Huber spaces-1

One of the motivating examples for Huber spaces comes from some application of valuation theory in Riemann surfaces.

Suppose that k is a algebraically closed field and k\subset K is a field extension of transcendental degree 1. Then there is a very famous result in the theory of Riemann surfaces which says that each such extension K corresponds (in some sense uniquely) to a Riemann surface over k. One fascinating proof to this result is to utilize valuation theory.

We say a function v:K\rightarrow \mathbb{Z}\bigcup \{\infty\} is a discret valuation if v(0)=\infty,v(1)=1, and v|_{K^{*}} is a group homomorphism to the group \mathbb{Z}. What is more, v(a+b)\geq min(v(a),v(b))(here the infinity should be understood as the positive infinity). Then we write V(K/k) as the set of all such discret valuations. And for any k-algebra contained in K, we define U(B)=\{v\in V(K)|v(B)\subset \mathbb{N}\bigcup \{\infty\}\}. Then we let B run over all the k-algebras contained in K and take all these U(B) as a basis of topology, so that the topology thus generated is verified to give a Riemann surface structure to V(K/k). This is the essential part of the proof.

So here we want to generalize the pair (K,B), and that is where Huber spaces arise.

First we say something on topological rings. A commutative ring (A,+,*) (with unit) with some topology said to be a topological ring if the A\times A\rightarrow A, (a,a')\mapsto a-a'; A\times A\rightarrow A,(a,a')\mapsto a*a' are both continuous. Some typical examples are the p-adic integers \mathbb{Z}_p.

In non-archimedean analysis, we often require that the topology on A is given by some ‘nice’ sets, in other words, one fundamental system of open sets of 0 can be a system of subgroups of (A,+). Note that, this is not at all a trivial requirement since that the most common spaces like the real numbers \mathbb{R} 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 (A,+) can give a topology to A such that it becomes a topological ring. In the following, we will write S\cdot T as the subgroup generated by elements of the form st(s\in S,t\in T) in A.

Suppose that \mathfrak{G} is a set consisting of subgroups of (A,+), then \mathfrak{G} defines a fundamental system of neighborhood of 0 such that A becomes a topological ring if and only if \mathfrak{G} satisfies the following three points:(1)for any G,G'\in\mathfrak{G}, there is a H\in\mathfrak{G} such that H\subset G\bigcap G';(2)for any x\in A and any G\in\mathfrak{G}, there exists a H\in\mathfrak{G} such that xH\subset G;(3)for any G\in\mathfrak{G}, there is a H\in\mathfrak{G} such that H\cdot H\subset G.

There is an important remark to make. The notion S\cdot T of two sets is in some sense the same as ST in the setting of non-archimedean analysis, since we can always find a system of subgroups as a fundamental system of neighborhood of 0. 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 A is called an adic ring if there is an ideal I\subset A such that the system (I^n)_{n\in\mathbb{N}} defines a fundamental system of neighborhood of 0.

Sometimes for precision, we call a ring an I-adic ring if the ideal in the definition is I. Note that an ideal J\subset A is open in the I-adic topology if there is a positive integer n such that I^n\subset J. So the I-adic topology and J-adic topology are the same if and only if there exist two positive integers n,m such that I^n\subset J,J^m\subset I. This amounts to say that these two ideals are comparable. In particular, I^i-adic topology and I^j-adic topology are always the same(i,j>0).

Suppose that A'\subset A is an open subring, then the I-adic topology on A gives an induced topology on A'. Since A' is open, then I^N\subset A' for some integer N. Then it is easy to see that the induced subspace topology on A' and the I^N-adic topology on A' 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.

A is a topological ring, and B\subset A is a subset. We say that B is bounded in A if for any open neighborhood U of 0 in A, there exists another open neighborhood of 0 in A such that B\cdot V\subset U.

Here if A is a non-archimedean topological ring, then we can replace B\cdot V by BV, and the definition remains unchanged. And we call an element a\in A is power-bounded if \{a^n\} is a bounded set. a is called topologically nilpotent if a^n\rightarrow 0. And we write A^o as the set of power-bounded elements of A and A^{oo} as the set of topologically nilpotent elements of A.

A little remark is that any element a\in A as a single-element set \{a\} in a topological ring is bounded due to the very simple fact that A\times A\rightarrow A,(a,a')\mapsto aa' is continuous.

In the next post we will enlarge the notion of adic rings to f-adic rings.


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