Contramodule as direct limit of its finitely generated subcontramodules

Question

$\DeclareMathOperator\Hom{Hom}$Let $K$ be a field. Let $C$ be a $K$-coalgebra. A contramodule $M$ over $C$ is a $K$-space with a $K$-linear map $\pi_M:\Hom_K(C,M)\longrightarrow M$ such that $\pi_M \circ \Hom(\varepsilon_C,M)= id_M$ and $\pi_M \circ Hom(\Delta_C,M)=\pi \circ Hom(C,\pi)$ where $\Delta_C$ denotes the comultiplication in $C$. My primary references for contramodules are Positselski - Contramodules and Positselski - Contramodules over pro-perfect topological rings.

The category of $C$-contramodules is cocomplete (see, for instance, §1.7, Positselski - Contramodules over pro-perfect topological rings). However, colimit in the $C$-contramodule category doesn't appear to be the one coming from the underlying vector space structure. It is not clear to me how the colimit in this category is defined. Any help will be great! Further, I want to understand if a contramodule over $C$ is the direct limit of its finitely generated subcontramodules. We know that similar statement holds for a module over a ring and for a comodule over a coalgebra over a field. But unlike the category of modules and the category of comodules (over a coalgebra over a field), the $C$-contramodule category is not locally finitely generated.

Answer 1

This answer concerns contramodules that are finitely generated as objects of the category of contramodules, in the sense of the abstract category-theoretic definition of "finitely generated". The examples below in this answer tend to suggest that there are too few such finitely generated contramodules to hope for an interesting theory.

Let $\mathbf{K}$ be a cocomplete category (i.e., a category with set-indexed colimits). An object $Q\in\mathbf{K}$ is said to be finitely generated if, for any diagram $(E_\gamma)_{\gamma\in\Gamma}$ of objects in $\mathbf{K}$, indexed by a directed poset $\Gamma$, with monomorphisms $E_\gamma\to E_\delta$ for $\gamma<\delta\in\Gamma$, the natural map $$ \varinjlim\nolimits^{\mathbf{Sets}}_{\gamma\in\Gamma} \operatorname{Hom}_{\mathbf{K}}(Q,E_\gamma) \longrightarrow \operatorname{Hom}_{\mathbf{K}}(Q,\, \varinjlim\nolimits^{\mathbf{K}}_{\gamma\in\Gamma}E_\gamma) $$ is an isomorphism (bijection) of sets.

The following example of a direct limit is helpful to consider. In any cocomplete category $\mathbf{K}$, for any family of elements $(P_x\in\mathbf{K})_{x\in X}$, the coproduct of all the objects $P_x$ in $\mathbf{K}$ is the direct limit of the coproducts of $(P_z)_{z\in Z}$ taken over all the finite subsets $Z\subset X$. So let $\Gamma$ denote the directed poset of all finite subsets $Z\subset X$, ordered by inclusion. Then we have $$ \coprod\nolimits^\mathbf{K}_{x\in X}P_x= \varinjlim\nolimits^{\mathbf K}_{Z\in\Gamma} \left(\coprod\nolimits^{\mathbf K}_{z\in Z}P_z\right). $$ In particular, when all the objects $P_x$ are the same, $P_x=P$ for all $x\in X$, we have $$ P^{(X)} = \varinjlim\nolimits^{\mathbf K}_{Z\in\Gamma} P^{(Z)}, $$ where the notation is $P^{(Y)}=\coprod^{\mathbf K}_{y\in Y}P$.

If $\mathbf{K}$ is a category with zero object then, for any objects $E$ and $F\in\mathbf{K}$, the coproduct injections $E\to E\sqcup F$ and $F\to E\sqcup F$ are monomorphisms. So in particular, in the above example, the $\Gamma$-indexed directed diagram of finite copowers $P^{(Z)}$ of an object $P$ is a diagram of monomorphisms.

