The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below.
Let $k$ be a field. The following are equivalent for a $k$-algebra $A$:
- $A$ is pseudocompact,
- $A$ is profinite,
- $A\cong C^*$ where $C$ is a coalgebra (and $(-)^*=\operatorname{Hom}_k(-,k)$).
The questions I have are about uniqueness:
- What is an explicit example of a profinite algebra where $C$ is not unique, i.e. two coalgebras $C$ and $C'$ whose duals are isomorphic as $k$-algebras (but of course not as pseudocompact algebras as the categories of coalgebras and pseudocompact algebras are dual.
- Formulated another way: What is an example of an algebra $A$ with two pseudocompact topologies on it?
- What about (infinitely generated) modules over a (noetherian) pseudocompact algebra.
Some background:
A pseudocompact $k$-algebra is a Hausdorff linear topological $k$-algebra ($k$ here has the discrete topology) $A$ having a basis $\mathcal{F}$ consisting of ideals of finite codimension such that the natural morphism $A\to \varprojlim A/I$ is an isomorphism.
There is a duality of categories between the category of $k$-algebras and the category of coalgebras giving by the $k$-dual in one direction and the continuous dual in the other direction.
Given any profinite algebra $A$, i.e. an algebra which can be written as $\varprojlim A_i$ for finite dimensional algebras $A_i$, the initial topology for $A\to \varprojlim A_i$ (where the $A_i$ are given the discrete topology) gives $A$ a pseudocompact topology.
A priori there is no reason that an infinite dimensional algebra should not have more than one pseudocompact topology. Definitely a profinite algebra can usually be written as $\varprojlim$ in many different ways.
In [Gastel-Van den Bergh: Graded modules of Gelfand-Kirillov dimension one over three-dimensional Artin-Schelter regular algebras, 1997, Proposition 3.21] I found the result that for a noetherian pseudocompact ring there is a unique pseudocompact topology, namely the $J$-adic topology where $J$ denotes the Jacobson radical. I assume that his works in the $k$-enriched setting as well. Thus an answer to the second question should be non-noetherian.
Similarly in [Schneider: p-adic Lie groups, Corollary 22.4] I found that a similar uniqueness result holds for finitely generated pseudocompact modules over noetherian pseudocompact rings.
I also found that the similar case of profinite groups was only settled in 2007 in [Nikolov-Segal: On finitely generated profinite groups, I: strong completeness and uniform bounds] proving that there is a unique pseudocompact topology on each finitely generated profinite group. In this area there are some explicit counterexamples, see e.g. this mathoverflow question.
