of a Tate vector space $V$ has a canonical central extension
$\operatorname{\mathfrak{gl}}(V)^{~}$ with the kernel $k$ defined in terms of
the trace functional on the space of all continuous linear operators
$V\longrightarrow V$ of finite rank [7, Sections 2.7.8 and 3.8.17–18] (see
also [52, Sections D.1.6–8]; a historical discussion can be found in [7, the
beginning of Section 2.7]). The central extension
$\gamma_{0}\colon\operatorname{\mathfrak{gl}}(V)^{~}\longrightarrow\operatorname{\mathfrak{gl}}(V)$
can be characterized by the property that there is a well-defined linear
action of the Lie algebra $\operatorname{\mathfrak{gl}}(V)^{~}$ in the space
$\bigwedge^{\infty/2}(V)$ of semi-infinite exterior forms over $V$ lifting the
projective action of the Lie algebra $\operatorname{\mathfrak{gl}}(V)$ (see
[29], [40, Lecture 4], and [8, Sections 4.2.13 and 7.13.16]). One chooses the
canonical basis element $1_{\operatorname{\mathfrak{gl}}^{~}}\in
k\subset\operatorname{\mathfrak{gl}}(V)^{~}$ so that it acts by the identity
operator in the space of semi-infinite forms; abusing the terminology, we say
that $\bigwedge^{\infty/2}(V)$ is a “$\operatorname{\mathfrak{gl}}(V)$-module
with the central charge $\gamma_{0}$”.

Pulling back the central extension
$\operatorname{\mathfrak{gl}}({\mathfrak{g}})^{~}\longrightarrow\operatorname{\mathfrak{gl}}({\mathfrak{g}})$
with respect to the adjoint representation
$\operatorname{\mathfrak{ad}}_{\mathfrak{g}}\colon{\mathfrak{g}}\longrightarrow\operatorname{\mathfrak{gl}}({\mathfrak{g}})$,
one obtains the canonical central extension
$\varkappa_{0}\colon{\mathfrak{g}}^{~}\longrightarrow{\mathfrak{g}}$ of an
arbitrary Tate Lie algebra ${\mathfrak{g}}$. The above convention for the
choice of the canonical basis element $1_{\operatorname{\mathfrak{gl}}^{~}}$
and the consequent choice of the basis element $1_{{\mathfrak{g}}^{~}}\in
k\subset{\mathfrak{g}}$ provide it that for any discrete module $M$ over the
Lie algebra ${\mathfrak{g}}^{~}$ where the element $1_{{\mathfrak{g}}^{~}}$
acts by minus the identity operator (“a discrete ${\mathfrak{g}}$-module with
the central charge $-\varkappa_{0}$”) there is a well-defined discrete action
of the original Lie algebra ${\mathfrak{g}}$ in the tensor product
$\bigwedge^{\infty/2}(V)\otimes_{k}M$, allowing to define the _semi-infinite
homology_ of ${\mathfrak{g}}$ with coefficients in $M$ [7, Sections 3.8.19–22]
as the homology of a natural differential on
$\bigwedge^{\infty/2}(V)\otimes_{k}M$ (see [7, the beginning of Section 3.8]
for a historical discussion). In addition, in [52, Sections D.5.5–6], the
semi-infinite _cohomology_ of a Tate Lie algebra ${\mathfrak{g}}$ with
coefficients in any _${\mathfrak{g}}^{~}$ -contramodule_ ${\mathfrak{P}}$
where $1_{{\mathfrak{g}}^{~}}$ acts by the identity (“a
${\mathfrak{g}}$-contramodule with the central charge $\varkappa_{0}$”) is
also defined as the cohomology of a natural differential on the space
$\operatorname{Hom}_{k}(\bigwedge^{\infty/2}(V),\medspace{\mathfrak{P}})$.

In particular, for the Lie algebra ${\mathfrak{g}}=k((z))d/dz$ of vector
fields on the formal circle, the canonical central extension
${\mathfrak{g}}^{~}$ is the Virasoro algebra $\mathbb{V}\mathrm{ir}$ (see
Section 1.7). The canonical basis element $1\in k\subset\mathbb{V}\mathrm{ir}$
is $-C/26$, i. e., the central element $C\in\mathbb{V}\mathrm{ir}$ acts by the
scalar $-26$ in the space of semi-infinite forms
$\bigwedge^{\infty/2}\big{(}k((z))d/dz\big{)}$. So the semi-infinite homology
and cohomology is defined for any discrete $\mathbb{V}\mathrm{ir}$-module (or,
as we will say, “$k((z))d/dz$-module”) with the central charge $C=26$ and any
$\mathbb{V}\mathrm{ir}$-contramodule (“$k((z))d/dz$-contramodule”) with the
central charge $-26$.