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Now let $C$ be a coassociative, counital coalgebra over a field $k$. We are interested in the category of left $C$-contramodules $\mathbf{K}=C{-}\mathbf{contra}$. Given a vector space $V$, the $C$-contramodule $F=\operatorname{Hom}_k(C,V)$ (with the contraaction map $\pi_F$ induced by the comultiplication map $\Delta_C$) is called the free $C$-contramodule generated by $V$. Let $X$ be a set indexing a basis in $V$.

We will use the notation $A[X]=\bigoplus_{x\in X}^{\mathbf{Ab}}A$ for the direct sum of copies of the same object in the category of abelian groups (or vector spaces). So, in particular, we have $V=k[X]$. The $C$-contramodule $\operatorname{Hom}_k(C,k[X])$ can be also called the free $C$-contramodule generated by set $X$. In particular, consider $P=C^*=\operatorname{Hom}_k(C,k)$; this is the free $C$-contramodule with one generator. The notation $C^*[[X]]=\operatorname{Hom}_k(C,k[X])$ is sometimes used.

In the category of $C$-contramodules $\mathbf{K}=C{-}\mathbf{contra}$, we have $$ \operatorname{Hom}_k(C,k[X])=\coprod\nolimits_{x\in X}^{\mathbf K}C^*=P^{(X)}. $$ Hence, following the above discussion of an arbitrary category $\mathbf{K}$, $$ \operatorname{Hom}_k(C,k[X])= \varinjlim\nolimits_{Z\in\Gamma}^{\mathbf K} \operatorname{Hom}_k(C,k[Z]), $$ where $\Gamma$ is the directed poset of all finite subsets $Z\subset X$.

Now let $Q=C^*$ be also the free $C$-contramodule with one generator. Then the functor $\operatorname{Hom}_{\mathbf K}(Q,{-})\colon C{-}\mathbf{contra}\to\mathbf{Sets}$ is simply the forgetful functor $C{-}\mathbf{contra}\to\mathbf{Sets}$ (assigning to a $C$-contramodule $M$ its underlying set $M$).

Hence we have \begin{multline*} \varinjlim\nolimits^{\mathbf{Sets}}_{Z\in\Gamma} \operatorname{Hom}_{\mathbf{K}}(Q,P^{(Z)}) = \varinjlim\nolimits^{\mathbf{Sets}}_{Z\in\Gamma} \operatorname{Hom}_k(C,k[Z]) \\ = \varinjlim\nolimits^{\mathbf{Sets}}_{Z\in\Gamma} (C^*\otimes_k k[Z]) = C^*\otimes_k k[X]=C^*[X]. \end{multline*}

On the other hand, $$ \operatorname{Hom}_{\mathbf{K}}(Q,\, \varinjlim\nolimits^{\mathbf{K}}_{Z\in\Gamma} P^{(Z)}) = \operatorname{Hom}_{\mathbf{K}}(Q,P^{(X)})= \operatorname{Hom}_k(C,k[X])=C^*[[X]]. $$

Whenever the coalgebra $C$ is infinite-dimensional over $k$ and the set $X$ is infinite, the natural (injective) map $C^*\otimes_k k[X]\to\operatorname{Hom}_k(C,k[X])$ is not an isomorphism. Thus, for any infinite-dimensional coalgebra $C$, the free $C$-contramodule with one generator $Q=C^*$ is not a finitely generated object in $\mathbf{K}=C{-}\mathbf{contra}$.

This is too long already, so I will perhaps continue with a specific counterexample in a comment.

Answer 2

Let us say that a contramodule $P$ is set-theoretically finitely generated if there exists a finite set of elements $p_1,\dotsc,p_n\in P$ such that any subcontramodule of $P$ containing $p_1,\dotsc,p_n$ coincides with $P$. So the set-theoretical finite generatedness is not an inner property of an object of a contramodule category (rather, it can be defined in terms of the category of contramodules and the forgetful functor from it to the category of sets).

