On the definition of a compactoid

This is a paper about locally convex modules over the valuation ring of a nonarchimedean valued field. We will discuss the definition of a compactoid module and we will give some relations between compactoidity and other compact-like properties. 1991 Mathematics subject classification: 46S10. 0. Introduction In this paper K is a non-archimedean complete valued field with a non-trivial valuation I. BK is the valuation ring of K, i.e. BK = {03BB E K 1}. One problem in defining convex-compact sets in a locally convex space over K is that K is in general not locally compact, which means that ’convex-compact’ becomes a trivial notion. To overcome this problem, Springer proposed in 1965 in [11] the notion of c-compactness, which is based on a convexification of the intersection property of compactness. However, this notion is only suitable for spherically complete K. Later, in 1974, Gruson and van der Put [4] introduced a new notion: compactoidity, based upon a convexification of precompactness. This notion is meaningful for general (non-archimedean) fields K. Gruson proved in [3] that in locally convex spaces over a spherically complete K, ’c-compact & bounded’ is the same as ’complete compactoid’. In this paper we will discuss how to extend the definition of compactoidity to arbitrary locally convex modules over the valuation ring. This is the content of section 2. Section 1 is introductory while in section 3 modules of finite rank will be treated. These modules play an important role in the definition of locally compactoid modules and modules of finite type in section 4. In this last section we also compare the different types of compact-like properties. No proofs are given here. Proofs of the theorems stated in this note will be published elsewhere.


Introduction
In this paper K is a non-archimedean complete valued field with a non-trivial valuation I. BK is the valuation ring of K, i.e. BK = {03BB E K 1}.
One problem in defining convex-compact sets in a locally convex space over K is that K is in general not locally compact, which means that 'convex-compact' becomes a trivial notion.
To overcome this problem, Springer proposed in 1965 in [11] the notion of c-compactness, which is based on a convexification of the intersection property of compactness. However, this notion is only suitable for spherically complete K.
Later, in 1974, Gruson and van der Put [4] introduced a new notion: compactoidity, based upon a convexification of precompactness. This notion is meaningful for general (non-archimedean) fields K. Gruson proved in [3] that in locally convex spaces over a spherically complete K, 'c-compact & bounded' is the same as 'complete compactoid'. In this paper we will discuss how to extend the definition of compactoidity to arbitrary locally convex modules over the valuation ring. This is the content of section 2. Section 1 is introductory while in section 3 modules of finite rank will be treated. These modules play an important role in the definition of locally compactoid modules and modules of finite type in section 4. In this last section we also compare the different types of compact-like properties. No proofs are given here. Proofs of the theorems stated in this note will be published elsewhere. 1  (i) Each Hausdorff locally convex space over K is a locally convex BK-module. (ii) If A is a locally convex BK-module and B is a submodule of A, then (B, |B) is a locally convex BK-module. Theorem 1.16 Let (A, r) be a locally convex BK-module. Then: (A, r) is topologically embeddable in a locally convez space ~ There exists a separating collection of faithful seminorms on A generating r. Definition 1.17 A normed BK-module (A, ~~ ~~ ) is called of countable type if there exists a countable subset X of A such that co X is dense in A.
A locally convex B K-module is called of countable type if for every continuous seminorm p on A the normed space (A/Ker p, p) is of countable type.
In the rest of this paper we will call a BK-module shortly a module.

