In the previous post we have introduced the adic rings, in this post we will introduce a type of ring which includes adic rings as open subrings.
A topological ring is called adic ring if it contains an open subring for which the induced subspace topology is an adic topology where is a finitely-generated ideal of .
The in the definition is called a ring of definition for and the ideal is called an ideal of definition for . We have three equivalent characterization for the ring of definition. That is
If is an adic ring, then the following three statements are equivalent: (1) is a ring of definition; (2) is open and an adic ring; (3) is open and bounded;
Proof: Before we show (1) implies (2), we have a little remark: defines a fundamental system of neighborhoods of of (note here, not of ). This is rather trivial once we notice that each element of is bounded and are all open in . The implication of (1) to (2) is trivial. That (2) implies (3) is also trivial. So it remains to show the implication of (3) to (1). Suppose a ring of definition for is and an ideal of definition for in is , since is open, we have that for some . So, if is the finite set of generators of , then , too. So, we define to be the ideal genertated by (), which is finitely generated. Moreover, , and (here we use a rather simple yet important fact, that is, , which garantes that is open, otherwise it will be very difficult to show this point), thus proving that the -adic topology and the induced subspace topology on are the same. What is more, is finitely generated, thus is a ring of definition.
The followings are some simple properties of adic rings:
is an adic ring, (1)if are two rings of definitions for , then , are again rings of definition;(2) Subset of is bounded and contains , and subset of is an open subring, what is more, , then there exists a ring of definition for such that ;(3) is the union of rings of definition for .
The proofs are not very difficult. For the first point, it is sufficient to show that are open and bounded(to show that is bounded, note that for any an open neighborhood of in , there exists another open neighborhood such that . Note that, is an open set! Thus there exists a third open neighborhood of in such that , thus we have that , thus showing the boundedness of ). For the second point, we can take a ring of definition, say for , then noting that since is an open subring, is a ring of definition for , making into an adic ring. It is easy to verify that is also a ring of definition for , thus also for (since is open in ); For the third point, since any is power-bounded, thus there exists a ring of definition containing . So is contained in the union of rings of definition. Conversely, since any ring of definition is bounded, so is any element in it, thus showing that the union of rings of definition is contained in , and this finishes the proof for the third point.
The following result says something on and . Combining the first and the third points, we get that is actually an open subring of .
is adic ring, then is integrally closed; for any subset , is open if and only if .
For the first point, we observe that if satisfies with , we choose a ring of definition containing , then . Yet the second set is a finite set, thus bounded, and so is bounded. It is clearly open, thus it is a ring of definition, and this shows that is power-bounded, thus , so is integrally closed. For the second point, if is open, then for any , , so , thus , and so . If , then note that since defines a fundamental system of neighborhoods of , so each element in is topologically nilpotent, thus . That is to say, . Yet is finitely generated, so there exists an integer such that , thus showing that is open.
In the next post, I will say something about Tate-rings, which are a special kind of adic rings.