When a Tate vector space $V$ is endowed with a continuous coaction of a
commutative Hopf algebra ${\mathcal{C}}$, the topological Lie algebra
$\operatorname{\mathfrak{gl}}(V)^{~}$ acquires the induced continuous coaction
of ${\mathcal{C}}$ [52, Sections D.1.6–7]. Hence a continuous coaction of a
commutative Hopf algebra ${\mathcal{C}}$ in a Tate Lie algebra
${\mathfrak{g}}$ always lifts naturally to a continuous coaction of
${\mathcal{C}}$ in the canonical central extension ${\mathfrak{g}}^{~}$ of
${\mathfrak{g}}$. Furthermore, the canonical central extension
${\mathfrak{g}}^{~}\longrightarrow{\mathfrak{g}}$ splits naturally over any
compact open Lie subalgebra ${\mathfrak{h}}\subset{\mathfrak{g}}$ [7, Section
3.8.19], [52, Section D.1.8] (warning: over a pair of embedded compact open
Lie subalgebras
${\mathfrak{h}}^{\prime}\subset{\mathfrak{h}}^{\prime\prime}\subset{\mathfrak{g}}$,
these splittings do _not_ agree, but rather differ by a relative adjoint trace
character). For any Tate Harish-Chandra pair $({\mathfrak{g}},{\mathcal{C}})$,
this allows to extend the canonical central extension
$\varkappa_{0}\colon{\mathfrak{g}}^{~}\longrightarrow{\mathfrak{g}}$ of the
Tate Lie algebra ${\mathfrak{g}}$ to a canonical central extension of Tate
Harish-Chandra pairs
$({\mathfrak{g}}^{~},{\mathcal{C}})\longrightarrow({\mathfrak{g}},{\mathcal{C}})$,
which we will denote also by $\varkappa_{0}$.

Let
$\varkappa\colon({\mathfrak{g}}^{\prime},{\mathcal{C}}^{\prime})\longrightarrow({\mathfrak{g}},{\mathcal{C}})$
be a central extension of Tate Harish-Chandra pairs with the kernel $k=k\cdot
1_{{\mathfrak{g}}^{\prime}}$. By a _Harish-Chandra module_ ${\mathcal{M}}$
_over $({\mathfrak{g}},{\mathcal{C}})$ with the central charge $\varkappa$_ we
mean a Harish-Chandra module over the Tate Harish-Chandra pair
$({\mathfrak{g}}^{\prime},{\mathcal{C}})$ such that the central element
$1_{{\mathfrak{g}}^{\prime}}\in{\mathfrak{g}}^{\prime}$ acts by the identity
operator in ${\mathcal{M}}$. Similarly, a _Harish-Chandra contramodule_
${\mathfrak{P}}$ over a Tate Harish-Chandra pair
$({\mathfrak{g}},{\mathcal{C}})$ _with the central charge $\varkappa$_ is a
Harish-Chandra contramodule over $({\mathfrak{g}}^{\prime},{\mathcal{C}})$
such that the central element $1_{{\mathfrak{g}}^{\prime}}$ contraacts by the
identity operator in ${\mathfrak{P}}$, that is the composition

${\mathfrak{P}}\simeq
k\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{g}}^{\prime}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$

of the map induced by the choice of the element
$1_{{\mathfrak{g}}^{\prime}}\in{\mathfrak{g}}^{\prime}$ with the
${\mathfrak{g}}^{\prime}$-contraaction map is equal to the identity map
${\mathfrak{P}}\longrightarrow{\mathfrak{P}}$.

The category ${\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$ of
Harish-Chandra modules over $({\mathfrak{g}},{\mathcal{C}})$ with the central
charge $\varkappa$ is abelian; the fully faithful embedding functor
${\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})\longrightarrow{\mathsf{O}}({\mathfrak{g}}^{\prime},{\mathcal{C}})$
is exact and preserves both the infinite direct sums and products. The
category
${\mathsf{O}}^{\mathsf{ctr}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$ of
Harish-Chandra contramodules over $({\mathfrak{g}},{\mathcal{C}})$ with the
central charge $\varkappa$ is also abelian; the fully faithful embedding
functor
${\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})\longrightarrow{\mathsf{O}}({\mathfrak{g}}^{\prime},{\mathcal{C}})$
is exact and preserves the infinite direct sums and products. There are enough
injective objects in the category
${\mathsf{O}}_{\varkappa}({\mathfrak{g}}^{\prime},{\mathcal{C}})$ and enough
projective objects in the category
${\mathsf{O}}^{\mathsf{ctr}}_{\varkappa}({\mathfrak{g}}^{\prime},{\mathcal{C}})$;
we will see below in Section 3.5 how one can construct them.

