in this post we will say something on Tate rings.
A ring is a Tate ring if it is an adic ring and there is a unit element such that .
Note that for any ring of definition for a Tate ring , is open, thus , yet according to the definition, , so contains a unit element.
Suppose that , then the morphism is an isomorphism. So, it sends the open set to an open set . Thus is open in , too. Suppose one ideal of definition associated to for is , then for some . Yet on the other hand, , so for some , thus , this shows that the adic topology on and the adic topology on are the same. So, the ideals defines a fundamental system of neighborhoods of for . What is more, for any open neighborhood of in , and any element , there is another neighborhood of in such that . Yet there is some positive integer such that , thus , this shows that . So, for some . This means that . It is not difficult to show that
A ring is Tate ring if and only if there is a ring of definition with an ideal of definition such that .
So, in this sense, Tate rings are very special adic rings: the ideal of definition is a principal ideal, and in some sense, we can define its all integer powers, such that they form a fundamental system of neighborhoods of and .
In the next post, we will define morphisms between adic rings.