystyle\left|\frac{x}{a}\right|<\frac{r_{1}}{\left|a\right|}+1$ and $\displaystyle\left|\frac{y}{b}\right|<\frac{r_{2}}{\left|b\right|}+1$, which implies that

$\begin{align*} \left|\frac{x}{a}\cdot\frac{y}{b}\right|=\frac{\left|xy\right|}{\left|ab\right|}&<(\frac{r_{1}}{\left|a\right|}+1)(\frac{r_{2}}{\left|b\right|}+1) \\ \left|xy\right|&<\left|ab\right|(\frac{r_{1}}{\left|a\right|}+1)(\frac{r_{2}}{\left|b\right|}+1) \\ &=r_{1}r_{2}+\left|b\right|r_{1}+\left|a\right|r_{2}+\left|ab\right|. \end{align*}$

So the multiplication of two open balls is the open ball $$ \mathbb{B}_{1}\cdot\mathbb{B}_{2}=\{xy\mid\left|xy-ab\right|<r_{1}r_{2}+\left|b\right|r_{1}+\left|a\right|r_{2}\}.$$

Finally, consider the inverse map. We have $\left|\left|y\right|-\left|b\right|\right|<r_{2}$, which implies that $-\left|y\right|+\left|b\right|<r_{2}$ or $\left|y\right|-\left|b\right|<r_{2}$. So we have $$ \frac{1}{r_{2}+\left|b\right|}<\frac{1}{\left|y\right|}<\frac{1}{\left|b\right|-r_{2}},$$ which shows that the inverse of an open ball is the open set $$ \mathbb{B}_{2}^{-1}=\{y^{-1}\mid\frac{1}{r_{2}+\left|b\right|}<\left|y^{-1}\right|<\frac{1}{\left|b\right|-r_{2}} \}. $$

Consequently, for every neighborhood $U=\{z\mid \left|z-(a-b)\right|<r\}$ of $a-b$, there exist $a\in\mathbb{B}_{1}',b\in\mathbb{B}_{2}'$ such that $\mathbb{B}_{1}'-\mathbb{B}_{2}'\subset U$. And for every neighborhood $V=\{z\mid\left|z-ab\right|<r\}$ of $ab$, there exist $a\in\mathbb{B}_{1}',b\in\mathbb{B}_{2}'$ such that $\mathbb{B}_{1}'\mathbb{B}_{2}'\subset V$. For every neighborhood $W=\{z\mid\left|z-b^{-1}\right|<r\}$ of $b^{-1}$, there exists $b\in\mathbb{B}_{2}'$ such that $\mathbb{B}'^{-1}_{2}\subset W$.

Definition 2.14. A (non-archimedean) absolute value on a field $K$ is a map $\left| \cdot \right| : K \rightarrow \mathbb{R}_{\geq0}$, such that for all $x,y\in K$ the following conditions are verified:

1. $\left| x \right| = 0 \Leftrightarrow x=0$,
2. $\left| xy \right| = \left| x \right| \left| y \right|$,
3. $\left| x+y \right| \leq \max\{\left| x \right|, \left| y \right|\}$.

Then we say that $(K,\left|\cdot\right|)$ is a valued field. Using absolute value, we can define Cauchy sequence in the usual way. And $K$ is complete if every Cauchy sequence converges in $K$.

A field that is complete with respect to a non-archimedean absolute value is called a non-archimedean field. Similarly, a field that is complete under an archimedean absolute value is called an archimedean field.

Remark 2.15. Note that, in Peter Scholze's thesis, [16], non-archimedean fields need not to be complete. And instead, a non-archimedean field is defined to be a topological field $k$ endowed with a non-trivial valuation of rank 1 inducing the topology.

When a field is endowed with a non-archimedean absolute value, something amazing happens! We get some beautiful results that are distinct from the archimedean case. In this section, $K$ will be always equipped with a non-archimedean absolute value. We will show some peculiarities of $K$ when equipped with a non-archimedean absolute vale.

Proposition 2.16. Let $a,b\in K$. If $a\neq b$, then $$ \left|a+b\right|=\max\{\left|a\right|,\left|b\right|\}.$$

Proof. Assume that $\left| b \right|<\left| a \right|$, then we have $\left| a+b \right|\leq\left| a \right|$ by definition. We assume further $\left| a+b \right|\neq\left| a \right|$, then we have $\left| a+b \right|<\left| a \right|$, which implies $$ \left| a \right|=\left| (a+b)-b \right|\leq \max\{\left| a+b \right|,\left| b \right|\}<\left| a \right|, $$ which is contradictory! So we must have $\left| a+b \right|=\left| a \right|=\max\{\left|a\right|,\left|b\right|\}$.

Lemma 2.17. A series $\sum_{v=0}^{\infty}a_{v}$ in $K$ is a Cauchy sequence if and only if the coefficients $a_{v}$ form a zero sequence, i.e. $\lim_{v\to\infty}\left|a_{v}\right|=0$.

