Modules over the dual of an infinite dimensional coalgebra Let $k$ be a field and let $A$ be a finite dimensional (unital, associative, not necessarily commutative) $k$-algebra. The $k$-linear dual of $A$ is a coalgebra, and viceversa, the $k$-linear dual of a finite dimensional coalgebra is an algebra. Let $V$ be a finite dimensional $k$-vector space. The isomorphisms
$$\mathrm{Hom}_k(V \otimes A, V) \cong \mathrm{Hom}_k(V, \mathrm{Hom}_k(A,V)) \cong \mathrm{Hom}_k(V, V \otimes A^\vee)$$
and
$$\mathrm{Hom}_k(V \otimes A, V) \cong \mathrm{Hom}_k(V^\vee, (V \otimes A)^\vee) \cong \mathrm{Hom}_k(V^\vee, V^\vee \otimes A^\vee)$$
provide bijections between all of the following:

*

*The $A$-module structures on $V$

*The $A^\vee$-comodule structures on $V$

*The $A$-module structures on $V^\vee$

*The $A^\vee$-comodule structures on $V^\vee$
In particular, there is an equivalence of categories $\mathbf{Mod}_A \cong \mathbf{Comod}_{A^\vee}$. (Here, (co)modules are always assumed to be finitely generated and to carry a left (co)action).
Many of the above identifications exploit finite dimensionality of $A$. If $A$ is infinite dimensional, then the induced map $A^\vee \to (A \otimes A)^\vee$ does not, in general, land in the proper subspace $A^\vee \otimes A^\vee$ and therefore does not define a coalgebra structure on $A^\vee$. Therefore, the dual of an algebra is not always a coalgebra. But the dual of a coalgebra is always an algebra! Hence, it is natural to ask:

Is it true that for any (also infinite dimensional) coalgebra $B$, we still have an equivalence of categories $\mathbf{Comod}_B \cong \mathbf{Mod}_{B^\vee}$? Do any of the above identifications of Hom-spaces still hold in the infinite dimensional case (using $B$ instead of $A^\vee$ and $B^\vee$ instead of $A$)?

 A: The answer, in general, is no. Let me assume that $\Bbbk$ is a field, for the sake of simplicity. There is an equivalence of categories between the category of (say, right) comodules over a $\Bbbk$-coalgebra $C$ and the subcategory of left rational $C^*$-modules, where $C^* = \mathsf{Hom}_{\Bbbk}(C,\Bbbk)$.
Let me try to sketch how things go in the infinite-dimensional case.
Lemma.
For every $\Bbbk $-module $M$ and every $\Bbbk $-coalgebra $C$, every linear map $\delta :M\rightarrow M\otimes C$ induces a linear map $\mu _{\delta }:C^{\ast }\otimes M\rightarrow M$ given by
$$\mu _{\delta }\left( f\otimes m\right) =\left( M\otimes f\right) \left(
\delta \left( m\right) \right) \label{eq:mudelta} \tag{1}$$
for all $f\in C^{\ast }$, $m\in M$. Moreover, $\delta $ is a coassociative
and counital coaction if and only if $\mu _{\delta }$ is an associative and
unital action.
This provides for us a functor $\mathcal{L}:\mathfrak{M}^{C}\rightarrow
\left. _{C^{\ast }}\mathfrak{M}\right. $ from the category of right $C$-comodules $\mathfrak{M}^{C}$ to the one of left $C^{\ast }$-modules $\left.
_{C^{\ast }}\mathfrak{M}\right. $.
Definition.
A $C^{\ast }$-module $\left( M,\mu \right) $ is said to be rational
if there exists a linear map $\delta :M\rightarrow M\otimes C$, called the
associated coaction, such that $\mu =\mu _{\delta }$.
For every $\Bbbk $-module $M$ and every $\Bbbk $-coalgebra $C$ we may consider, on the one hand, the following map
$$
\alpha _{M}:M\otimes C\rightarrow \mathsf{Hom}_{\Bbbk }\left( C^{\ast
},M\right), \qquad m\otimes c\mapsto \left[ f\mapsto mf\left( c\right) \right] ,
$$
that is to say,
\begin{equation}
\alpha _{M}\left( m\otimes c\right) \left( f\right) =\left( M\otimes
f\right) \left( m\otimes c\right), \tag{2} \label{eq:alpha}
\end{equation}
which is always injective.
On the other hand, for every $C^{\ast }$-module $M$ we may consider the
assignment
$$
\beta _{M}:M\rightarrow \mathsf{Hom}_{\Bbbk }\left( C^{\ast },M\right)
, \qquad m\mapsto \left[ f\mapsto f\cdot m\right] ,
$$
where $\cdot $ denotes the $C^{\ast }$ action.
Remark.
Note that $\mathsf{Hom}_{\Bbbk }\left( C^{\ast },M\right) $ is a $C^{\ast }$-module with action
$$
C^{\ast }\otimes \mathsf{Hom}_{\Bbbk }\left( C^{\ast },M\right) \rightarrow 
\mathsf{Hom}_{\Bbbk }\left( C^{\ast },M\right) , \qquad f\otimes \psi \mapsto \left[
\left( f\cdot \psi \right) :g\mapsto \psi \left( g\ast f\right) \right] .
$$
and both $\alpha _{M}$ and $\beta _{M}$ are morphisms of $C^{\ast }$-modules.
Moreover, they are natural transformations.
Proposition.
The following are equivalent for a $C^{\ast }$-module $M$:

*

*there exists $\delta :M\rightarrow M\otimes C$ such that $\alpha_{M}\circ \delta =\beta _{M}$;

*there exists $\delta :M\rightarrow M\otimes C$ such that $\mu_{M}=\mu _{\delta }$ (i.e. $M$ is a rational $C^{\ast }$-module).