Compactoidity
For an absolutely convex subset of a locally convex space we know the following definition of compactoidity [2]. Definition 2.1 Let (E, r) be a locally convex space and A an absolutely convex subset of E. A is called compactoid if for every zero neighbourhood U of E there exist n E N and E E such that A C This definition of compactoid seems to depend on the space (E, r) of which A is a subset, but that is only partially true. It only depends on the restricted topology riA on A. Katsaras has proved: Theorem 2.2 (Katsaras' Theorem [5]) A C E is compactoid ~ For every a E K with |03BB| > 1 and every zero neighbourhood U of E there exist n E N and xl, ... , xn E 03BBA such that A C U + co{xr, ... , xn}.
Combining Katsaras' Theorem and the proof of lemma 10.5 in [6], saying that for every A E K with ~a~ > 1, T~A can on only one way be extended to a topology v on ~A that is induced by a locally convex topology on a vector space, one can prove: Theorem 2.3 A is compactoid in (E,T) ~ A is compactoid in every locally convex space (F, v) in which (A, |A) can be topologically embedded.
A first attempt to generalize the notion of compactoidity to arbitrary locally convex modules is a translation of Definition 2.1. Definition 2.4 Let A be a submodule of a locally convex module (B, T). A is called a compactoid in B if for every zero neighbourhood U in B there exist n E N and x1, ... , , xn E B such that A C U A disadvantage of this definition is that it depends on the module B. For instance: If the valuation on K is dense, then B~. := {a E K 1~ is a compactoid in K, but not in BK.
To overcome this obstruction we could try the following definition. Definition 2.5 A locally convex module is called a compactoidl -module if A is compactoid in every locally convex module in which it can be topologically embedded.
If (A,r) is such a compactoid module, then, in particular, A is compactoid in itself. And it is not hard to see that the converse is also true. So our new definition of compactoidity is equivalent to the following. Definition 2.6 A locally convex module (A, r) is called a compactoid1-module if for every zero neighbourhood U in A there exist n E N and x1, ..., xn E A such that A C U + CO{x~, ... , ~n~.
This definition of a compactoid module now depends only on the module A itself, but it is too restricted for being a generalization of the notion of a compactoid for absolutely convex subsets of locally convex spaces. For example, if the valuation on K is dense, then provided with the valuation topology is a compactoid in the sense of Definition 2.1, but not a compactoidl-module. Definition 2.4 gives rise to yet another definition of compactoidity that also avoids the dependence on the module B; we simply replace in Definition 2.5 'every' by 'some': Definition 2.7 A locally convex module (A, T) is called a compactoid2-module if A is compactoid in some locally convex module in which it can be topologically embedded.
This definition turns out to be equivalent with our final definition of compactoidity (2.12) at the end of this section.
A second attempt for a generalization of compactoidity is based on Katsaras' Theorem.
This theorem does not directly carry over to a definition that is suitable for arbitrary locally convex modules, since for a module A the set AA, where A ~ K with |03BB| > 1, is not defined.
A slight modification in the formulation of Katsaras' Theorem gives us a workable definition of a compactoid module. Definition 2.8 Let (A, T) be a locally convex module. (A, r) is called a compactoid3module if for every a E .8h-, with |03BB| 1 and every zero neighbourhood U in A there exist n ~ N and x1,...,xn ~ A such that 03BBA ~ U + co{x1,...,xn}. This definition is a good one in the sense that it only depends on the module A and the topology r on A. Moreover, it is a generalization of the notion of compactoid for absolutely convex subsets of locally convex spaces, as we can see in the following theorem. Theorem 2.9 Let A be an absolutely convex subset of a locally convex space (E, T). Then: The definition of a compactoid3-module has been used by Wim Schikhof in [10]. Yet we propose a slight modification of this definition, because of the following reasons. 1) If A is an absolutely convex subset of a locally convex space (E, r), then the linear span of A is of countable type, hence the module (A, riA) is also of countable type. However, a locally convex compactoid3-module need not be of countable type.
2) If A is an absolutely convex subset of a locally convex metrizable space (E, T), then ( A, is topologicaily embeddable in a compactly generated module, that is a locally convex module (B, v), such that there exists a compact subset X of B with B = coX. . However, a locally convex metrizable compactoid3-module can not always be topologically embedded in a compactly generated module. For example, let (E, ~~ ~~) be a Banach space with an orthonormal base, which is not of countable type. Let  For an open submodule U of E we obtain the following diagram.
It is not hard to see that any submodule of A that is open in the restricted topology has the form A n U, where U is an open submodule of E. Now one can prove the following. Theorem 2.11 Let A be an absolutely convex subset of a locally convex space. Then: A is a compactoid ~ For every submodule V of A that is open in the restricted topology, A/V is embeddable in a finitely generated module.
These observations lead to the following definition, meaningful for arbitrary locally convex modules.

Definition 2.12 (Final definition of compactoidity) Let (A, T) be a locally convex module. (A, T) is called a compactoid module if for every open submodule U of A, the quotient A/U is embeddable in a finitely generated module.
From Theorem 2.11 it is clear that this definition is a generalization of the notion of compactoidity for absolutely convex subsets of locally convex spaces. Moreover, this definition is beautiful, because it does not involve embeddings into other modules like in Definition 2.1,2.5and2.7. This definition of compactoidity was also proposed by Caenepeel in an unpublished note. The basic theory of compactoidity in locally convex space can be generalized to locally convex compactoid modules, and also the difficulties we had with the compactoid3-modules, mentioned after Theorem 2.9 vanish (see Theorem 4.4).
We will conclude with some theorems comparing the various definitions of compactoidity given in this section.

Theorem 2.13 Let A be a submodule of a locally convex module (B, T), such that A is compactoid in B (in the sense of Definition 2.4). Then (A, |A) is a compactoid module.
Theorem 2.14 Let (A, r) be a locally convex module, then: (A, r) is a compactoid module ~ There exists a locally convex module (B, v), in which (A, T) can be topologically embedded, such that A is compactoid in B (i. e. (A, T) is a compactoid2-module ).

