One of the motivating examples for Huber spaces comes from some application of valuation theory in Riemann surfaces.
Suppose that is a algebraically closed field and is a field extension of transcendental degree . Then there is a very famous result in the theory of Riemann surfaces which says that each such extension corresponds (in some sense uniquely) to a Riemann surface over . One fascinating proof to this result is to utilize valuation theory.
We say a function is a discret valuation if , and is a group homomorphism to the group . What is more, (here the infinity should be understood as the positive infinity). Then we write as the set of all such discret valuations. And for any algebra contained in , we define . Then we let run over all the algebras contained in and take all these as a basis of topology, so that the topology thus generated is verified to give a Riemann surface structure to . This is the essential part of the proof.
So here we want to generalize the pair , and that is where Huber spaces arise.
First we say something on topological rings. A commutative ring (with unit) with some topology said to be a topological ring if the are both continuous. Some typical examples are the adic integers .
In non-archimedean analysis, we often require that the topology on is given by some ‘nice’ sets, in other words, one fundamental system of open sets of can be a system of subgroups of . Note that, this is not at all a trivial requirement since that the most common spaces like the real numbers do not have this property(and that is why there is non-archimedean analysis). Yet this requirement is on the other hand rather natural for the non-archimedean analysis.
Here we consider when such a system of subgroups of can give a topology to such that it becomes a topological ring. In the following, we will write as the subgroup generated by elements of the form in .
Suppose that is a set consisting of subgroups of , then defines a fundamental system of neighborhood of such that becomes a topological ring if and only if satisfies the following three points:(1)for any , there is a such that ;(2)for any and any , there exists a such that ;(3)for any , there is a such that .
There is an important remark to make. The notion of two sets is in some sense the same as in the setting of non-archimedean analysis, since we can always find a system of subgroups as a fundamental system of neighborhood of . In the following we shall use frequently this remark.
One important class of subgroups of a ring is the ideals. And this introduces the adic rings.
Definition: A ring is called an adic ring if there is an ideal such that the system defines a fundamental system of neighborhood of .
Sometimes for precision, we call a ring an adic ring if the ideal in the definition is . Note that an ideal is open in the adic topology if there is a positive integer such that . So the adic topology and adic topology are the same if and only if there exist two positive integers such that . This amounts to say that these two ideals are comparable. In particular, adic topology and adic topology are always the same().
Suppose that is an open subring, then the adic topology on gives an induced topology on . Since is open, then for some integer . Then it is easy to see that the induced subspace topology on and the -adic topology on are the same. We will also often use this fact.
Next is a very important notion in analysis, the boundedness. For a metric space, this is a rather clear concept, yet if the space has no metric, things becomes not so easy. Here we adopt the definition of boundedness from functional analysis.
is a topological ring, and is a subset. We say that is bounded in if for any open neighborhood of in , there exists another open neighborhood of in such that .
Here if is a non-archimedean topological ring, then we can replace by , and the definition remains unchanged. And we call an element is power-bounded if is a bounded set. is called topologically nilpotent if . And we write as the set of power-bounded elements of and as the set of topologically nilpotent elements of .
A little remark is that any element as a single-element set in a topological ring is bounded due to the very simple fact that is continuous.
In the next post we will enlarge the notion of adic rings to adic rings.