Our next goal is to define semialgebras
${\boldsymbol{\mathcal{S}}}^{l}={\boldsymbol{\mathcal{S}}}^{l}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$
and
${\boldsymbol{\mathcal{S}}}^{r}={\boldsymbol{\mathcal{S}}}^{r}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$
such that the categories of Harish-Chandra modules and contramodules over a
Tate Harish-Chandra pair $({\mathfrak{g}},{\mathcal{C}})$ with the central
charge $\varkappa$ could be identified with the categories of semimodules and
semicontramodules over ${\boldsymbol{\mathcal{S}}}^{l}$ and
${\boldsymbol{\mathcal{S}}}^{r}$. Let us start with the related elementary
construction of the modified universal enveloping algebra corresponding to a
central extension of (nontopological) Lie algebras.

Given a central extension
$\varkappa\colon{\mathfrak{g}}^{\prime}\longrightarrow{\mathfrak{g}}$ with the
kernel $k=k\cdot 1_{{\mathfrak{g}}^{\prime}}$, the algebra
$U_{\varkappa}({\mathfrak{g}})$ is constructed as the quotient algebra
$U({\mathfrak{g}}^{\prime})/(1_{U({\mathfrak{g}}^{\prime})}-1_{{\mathfrak{g}}^{\prime}})$
of the enveloping algebra $U({\mathfrak{g}}^{\prime})$ by the ideal generated
by the difference between the unit element $1_{U({\mathfrak{g}}^{\prime})}$ of
the associative algebra $U({\mathfrak{g}}^{\prime})$ and the fixed basis
vector $1_{{\mathfrak{g}}^{\prime}}$ in the kernel of the central extension.
Then the category of left $U_{\varkappa}({\mathfrak{g}})$-modules is
isomorphic to the category of “${\mathfrak{g}}$-modules with the central
charge $\varkappa$”, i. e., ${\mathfrak{g}}^{\prime}$-modules where the
element $1_{{\mathfrak{g}}^{\prime}}$ acts by the identity operator, while the
category of right $U_{\varkappa}({\mathfrak{g}})$-modules is isomorphic to the
category of ${\mathfrak{g}}$-modules with the central charge $-\varkappa$.

Now the semialgebras
${\boldsymbol{\mathcal{S}}}^{l}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$ and
${\boldsymbol{\mathcal{S}}}^{r}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$ are
constructed as the tensor products

${\boldsymbol{\mathcal{S}}}^{l}=U_{\varkappa}({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}\quad\text{and}\quad{\boldsymbol{\mathcal{S}}}^{r}={\mathcal{C}}\otimes_{U({\mathfrak{h}})}U_{\varkappa}({\mathfrak{g}}).$

The datum of a left ${\boldsymbol{\mathcal{S}}}^{l}$-semimodule structure on a
given $k$-vector space ${\boldsymbol{\mathcal{M}}}$ is equivalent to that of a
Harish-Chandra module structure over $({\mathfrak{g}},{\mathcal{C}})$ with the
central charge $\varkappa$, while the datum of a left
${\boldsymbol{\mathcal{S}}}^{r}$-semicontramodule structure on a given
$k$-vector space ${\boldsymbol{\mathfrak{P}}}$ is equivalent to that of a
Harish-Chandra contramodule structure over $({\mathfrak{g}},{\mathcal{C}})$
with the central charge $\varkappa$ [52, Sections D.2.2, D.2.5 and D.2.8]. The
following theorem, when its condition is satisfied, allows to describe left
${\boldsymbol{\mathcal{S}}}^{r}$-semimodules.

######  Theorem.

Assume that the pairing
$\phi\colon{\mathcal{C}}\otimes_{k}U({\mathfrak{h}})\longrightarrow k$ is
nondegenerate in ${\mathcal{C}}$. Then there is a natural isomorphism of
semialgebras

${\boldsymbol{\mathcal{S}}}^{r}_{\varkappa+\varkappa_{0}}({\mathfrak{g}},{\mathcal{C}})\simeq{\boldsymbol{\mathcal{S}}}^{l}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$

over the coalgebra ${\mathcal{C}}$.

