Counterexample to the fundamental theorem of comodules

Question

The Fundamental Theorem of Comodules (aka the Finiteness Theorem for Comodules) states that if $\Bbbk$ is a field, then any element of a comodule over a $\Bbbk$-coalgebra lies in a finite-dimensional subcomodule. I know that this is a distinctive property of $\Bbbk$ being a field.

Some years ago I run into a paper/book in which the author showed that this is not true anymore already when $\Bbbk$ is a commutative ring, but at the present moment I am not able to find that reference again.

Since I need it for didactic reasons, I would like to ask if anybody is aware of some reference in which the topic is treated with some details and, in particular, where I may find a counterexample when $\Bbbk$ is not a field.

Answer 1

I finally found the example I saw a long time ago. It comes from Bergman, "Cogroups and Co-rings in Categories of Associative Rings", §32.

Literally, take a commutative ring $\Bbbk$ which is a principal ideal domain and an irreducible element $p\in\Bbbk$. Consider a module $M$ over $\Bbbk$ presented by generators $x_i$, $i=0,1,2,\ldots$, and relations $p^2x_i=0$. Define a comultiplication $\Delta: M\to M\otimes_\Bbbk M$ by $\Delta\left(x_i\right) = px_{i+1}\otimes_\Bbbk x_{i+1}$. This is trivially coassociative, in view of the relations on $M$, but not counital. However, one can add a counit a posteriori (as one does for units to rings) and the outcome should be a coassociative and counital coalgebra which does not satisfy the Fundamental Theorem.