This answer concerns representing a contramodule as a direct limit of its set-theoretically finitely generated subcontramodules. This is a more interesting question, actually, then the one about category-theoretically finitely generated subcontramodules (discussed in the other answer).

The maximal natural generality for this question is that of algebras/modules over additive monads on the category of sets. This is very abstract, so let us choose an intermediate generality level of contramodules over a complete, separated topological ring $R$ with a base of neighborhoods of zero consisting of open right ideals (as per one of the references in the question). The particular case of contramodules over a coalgebra $C$ over a field $k$ corresponds to the choice of a linearly compact topological algebra $R=C^*$.

Consider the forgetful functor $\rho\colon R{-}\mathbf{contra}\to R{-}\mathbf{mod}$ from the category of left $R$-contramodules $R{-}\mathbf{contra}$ to the category of left $R$-modules $R{-}\mathbf{mod}$. The functor $\rho$ is faithful, exact, and preserves infinite products (but not coproducts). The functor $\rho$ has a left adjoint functor $\delta\colon R{-}\mathbf{mod}\to R{-}\mathbf{contra}$, which can be constructed as follows.

For every set $X$, denote by $R[X]$ the free left $R$-module spanned by $X$ and by $R[[X]]$ the free left $R$-contramodule spanned by $X$. The underlying set of $R[X]$ is the set of all finite linear combinations of elements of $X$ with the coefficients in $R$, while the underlying set of $R[[X]]$ is the set of all infinite linear combinations $\sum_{x\in X}r_xx$ of elements of $X$ with the families of coefficients $(r_x\in R)_{x\in X}$ converging to zero in $R$. The latter condition means that for every neighborhood of zero $U\subset R$ the set $\{x\in X\mid r_x\notin U\}$ is finite.

Then one has $\delta(R[X])=R[[X]]$. Furthermore, the functor $\delta$, being a left adjoint, preserves colimits, and in particular cokernels. Any $R$-module $M$ can be represented as the cokernel of a morphism of free $R$-modules $f\colon R[Y]\to R[X]$. The $R$-contramodule $\delta(M)$ can be computed as the cokernel of the induced morphism of free $R$-contramodules $\delta(f)\colon R[[Y]]\to R[[X]]$.

Notice that for a finite set $Z$ the $R$-modules $R[Z]$ and $R[[Z]]$ coincide, or more precisely $\rho(R[[Z]])=R[Z]$. So one has $\rho\delta(R[Z])=R[Z]$ and $\delta\rho(R[[Z]])=R[[Z]]$. The following lemma provides a generalization of this observation.

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Lemma. For any set-theoretically finitely generated $R$-contramodule $P$, the adjunction morphism $\delta\rho(P)\to P$ in $R{-}\mathbf{contra}$ is an isomorphism.

Proof. The functor $\rho$ is conservative, so it suffices to show that the underlying map $\rho\delta\rho(P)\to\rho(P)$ is an isomorphism in $R{-}\mathbf{mod}$, or equivalently, the adjunction morphism $\rho(P)\to\rho\delta\rho(P)$ in $R{-}\mathbf{mod}$ is an isomorphism.

Saying that $P$ is a set-theoretically finitely generated contramodule means that $P$ is a quotient contramodule of a free contramodule $R[[Z]]$ with a finite set of generators $Z$. As $\rho(R[[Z]])=R[Z]$, it follows that $\rho(P)$ is a finitely generated $R$-module. So the $R$-module $\rho(P)$ can be represented as the cokernel of a morphism of free $R$-modules $f\colon R[Y]\to R[Z]$ (where the set $Z$ is finite, while the set $Y$ may be infinite).