###### Proof.

This is one of the most difficult, and at the same time a singularly least
well-understood result of the book [52]. The precise formulation, where the
desired isomorphism is uniquely characterized by a certain list of conditions,
can be found in [52, Section D.3.1]. The lengthy proof, which is based on the
relative nonhomogeneous quadratic duality theory developed in [52, Chapter 11]
(see [52, Section 0.4] for an introduction), occupies the rest of [52, Section
D.3]. ∎

##  3\. Tensor Operations and Adjusted Objects

###  3.1. Comodules and contramodules over coalgebras over fields

Let ${\mathcal{C}}$ be a coassociative coalgebra over a field $k$. The
constructions of the _cotensor product_
${\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}}$
of a right ${\mathcal{C}}$-comodule ${\mathcal{N}}$ and a left
${\mathcal{C}}$-comodule ${\mathcal{M}}$ and the vector space of
_cohomomorphisms_
$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{M}},{\mathfrak{P}})$ from a left
${\mathcal{C}}$-comodule ${\mathcal{M}}$ to a left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ were already presented at the
end Section 2.5 and repeated at the beginning of Section 2.6. The main
function of these two tensor operations on comodules and contramodules in the
theories developed in the book [52] is to provide the tensor and module
category structures in whose terms the notion of a semialgebra and the
categories of semimodules and semicontramodules are subsequently defined.

Let us now introduce the definition of the _contratensor product_ of a right
${\mathcal{C}}$-comodule ${\mathcal{N}}$ and a left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, which plays a key role in the
comodule-contramodule correspondence constructions. The contratensor product
${\mathcal{N}}\odot_{\mathcal{C}}{\mathfrak{P}}$ is a $k$-vector space defined
as the cokernel of (the difference of) the pair of maps

$({\mathrm{id}}\otimes{\mathrm{ev}}_{\mathcal{C}})\circ(\nu_{N}\otimes{\mathrm{id}}),\
\,{\mathrm{id}}\otimes\pi_{\mathfrak{P}}\text{$\mskip
1.5mu$}\colon{\mathcal{N}}\otimes_{k}\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\rightrightarrows{\mathcal{N}}\otimes_{k}{\mathfrak{P}},$

one of which is the composition
${\mathcal{N}}\otimes_{k}\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathcal{N}}\otimes_{k}{\mathcal{C}}\otimes_{k}\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathcal{N}}\otimes_{k}{\mathfrak{P}}$
of the map induced by the ${\mathcal{C}}$-coaction in ${\mathcal{N}}$ and the
map induced by the evaluation map
${\mathrm{ev}}_{\mathcal{C}}\colon{\mathcal{C}}\otimes_{k}\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$,
while the other one is induced by the ${\mathcal{C}}$-contraaction in
${\mathfrak{P}}$. The functor of contratensor product of comodules and
contramodules over a coalgebra ${\mathcal{C}}$ is right exact. For any right
${\mathcal{C}}$-comodule ${\mathcal{N}}$ and $k$-vector space $V$ there is a
natural isomorphism of $k$-vector spaces

${\mathcal{N}}\odot_{\mathcal{C}}\operatorname{Hom}_{k}({\mathcal{C}},V)\simeq{\mathcal{N}}\otimes_{k}V.$

Furthermore, for any right ${\mathcal{C}}$-comodule ${\mathcal{N}}$, left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, and $k$-vector space $V$ there
is a natural isomorphism of $k$-vector spaces

$\operatorname{Hom}_{k}({\mathcal{N}}\odot_{\mathcal{C}}{\mathfrak{P}},\medspace
V)\simeq\operatorname{Hom}^{\mathcal{C}}({\mathfrak{P}},\operatorname{Hom}_{k}({\mathcal{N}},V)),$

where the vector space $\operatorname{Hom}_{k}({\mathcal{N}},V)$ is endowed
with a left ${\mathcal{C}}$-contramodule structure as explained in Section 1.2
[52, Sections 0.2.6 and 5.1.1].

