# Duality between coalgebras and (pseudocompact) algebras - uniqueness

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.