Now the $R$-contramodule $\delta\rho(P)$ is the cokernel of the morphism of free $R$-contramodules $\delta(f)\colon R[[Y]]\to R[[Z]]$, so the $R$-module $\rho\delta\rho(P)$ is the cokernel of the map $\rho\delta(f)\colon\rho(R[[Y]])\to R[Z]$. We have the adjunction morphism of morphisms (i.e., a commutative square) $f\to\rho\delta(f)$ in $R{-}\mathbf{mod}$, which reduces to a commutative triangle $R[Y]\to\rho(R[[Y]])\to R[Z]$. It follows that the adjunction map $\rho(P)\to\rho\delta\rho(P)$ is surjective.

As the composition $\rho(P)\to\rho\delta\rho(P)\to\rho(P)$ is an isomorphism, we can conclude that the map $\rho(P)\to\rho\delta\rho(P)$ is an isomorphism, too. This finishes the proof of the lemma.

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Let us introduce short notation $\mathbf{K}=R{-}\mathbf{contra}$ and $\mathbf{A}=R{-}\mathbf{mod}$ for the categories of left $R$-contramodules and left $R$-modules.

Given an $R$-contramodule $E$, we notice first of all that a set-theoretically finitely generated subcontramodule of $E$ is the same thing as a finitely generated $R$-submodule of $E$. In other words, the forgetful functor $\rho$ induces a bijection between the set of all set-theoretically finitely generated subcontramodules of $E$ and the set of all finitely generated submodules of $\rho(E)$. This follows from the fact that $\rho(R[[Z]])=R[Z]$ for any finite set $Z$.

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Theorem. Let $E$ be an $R$-contramodule and $\Gamma$ be a subset in the set of all set-theoretically finitely generated subcontramodules in $E$ such that $\Gamma$ is a directed poset in the inclusion order and no proper $R$-submodule of $E$ contains all the subcontramodules of $E$ belonging to $\Gamma$. (For example, taking $\Gamma$ to be the set of all set-theoretically finitely generated subcontramodules of $E$ satisfies these conditions.) Then one has a natural isomorphism in $\mathbf{K}=R{-}\mathbf{contra}$ $$ \varinjlim\nolimits^{\mathbf K}_{P\in\Gamma}P\simeq\delta\rho(E). $$

Proof. By assumptions, we have $$ \varinjlim\nolimits^{\mathbf A}_{P\in\Gamma}\rho(P)=\rho(E) $$ in $\mathbf{A}=R{-}\mathbf{mod}$. Applying the functor $\delta$, which preserves all colimits as a left adjoint, we get $$ \varinjlim\nolimits^{\mathbf K}_{P\in\Gamma}\delta\rho(P)=\delta\rho(E) $$ in $\mathbf{K}=R{-}\mathbf{contra}$. It remains to use the lemma to the effect that $\delta\rho(P)=P$.

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So we see that the natural morphism $\varinjlim^{\mathbf K}_{P\in\Gamma}P\to E$ is an isomorphism if and only if the natural morphism $\delta\rho(E)\to E$ is. Quite generally, given a pair of adjoint functors $\delta$ and $\rho$, the adjunction morphism $\delta\rho\to\mathrm{Id}$ is an isomorphism if and only if the functor $\rho$ is fully faithful. Thus the map $\varinjlim^{\mathbf K}_{P\in\Gamma}P\to E$ is an isomorphism for every $R$-contramodule $E$ if and only if the forgetful functor $\rho\colon R{-}\mathbf{contra}\to R{-}\mathbf{mod}$ is fully faithful.

The latter property does not hold in general, of course (it is not difficult to come up with an example of a coalgebra $C$ over a field $k$ for which the forgetful functor $C{-}\mathbf{contra}\to C^*{-}\mathbf{mod}$ is not full). But it does hold for surprisingly many infinite-dimensional coalgebras $C$ or topological rings $R$. (In particular, it certainly holds for $C^*=k[[t]]$.)

Answer 3

At last, a question was raised about constructing colimits in contramodule categories generally. Once again, the maximal natural generality is that of accessible additive monads on the category of sets. But let us discuss the more specific case of contramodules over a topological ring $R$.