For any two coalgebras ${\mathcal{C}}$ and ${\mathcal{D}}$ over $k$, any
${\mathcal{C}}$-${\mathcal{D}}$-bicomodule ${\mathcal{K}}$, and any left
${\mathcal{C}}$-comodule ${\mathcal{M}}$, the vector space of
${\mathcal{C}}$-comodule homomorphisms
$\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}},{\mathcal{M}})$ has a natural
left ${\mathcal{D}}$-contramodule structure. One can define it by noticing
that $\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}},{\mathcal{M}})$ is a
subcontramodule in the left ${\mathcal{D}}$-contramodule
$\operatorname{Hom}_{k}({\mathcal{K}},{\mathcal{M}})$, whose contramodule
structure is induced by the right ${\mathcal{D}}$-comodule structure on
${\mathcal{K}}$. Similarly, for any ${\mathcal{C}}$-${\mathcal{D}}$-bicomodule
${\mathcal{K}}$ and left ${\mathcal{D}}$-contramodule ${\mathfrak{P}}$ the
vector space ${\mathcal{K}}\odot_{\mathcal{D}}{\mathfrak{P}}$ has a natural
left ${\mathcal{C}}$-comodule structure.

For any ${\mathcal{C}}$-${\mathcal{D}}$-bicomodule ${\mathcal{K}}$, left
${\mathcal{C}}$-comodule ${\mathcal{M}}$, and left
${\mathcal{D}}$-contramodule ${\mathfrak{P}}$ there is a natural isomorphism
of $k$-vector spaces

$\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}}\odot_{\mathcal{D}}{\mathfrak{P}},\medspace{\mathcal{M}})\simeq\operatorname{Hom}^{\mathcal{D}}({\mathfrak{P}},\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}},{\mathcal{M}})).$

In other words, the functor
${\mathcal{K}}\odot_{\mathcal{D}}{-}\colon{\mathcal{D}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
is left adjoint to the functor
$\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}},{-})\colon{\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow{\mathcal{D}}{\operatorname{\mathsf{--contra}}}$
[52, Section 5.1.2]. The adjoint functors
$\operatorname{Hom}_{\mathcal{C}}({\mathcal{C}},{-})$ and
${\mathcal{C}}\odot_{\mathcal{C}}{-}$ of comodule homomorphisms from and
contratensor product with the ${\mathcal{C}}$-${\mathcal{C}}$-bicomodule
${\mathcal{K}}={\mathcal{C}}$ restricted to the additive subcategories of
injective ${\mathcal{C}}$-comodules and projective
${\mathcal{C}}$-contramodules provide the equivalence of additive categories
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}_{\mathsf{inj}}\simeq{\mathcal{C}}{\operatorname{\mathsf{--contra}}}_{\mathsf{proj}}$
described at the end of Section 1.2.

Relations between (or rather, in the case of a coalgebra ${\mathcal{C}}$ over
a field $k$, the coincidences of) the following classes of adjusted objects,
together with the classes of injective and projective objects, in the comodule
and contramodule categories play an important role in the co/contramodule and
semico/semicontramodule theory. A discussion of these coincidences is the aim
of the remaining part of this section. We will see in the proof of Proposition
3.5 below how these results are being used.

A left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ is called _coflat_ if the
functor
${-}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}}\colon{\operatorname{\mathsf{comod--}}}{\mathcal{C}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ of cotensor product with ${\mathcal{M}}$ is
exact on the abelian category of right ${\mathcal{C}}$-comodules. A left
${\mathcal{C}}$-comodule ${\mathcal{M}}$ is called _coprojective_ if the
functor
$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{M}},{-})\colon{\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ is exact on the abelian category of left
${\mathcal{C}}$-contramodules.

Similarly, a left ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ is called
_contraflat_ if the functor
${-}\odot_{\mathcal{C}}\nobreak{\mathfrak{P}}\colon\allowbreak{\operatorname{\mathsf{comod--}}}{\mathcal{C}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ of contratensor product with
${\mathfrak{P}}$ is exact on the abelian category of right
${\mathcal{C}}$-comodules. A left ${\mathcal{C}}$-contramodule
${\mathfrak{P}}$ is called _coinjective_ if the functor
$\operatorname{Cohom}_{\mathcal{C}}({-},{\mathfrak{P}})\colon{\mathcal{C}}{\operatorname{\mathsf{--comod}}}^{\mathsf{op}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ is exact on the abelian category of left
${\mathcal{C}}$-comodules [52, Section 0.2.9].

######  Lemma.

Let ${\mathcal{C}}$ be a coassociative coalgebra over a field $k$. Then

(a) a ${\mathcal{C}}$-comodule is coflat if and only if it is coprojective and
if and only if it is injective;

(b) a ${\mathcal{C}}$-contramodule is contraflat if and only if it is
coinjective and if and only if it is projective.

