# Huber spaces-3

in this post we will say something on Tate rings.

A ring $A$ is a Tate ring if it is an $f-$adic ring and there is a unit element $a\in A^*$ such that $a\in A^{oo}$.

Note that for any ring of definition $A_0$ for a Tate ring $A$, $A_0\cdot A=A_0$ is open, thus $A^{oo}\subset \sqrt{A_0}$, yet according to the definition, $A^{oo}\bigcap A^*\neq\emptyset$, so $A_0$ contains a unit element.

Suppose that $a_0\in A_0\bigcap A^*\bigcap A^{oo}$, then the morphism $f: A\rightarrow A, a\mapsto a a_0$ is an isomorphism. So, it sends the open set $A_0$ to an open set $a_0A_0$. Thus $a_0A_0$ is open in $A_0$, too. Suppose one ideal of definition associated to $A_0$ for $A$ is $I_0$, then $I_0^n\subset a_0A_0$ for some $n$. Yet on the other hand, $a_0^k\rightarrow 0$, so $a_0^k\in I_0$ for some $k$, thus $a_0^kA_0\subset I_0A_0=I_0$, this shows that the $I_0-$adic topology on $A_0$ and the $a_0A_0-$adic topology on $A_0$ are the same. So, the ideals $(a_0^mA_0)$ defines a fundamental system of neighborhoods of $0$ for $A$. What is more, for any open neighborhood $U$ of $0$ in $A$, and any element $a\in A$, there is another neighborhood $V$ of $0$ in $A$ such that $aV\subset U$. Yet there is some positive integer $k$ such that $a_0k\in V$, thus $a_0^k a\in U$, this shows that $a_0^n a\rightarrow 0$. So, $a_0^N a\in A_0$ for some $N$. This means that $A=A_0[a_0^{-1}]$. It is not difficult to show that

A ring $A$ is Tate ring if and only if there is a ring of definition $A_0$ with an ideal of definition $a_0A_0$ such that $A=A_0[a_0]$.

So, in this sense, Tate rings are very special $f-$adic rings: the ideal of definition is a principal ideal, and in some sense, we can define its all integer powers, $I_0^n(n\in \mathbb{Z})$ such that they form a fundamental system of neighborhoods of $0$ and $A=\bigcup_{n\in\mathbb{Z}} I_0^n$.

In the next post, we will define morphisms between $f-$adic rings.