The example of the coherator functor in the quasi-coherent sheaf theory (mentioned in the comments to the question) is indeed instructive. The difference is that the coherator is a right adjoint (a coreflector), while in the contramodule theory we have a left adjoint.

Quite generally, let $\mathbf{A}$ be a category and $\mathbf{K}\subset\mathbf{A}$ be a full subcategory with the inclusion functor $\rho\colon\mathbf{K}\to\mathbf{A}$. The functor $\delta\colon\mathbf{A}\to\mathbf{K}$ left adjoint to $\rho$, if it exists, is called the reflector (of $\mathbf{A}$ onto $\mathbf{K}$). In this case, the full subcategory $\mathbf{K}\subset\mathbf{A}$ is said to be reflective.

Any reflective full subcategory $\mathbf{K}$ is closed under limits in its ambient category $\mathbf{A}$. Concerning the colimits, the colimits in $\mathbf{K}$ exist whenever they exist in $\mathbf{A}$, and can be computed as follows. Let $D\colon\Gamma\to\mathbf K$ be a diagram in $\mathbf K$. Then $\rho(D)$ is a diagram in $\mathbf{A}$. Furthermore, $\delta\rho(D)$ is a diagram in $\mathbf K$ naturally isomorphic to $D$ (as $\delta\rho\simeq\mathrm{Id}$, since $\rho$ is fully faithful). Now we have $$ \varinjlim\nolimits^{\mathbf K}_\Gamma D= \varinjlim\nolimits^{\mathbf K}_\Gamma\delta\rho(D)= \delta(\varinjlim\nolimits^{\mathbf A}_\Gamma\rho(D)), $$ as the functor $\delta$, being a left adjoint, preserves all colimits.

Simply put, in order to compute a colimit in $\mathbf K$, it suffices to compute the colimit of the same diagram in the ambient category $\mathbf A$, obtain an object of $\mathbf A$, and apply to it the reflector $\delta$.

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In the situation at hand, we always have a left adjoint functor $\delta\colon R{-}\mathbf{mod}\to R{-}\mathbf{contra}$ to the forgetful functor $\rho\colon \mathbf K=R{-}\mathbf{contra}\to R{-}\mathbf{mod}=\mathbf A$, but the functor $\rho$ is not always fully faithful.

When $\rho$ is fully faithful, the colimit of any diagram in $R{-}\mathbf{contra}$ can be computed by applying $\delta$ to the colimit of the same diagram in $R{-}\mathbf{mod}$.

When $\rho$ is not fully faithful, the colimits of diagrams in $R{-}\mathbf{contra}$ coming from diagrams in $R{-}\mathbf{mod}$ via $\delta$ can be computed in this way. This was done in the other answer in the specific case of a diagram of set-theoretically finitely generated subcontramodules.

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It may be worth mentioning that, for any topological ring $R$ (presumed complete, separated, with a base of open right ideals) one can construct a topological ring $S$ (with the same properties) such that the category $R{-}\mathbf{contra}$ is equivalent to $S{-}\mathbf{contra}$ and the forgetful functor $S{-}\mathbf{contra}\to S{-}\mathbf{mod}$ is fully faithful. However, the topological ring $S$ in this construction is much bigger than $R$ (it is some ring of infinite matrices with the entries in $R$; the equivalence $R{-}\mathbf{contra}\simeq S{-}\mathbf{contra}$ is a kind of infinitary Morita theory).

Also, when the topological ring $R$ is a linearly compact $k$-algebra, so $R=C^*$ for a coalgebra $C$ over a field $k$, the topological ring $S$ will not be of this form. So one cannot perform this construction for coalgebras, generally speaking, but only for topological rings. Still there are plenty of coalgebras $C$ for which the forgetful functor $C{-}\mathbf{contra}\to C^*{-}\mathbf{mod}$ is already fully faithful.