Definition.
For every $C^{\ast }$-module $M$ we define $M^{\mathsf{rat}}:=\beta
_{M}^{-1}\left( \alpha _{M}\left( M\otimes C\right) \right) $ and we call it
the rational part of $M$.
In what follows we are going to show that $M^{\mathsf{rat}}$ is always a
rational $C^{\ast }$-module and that it is the maximal rational $C^{\ast }$-module in $M$ (i.e. the biggest one whose induced $C^{\ast }$-action is
coming from a $C$-coation as in the first Lemma).
Lemma.
For every $C^{\ast }$-module $M$, $M^{\mathsf{rat}}$ is a $C^{\ast }$-submodule of $M$. In particular, it is a $C^{\ast }$-module. Moreover, $m\in M^{\mathsf{rat}}$ if and only if there exists a (necessarily unique) $\sum_{i=1}^{t}m_{i}\otimes c_{i}$ in $M\otimes C$ such that $f\cdot m=\sum_{i=1}^{t}m_{i}f\left( c_{i}\right) $ for every $f\in C^{\ast }$.
Furthermore, $\sum_{i=1}^{t}m_{i}\otimes c_{i}$ lives in $M^{\mathsf{rat}}\otimes C$. In particular, $M^{\mathsf{rat}}$ is a rational $C^{\ast }$-module and a $C$-comodule.
We are now ready to see why the functor $\mathcal{L}$ is not an equivalence in general. For every $C^{\ast }$-module $\left( M,\mu \right) $, consider its rational part $M^{\mathsf{rat}}$ together with the coaction $\delta_{\mu }:M^{\mathsf{rat}}\rightarrow M^{\mathsf{rat}}\otimes C$. Now, let $\varphi :\left( M,\mu \right) \rightarrow \left( N,\nu \right) $ be a morphism of $C^{\ast }$-modules and denote by $\varphi _{\ast }$ the $C^{\ast }$-linear morphism $\mathsf{Hom}_{\Bbbk }\left( C^{\ast },\varphi \right) :\mathsf{Hom}_{\Bbbk }\left( C^{\ast },M\right) \rightarrow \mathsf{Hom}_{\Bbbk }\left( C^{\ast },N\right) ,\psi \mapsto \varphi \circ \psi $.
Lemma.
The $C^{\ast }$-linear morphism $\varphi $ induces a $C $-colinear morphism $\varphi ^{\mathsf{rat}}:\left( M^{\mathsf{rat}},\delta _{\mu }\right) \rightarrow \left( N^{\mathsf{rat}},\delta _{\nu}\right) $.
Proposition.
The assigment $\mathcal{R}:\left. _{C^{\ast }}\mathfrak{M}\right. \rightarrow \mathfrak{M}^{C},\left( M,\mu \right)
\mapsto \left( M^{\mathsf{rat}},\delta _{\mu }\right) $ is functorial and it
is right adjoint to the functor $\mathcal{L}:\mathfrak{M}^{C}\rightarrow
\left. _{C^{\ast }}\mathfrak{M}\right. :\left( N,\delta \right) \mapsto
\left( N,\mu _{\delta }\right) $. The unit is given by the identity morphism
and the counit by the canonical inclusion $M^{\mathsf{rat}}\subseteq M$.
Theorem.
The functor $\mathcal{L}:\mathfrak{M}^{C}\rightarrow \left. _{C^{\ast }}\mathfrak{M}\right. :\left( N,\delta \right) \mapsto \left( N,\mu _{\delta
}\right) $ is an equivalence of categories (in fact, an isomorphism) if and
only if the coalgebra $C$ is finite-dimensional.
Denote by $\mathfrak{Rat}\left( \left. _{C^{\ast }}\mathfrak{M}\right.
\right) $ the full subcategory of rational $C^{\ast }$-modules. We can
consider the corestriction $\mathcal{L}^{\prime }:\mathfrak{M}%
^{C}\rightarrow \mathfrak{Rat}\left( \left. _{C^{\ast }}\mathfrak{M}\right.
\right) $ of the functor $\mathcal{L}$ and the restriction $\mathcal{R}
^{\prime }:\mathfrak{Rat}\left( \left. _{C^{\ast }}\mathfrak{M}\right.
\right) \rightarrow \mathfrak{M}^{C},\left( M,\mu \right) \mapsto \left(
M,\delta _{\mu }\right) $ of the functor $\mathcal{R}$.
Theorem.
The functors $\mathcal{L}^{\prime }$ and $\mathcal{R}^{\prime }$ are
quasi-inverses, giving an equivalence of categories $\mathfrak{M}^{C}\cong 
\mathfrak{Rat}\left( \left. _{C^{\ast }}\mathfrak{M}\right. \right) $.
For further details and a more exhaustive treatment, I would suggest Chapter 2, Section 2.2 of:
Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Şerban, Hopf algebras. An introduction, Pure and Applied Mathematics, Marcel Dekker. 235. New York, NY: Marcel Dekker. ix, 401 p. (2001). ZBL0962.16026.