Modules of finite rank
To know more about the structure of compactoid modules we need to investigate those modules that are embeddable in a finitely generated module.
Examples of these modules are bounded absolutely convex subsets of finite-dimensional vector spaces. The module BK/B-K is generated by one element and not absolutely convex (for it is a torsion module). The class of all modules that are embeddable in a finitely generated module is denoted BK. This last description of ,~x is very helpful as it directly links ~x to the well-known class of finite-dimensional absolutely convex sets.
Another class of modules, related to BK, is the class FK of all modules that are quotient of a (not necessarily bounded) finite-dimensional absolutely convex set.
The module K is a member of .~'x, but not of I~x.
Theorem 3.2 ,~'x is the smallest class C of modules for which: 1. K E C 2. C is closed with respect to submodules 3. C is closed with respect to finite direct sums 4. C as closed with respect to quotients.
The class plays an important role in the definition of a locally compactoid module in the following section. If A is a module of finite Fleischer rank, then the Fleischer rank of A is the minimal among all n E N for which there exist a torsion-free module B, with rank B = n and a surjective homomorphism 03C6 : B ~ A.

Compact-like modules
Besides the definition of a compactoid module we will discuss in this section the definitions of locally compactoid modules, modules of finite type, c-compact modules and some relations between these types of modules.
First we recall our final definition of compactoidity from section 2.
A locally convex module (A, T) is called a compactoid module if for every open submodule U of A the quotient A/U is embeddable in a finitely generated module.
This definition can now be modified into: (iii) A product of compactoid modules is a compactoid module. (iv) Each compactoid module is topologically embeddable in a product of finitely generated discrete torsion modules.
From (iv) of this theorem we obtain the following corollary.  (For the definition of pure compactoid in vector space see E7~).
A submodule of a pure compactoid module need not be a pure compactoid module. However, the following can be proved. (ii) A product of pure compactoid module3 is a pure compactoid module.
(iii) The completion of a pure compactoid module is a pure compactoid module.
(iv) If the completion of a locally convex module is a pure compactoid module, then the module itself is a pure compactoid module.
The following equivalence theorem about pure compactoid modules can also be proved. Interesting is the equivalence between (,Q) and (y). Consider the case that A is an absolutely convex subset of a locally convex space (K, r). Let us call a seminorm on A restricted, if it is the restriction of a faithful seminorm on E. In [8] it is proved that the following two assertions are equivalent.
(a') (A, is a pure compactoid module (03B2') Every continuous restricted seminorm on A has a maximum. But (,Q') is not equivalent with (,') Every continuous re3tricted seminorm on A is bounded. For example, J?~ is an absolutely convex subset of the locally convex space (E, ] ~), and every continuous restricted seminorm on 2?~ is bounded, for if p is a restricted seminorm on BK, then there exists a seminorm q on K such that p = q|B-K. Then p q( 1 ) on B-K.
But the valuation, which is of course a continuous restricted seminorm, has no maximum on B-K.
The final theorem about compactoid modules we will give is the following.
Theorem 4.10 Every compactoid module is topologically embeddable in a pure compactoid module. Now we will discuss the notion of a locally compactoid module. We can generalize the definition of a locally compactoid absolutely convex subset of a locally convex space, as is given in [9], to arbitrary locally convex modules in the same spirit as we did it for compactoids by replacing BK by FK. This definition is a generalization of the notion of locally compactoidity in locally convex space in the following sense. Theorem 4.12 Let A be an absolutely convex subset of a locally convex space (E, r). . Then the following two assertions are equivalent. (a) a locally compactoid module. There exists a locally convex space (F, v), in which (A, |A) can be topologically ernbedded, such that A is a locally compactoid absolutely convex subset of F (in the sense of (9j ).
In general it is not true that if A is an absolutely convex subset of a locally convex space (E, T) such that (A, is a locally compactoid module, then A is a locally compactoid subset of E. In [9] Schikhof gives a counterexample. The following theorem is not very suprising. Again with the aid of properties of modules in one can prove: Theorem 4.14 (i) A 3ubmodule of a locally compactoid module is a locally compactoid module.
(ii) A continuous homomorphic image of a locally compactoid module is a locally compactoid module.
(iii) A product of locally compactoid modules is a locally compactoid module.
From the theory of locally convex spaces we also know the notion of a space of finite type. Contrary to the situation in locally convex space, not all continuous seminorms on a module of finite type need to be of finite type. We even have the following.
Theorem 4.17 Let (A, T) be a locally convex module 3uch that every continuous seminorm on A is of finite type. Then A E FK.
We have the following representation theorem. (a) (A, T) is of finite type.