###### Proof.

Part (a): it is clear from the natural isomorphism
$\operatorname{Hom}_{k}({\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}},\medspace
V)\simeq\operatorname{Cohom}_{\mathcal{C}}({\mathcal{M}},\operatorname{Hom}_{k}({\mathcal{N}},V))$
for any right ${\mathcal{C}}$-comodule ${\mathcal{N}}$, left
${\mathcal{C}}$-comodule ${\mathcal{M}}$, and $k$-vector space $V$ (see
Section 2.6) that any coprojective left ${\mathcal{C}}$-comodule
${\mathcal{M}}$ is coflat, and from the natural isomorphism
$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{C}}\otimes_{k}V,\medspace{\mathfrak{P}})\simeq\operatorname{Hom}_{k}(V,{\mathfrak{P}})$
for any $k$-vector space $V$ and left ${\mathcal{C}}$-contramodule
${\mathfrak{P}}$ (see Section 2.5) that any injective left
${\mathcal{C}}$-comodule ${\mathcal{M}}$ is coprojective.

Conversely, by a comodule version of Baer’s criterion, a left
${\mathcal{C}}$-comodule ${\mathcal{M}}$ is injective whenever the functor
$\operatorname{Hom}_{\mathcal{C}}({-},{\mathcal{M}})$ is exact on the category
of _finite-dimensional_ left ${\mathcal{C}}$-comodules. Indeed, a
${\mathcal{C}}$-comodule morphism into ${\mathcal{M}}$ from a subcomodule of
any ${\mathcal{C}}$-comodule can be successively extended to larger and larger
subcomodules in the way of a transfinite induction or Zorn’s lemma, and one
only has to deal with subcomodules of finite-dimensional
${\mathcal{C}}$-comodules in the process. It remains to notice the natural
right ${\mathcal{C}}$-comodule structure on the dual vector space
${\mathcal{L}}^{*}$ to any finite-dimensional left ${\mathcal{C}}$-comodule
${\mathcal{L}}$, and the natural isomorphism

$\operatorname{Hom}_{\mathcal{C}}({\mathcal{L}},{\mathcal{M}})\simeq{\mathcal{L}}^{*}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}}$

for any finite-dimesional left ${\mathcal{C}}$-comodule ${\mathcal{L}}$ and
any left ${\mathcal{C}}$-comodule ${\mathcal{M}}$, in order to conclude that
any coflat left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ is injective.

Part (b): since any ${\mathcal{C}}$-comodule is a union of its finite-
dimensional subcomodules, and the functor of contratensor product
${-}\odot_{\mathcal{C}}{\mathfrak{P}}$ preserves inductive limits (in its
comodule argument), a left ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ is
contraflat whenever the functor ${-}\odot_{\mathcal{C}}{\mathfrak{P}}$ is
exact on the category of finite-dimensional right ${\mathcal{C}}$-comodules.
It remains to notice the natural isomorphism

$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{L}},{\mathfrak{P}})\simeq{\mathcal{L}}^{*}\odot_{\mathcal{C}}{\mathfrak{P}}$

for any finite-dimensional left ${\mathcal{C}}$-comodule ${\mathcal{L}}$ and
any left ${\mathcal{C}}$-comodule ${\mathfrak{P}}$ in order to conclude that
any coinjective left ${\mathcal{C}}$-comodule is coflat. The assertion that
every projective ${\mathcal{C}}$-contramodule is coinjective follows
immediately from the natural isomorphism
$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{M}},\operatorname{Hom}_{k}({\mathcal{C}},V))\simeq\operatorname{Hom}_{k}({\mathcal{M}},V)$
for any left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ and $k$-vector space $V$
(see Section 2.5).

Showing that every contraflat ${\mathcal{C}}$-contramodule is projective is
much more difficult. This assertion was formulated as a conjecture in [51] and
eventually proven in [52, Section A.3]. The result in question is a
generalization of the classical theorem that flat modules over a finite-
dimensional associative algebra are projective [3, Theorem P and Examples
I.3(1)]. The argument is [3] is based on the structure theory of Artinian
rings. Similarly, the proof of projectivity of contraflat contramodules in
[52] is based on the structure theory of coalgebras over fields [62, Sections
9.0–1] and the structure theory of contramodules over them developed in [52,
Section A.2], and first of all, on the contramodule Nakayama lemma (see
Section 2.1).