In particular, when $K$ is complete, the series $\sum_{v=0}^{\infty}a_{v}$ is convergent if and only if $\lim_{v\to\infty}\left|a_{v}\right|=0$.

Proof. cf.([7], Lemma 3, p 10).

Remark 2.18. Note that a series $\sum_{v=0}^{\infty}a_{v}$ can be viewed as a Cauchy sequence if we consider every partial sum $\sum_{v=0}^{n}a_{v}$ as a term. And $\lim_{v\to\infty}\left|a_{v}\right|=0$ is equivalent to $\lim_{v\to\infty}a_{v}=0$, since $\left|a_{v}-0\right|=\left|\left|a_{v}\right|-0\right|$ in this case.

Clearly, the non-archimedean absolute value on $K$ induces a distance on $K$ by $d(a,b)=\left|a-b\right|$, so there is an associated topology on $K$. Consider the distance between points in $K$. Let $x,y,z\in K$, since $\left|y-z\right|=\left|(x-y)+(z-x)\right|$, the non-archimedean inequality implies $$d(y,z)\leq\max\{d(x,y),d(x,z)\}.$$

By Proposition 2.16, if $d(x,y)\neq d(x,z)$, then $d(y,z)=\max\{d(x,y),d(x,z)\}$. In other words, consider a triangle in $K$, if two sides are not equal in length, then the longer one has the same length as the third side, which implies that any triangle in $K$ is isosceles.

Next, we consider the disks in $K$ as an example.

For a centre $a\in K$ and a radius $r\in \mathbb{R}_{> 0}$, we define the disk without boundary or open disk to be the set $$D^{-}(a,r) = \{ x \in K\mid d(x,a)<r \}. $$

Similarly, we define the disk with boundary or closed disk to be the set $$ D^{+}(a,r) = \{ x \in K\mid d(x,a)\leq r\}. $$

In addition, we can define the boundary or periphery of the disk $$ \partial D(a,r) = \{ x \in K\mid d(x,a)= r\}.$$

Without any loss of generality, we consider only the open disk. Let $b\in D^{-}(a,r)$. Since we have $\left|x-a\right|<r$, then $$ \left|x-b\right|=\left|(x-a)+(a-b)\right|\leq\max\{\left|x-a\right|,\left|a-b\right|\}<r. $$

This indicates that every point in a disk is a center. Moreover, if the intersection of two disks is not empty, then the two disks are concentric.

The following proposition shows another peculiarity related to the topology of $K$.

Proposition 2.19. The topology of $K$ is totally disconnected, i.e. the maximal connected components in $K$ are singletons.

Proof. cf. [7], Proposition 4, p11.

Next, we introduce the so-called valuations, which were first published by Krull in 1932. They generalize non-archimedean absolute values and have great applications in algebraic geometry. Their values are not constrained in real numbers and they can have more general values than non-archimedean absolute values.

There are two kinds of valuations, each of which is in different notations. We define valuations in additive notation initially.

Definition 2.20. Let $\Gamma$ be an abelian totally ordered group, we extend $\Gamma$ to $\Gamma \cup \{\infty\}$, such that $\alpha < \infty$  and $\alpha + \infty = \infty + \alpha = \infty$ for all $\alpha \in \Gamma$. A valuation in additive notation (or an additive valuation) on a field $K$ is a map $\upsilon: K \rightarrow \Gamma \cup \{\infty\}$ such that for all $x,y\in K$ the following hold:

1. $\upsilon(x) = \infty \Leftrightarrow x = 0$
2. $\upsilon(xy) = \upsilon(x) + \upsilon(y)$
3. $\upsilon(x+y) \geq \min\{\upsilon(x),\upsilon(y)\}$

Also we can define valuations in multiplicative notation.

Definition 2.21. Let $\Gamma$ be an abelian totally ordered group, we extend $\Gamma$ to $\Gamma \cup \{0\}$, such that $0 < \alpha$  and $\alpha \cdot 0 = 0 \cdot \alpha = 0$ for all $\alpha \in \Gamma$. A valuation in multiplicative notation (or a multiplicative valuation) on a field $K$ is a map $\upsilon: K \rightarrow \Gamma \cup \{0\}$ such that for all $x,y\in K$ the following hold:

1. $\upsilon(x) = 0 \Leftrightarrow x = 0$
2. $\upsilon(xy) = \upsilon(x)\upsilon(y)$
3. $\upsilon(x+y) \leq \max\{\upsilon(x),\upsilon(y)\}$

It is clear that non-archimedean absolute value is an example of multiplicative valuation when taking $(\Gamma, \cdot,\leq)$ as $(\mathbb{R}_{>0}, \cdot,\leq)$. In fact, we can show that, in this case, non-archimedean absolute value and additive valuation are equivalent. If we write $\upsilon_{0}$ for an additive valuation and $\upsilon_{1}$ for an absolute value. Then just let $\upsilon_{0}(x) = - \ln\upsilon_{1}(x)$ for all $x \in \textit{K}$, so we can pass from additive valuation back to multiplicative valuation by setting $\upsilon_{1}(x) = e^{-\upsilon_{0}(x)}$. This sets up a one-to-one correspondence or an isomorphism. In terms of this correspondence, we will make no difference between non-archimedean absolute value and real additive valuation.