In the exposition below, we restrict ourselves to proving that any coinjective
${\mathcal{C}}$-contramodule is projective. This is the result that has an
important application to semicontramodules that will be considered in Section
3.5. The argument that we describe here also has an advantage of being
generalizable to comodules and contramodules over corings over arbitrary
noncommutative rings [52, Lemma 5.2].

The assertion comes out as an unexpected consequence of one of the results
establishing mutual associativity of the cotensor product/cohomomorphism
operations $\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}$ or
$\operatorname{Cohom}_{\mathcal{C}}$ with the operations of contratensor
product, comodule homomorphisms, or contramodule homomorphisms
$\odot_{\mathcal{C}}$, $\operatorname{Hom}_{\mathcal{C}}$, or
$\operatorname{Hom}^{\mathcal{C}}$. The operations from the first and the
second list are exact on different sides, so they are only mutually
associative under certain adjustness conditions on the objects involved [52,
Section 5.2] (cf. [51], where these mutual associativity assertions were
formulated in a less general form insufficient for deducing the corollary
under discussion).

Let ${\mathcal{C}}$ and ${\mathcal{D}}$ be two coalgebras over a field $k$.

######  Proposition 1.

Let ${\mathcal{N}}$ be a right ${\mathcal{D}}$-comodule, ${\mathcal{K}}$ be a
${\mathcal{D}}$-${\mathcal{C}}$-bicomodule, and ${\mathfrak{P}}$ be a left
${\mathcal{C}}$-contramodule. Then there is a natural map of $k$-vector spaces

$({\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{\mathcal{K}})\odot_{\mathcal{C}}{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}({\mathcal{K}}\odot_{\mathcal{C}}{\mathfrak{P}}),$

which is an isomorphism whenever ${\mathfrak{P}}$ is a contraflat left
${\mathcal{C}}$-comodule or ${\mathcal{N}}$ is a coflat right
${\mathcal{C}}$-comodule.

######  Proposition 2.

Let ${\mathcal{L}}$ be a left ${\mathcal{D}}$-comodule, ${\mathcal{K}}$ be a
${\mathcal{C}}$-${\mathcal{D}}$-bicomodule, and ${\mathcal{M}}$ be a left
${\mathcal{C}}$-comodule. Then there is a natural map of $k$-vector spaces

$\operatorname{Cohom}_{\mathcal{D}}({\mathcal{L}},\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}},{\mathcal{M}}))\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{\mathcal{L}},\medspace{\mathcal{M}}),$

which is an isomorphism whenever ${\mathcal{M}}$ is an injective left
${\mathcal{C}}$-comodule or ${\mathcal{L}}$ is a coprojective left
${\mathcal{D}}$-comodule.

######  Proposition 3.

Let ${\mathfrak{P}}$ be a left ${\mathcal{C}}$-contramodule, ${\mathcal{K}}$
be a ${\mathcal{D}}$-${\mathcal{C}}$-bicomodule, and ${\mathfrak{Q}}$ be a
left ${\mathcal{D}}$-contramodule. Then there is a natural map of $k$-vector
spaces

$\operatorname{Cohom}_{\mathcal{D}}({\mathcal{K}}\odot_{\mathcal{C}}{\mathfrak{P}},\medspace{\mathfrak{Q}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Hom}^{\mathcal{C}}({\mathfrak{P}},\operatorname{Cohom}_{\mathcal{D}}({\mathcal{K}},{\mathfrak{Q}})),$

which is an isomorphism whenever ${\mathfrak{P}}$ is a projective left
${\mathcal{C}}$-contramodule or ${\mathfrak{Q}}$ is a coinjective left
${\mathcal{D}}$-contramodule.

###### Proof.

To construct the natural map in Proposition 1, one considers the composition

$({\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{\mathcal{K}})\otimes_{k}{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathcal{N}}\otimes_{k}{\mathcal{K}}\otimes_{k}{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathcal{N}}\otimes_{k}({\mathcal{K}}\odot_{\mathcal{C}}{\mathfrak{P}})$

and observes that it is has equal compositions with the two maps
$({\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{\mathcal{K}})\otimes_{k}\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\rightrightarrows({\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{\mathcal{K}})\otimes_{k}{\mathfrak{P}}$,
as well as with the two maps
${\mathcal{N}}\otimes_{k}({\mathcal{K}}\odot_{\mathcal{C}}{\mathfrak{P}})\rightrightarrows{\mathcal{N}}\otimes_{k}{\mathcal{D}}\otimes_{k}({\mathcal{K}}\odot_{\mathcal{C}}{\mathfrak{P}})$.
This map is an isomorphism when the ${\mathcal{C}}$-contramodule
${\mathfrak{P}}$ is contraflat, since the exact sequence