In the sequel, we will use multiplicative valuation, and simply call it valuation. A real valuation will mean a valuation with $\Gamma\subseteq \mathbb{R}_{>0}$.

A valuation is trivial if $\upsilon(x) = 1$ for all $x \neq 0$.

Definition 2.22. Let $v: K\rightarrow \Gamma\cup\{0\}$ be a valuation. The value group of $v$ is $\Gamma_{v}:=\{v(x)\in \Gamma\mid x\in K^{\times}\}$.

Definition 2.23. Let $v: K\rightarrow\Gamma\cup\{0\}$ be a valuation. The rank of $v$ is the convex rank or height of the value group $\Gamma_{v}$.

Remark 2.24. A non-trivial valuation $\upsilon$ on $K$ has rank 1 if $\Gamma \subset \mathbb{R}_{>0}$ as abelian totally ordered group, and it has higher rank if it is not of rank 1.

Remark 2.25. $K^{\times}$ is the multiplicative group of any field $K$. It can be observed that the valuation $\upsilon$ of rank 1 on $K$ is the same as the (non-archimedean) absolute value on $K$. Consequently, we can make no difference between rank-1 valuation and absolute value.

Definition 2.26. A field with a valuation is called a valuation field or a valued field.

Remark 2.27. The valued field in Definition 2.14 coincides with this.

Definition 2.28 ([4]). The valuation ring of a valuation $v$ on a field $K$ is $$ \mathscr{O}_{K}:= \{x\in K\mid v(x)\leq 1 \}.$$

Remark 2.29 ([4]). The valuation ring $\mathscr{O}_{K}$ is a subring of the field $K$. It has a unique maximal ideal $\mathfrak{m}: = \{ x\in K\mid v(x)<1\}$ and a group of units $\{x\in K\mid v(x) = 1\}$. The field $K$ is the fraction field of $\mathscr{O}_{K}$, that is we have $\textrm{Frac}(\mathscr{O}_{K}) \cong K$, so the valuation on $K$ is determined by its valuation ring.

Proposition 2.30. Let $R$ be a subring of the field $K$. The following conditions are equivalent :

1. $R$ is the valuation ring of some valuation on $K$.
2. $Q(R)$ = $K$ and ideals of $R$ form a chain.
3. For all $x\in K\backslash\{0\}$, $x\in R$ or $x^{-1}\in R$.

Proof. cf.([4], Chapter VI, Proposition 6.2).

Proposition 2.31. Let $\left|\cdot\right|_{v}$ and $\left|\cdot\right|_{w}$ be two absolute values on a field $K$. The following conditions are equivalent:

1. $\left|\cdot\right|_{v}$ and $\left|\cdot\right|_{w}$ induce the same topology on $K$;
2. $\left|x\right|_{v}<1$ if and only if $\left|x\right|_{w}<1$ for all $x\in K$;
3. there exist $c>0$ such that $\left|x\right|_{w} = (\left|x\right|_{v})^{c}$ for all $x\in K$.

Proof. cf.([4], Proposition 3.1, p240).

Definition 2.32. Two absolute values on a field are equivalent if they satisfy one of the conditions in Proposition 2.31.

Definition 2.33. Two valuations $v,w : K\rightarrow \Gamma\cup\{0\}$ are equivalent if there exists an order-preserving isomorphism $\alpha$ between $\Gamma_{v}$ and $\Gamma_{w}$ such that $w(x)=\alpha(v(x))$ for all $x\in K^{\times}$.

Remark 2.34. For each $c>0$, there exists an automorphism $x\mapsto x^{c}$ of $\mathbb{R}_{>0}$ which is order-preserving. So non-archimedean absolute values are equivalent as absolute values if and only if they are equivalent as valuations.

Definition 2.35 ([7], p154). The length of the maximal chain of prime ideals in a valuation ring $R$ is called the height or Krull dimension of $R$.

Example 2.36. Consider the non-archimedean absolute value $\left|\cdot\right|$ on a field $K$, the valuation ring $R$ of $\left|\cdot\right|$ is $\{x\in K\mid\left|x\right| \leq1\}$. The prime ideals in $R$ are the maximal ideal and the zero ideal. So the height of $R$ is 1.

Remark 2.37. A valuation ring $R$ has height 1 if and only if $R$ has convex rank 1.