$0\text{$\mskip 1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{\mathcal{K}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathcal{N}}\otimes_{k}{\mathcal{K}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathcal{N}}\otimes_{k}{\mathcal{D}}\otimes_{k}{\mathcal{K}}$

remains exact after applying the functor
${-}\odot_{\mathcal{C}}{\mathfrak{P}}$, and when the ${\mathcal{C}}$-comodule
${\mathcal{N}}$ is coflat, since the exact sequence

${\mathcal{K}}\otimes_{k}\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathcal{K}}\otimes_{k}{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathcal{K}}\odot_{\mathcal{C}}{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip 1.5mu$}0$

remains exact after applying the functor
${\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{-}$.

To construct the natural map in Proposition 3, one considers the composition

$\operatorname{Hom}_{k}({\mathcal{K}}\odot_{\mathcal{C}}{\mathfrak{P}},\medspace{\mathfrak{Q}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}({\mathcal{K}}\otimes_{k}{\mathfrak{P}},\medspace{\mathfrak{Q}})\\\
\text{$\mskip 1.5mu$}\simeq\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}({\mathfrak{P}},\operatorname{Hom}_{k}({\mathcal{K}},{\mathfrak{Q}}))\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}({\mathfrak{P}},\operatorname{Cohom}_{\mathcal{D}}({\mathcal{K}},{\mathfrak{Q}}))$

and observes that it has equal compositions with the two maps
$\operatorname{Hom}_{k}({\mathcal{D}}\otimes_{k}({\mathcal{K}}\odot_{\mathcal{C}}{\mathfrak{P}}),\medspace\allowbreak{\mathfrak{Q}})\rightrightarrows\operatorname{Hom}_{k}({\mathcal{K}}\odot_{\mathcal{C}}{\mathfrak{P}},\medspace{\mathfrak{Q}})$,
as well as with the two maps
$\operatorname{Hom}_{k}({\mathfrak{P}},\operatorname{Cohom}_{\mathcal{D}}({\mathcal{K}},{\mathfrak{Q}}))\allowbreak\rightrightarrows\operatorname{Hom}_{k}(\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}}),\operatorname{Cohom}_{\mathcal{D}}({\mathcal{K}},{\mathfrak{Q}}))$.
This map is an isomorphism when the ${\mathcal{C}}$-contramodule
${\mathfrak{P}}$ is projective, since the exact sequence

$\operatorname{Hom}_{k}({\mathcal{D}}\otimes_{k}{\mathcal{K}},\medspace{\mathfrak{Q}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}({\mathcal{K}},{\mathfrak{Q}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Cohom}_{\mathcal{D}}({\mathcal{K}},{\mathfrak{Q}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip 1.5mu$}0$

remains exact after applying the functor
$\operatorname{Hom}^{\mathcal{C}}({\mathfrak{P}},{-})$, and when the
${\mathcal{D}}$-contramodule ${\mathfrak{Q}}$ is coinjective, since the exact
sequence

${\mathcal{K}}\otimes_{k}\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathcal{K}}\otimes_{k}{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathcal{K}}\odot_{\mathcal{C}}{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip 1.5mu$}0$

remains exact after applying the functor
$\operatorname{Cohom}_{\mathcal{D}}({-},{\mathfrak{Q}})$. The proof of
Proposition 2 is similar. ∎

Now we can prove that any coinjective left ${\mathcal{C}}$-contramodule
${\mathfrak{P}}$ is projective. Let
$l\colon{\mathfrak{E}}\longrightarrow{\mathfrak{P}}$ be a surjective
${\mathcal{C}}$-contramodule morphism. According to the second assertion of
Proposition 3 applied to the coalgebras ${\mathcal{C}}={\mathcal{D}}$, the
bicomodule ${\mathcal{K}}={\mathcal{C}}$, and the contramodules
${\mathfrak{Q}}={\mathfrak{P}}$, there is a commutative diagram of maps of
$k$-vector spaces with a lower horizontal isomorphism

$\dgARROWLENGTH=4em\begin{diagram}$

The leftmost vertical map, being a quotient of the surjective map

$\operatorname{Hom}_{k}({\mathcal{C}}\odot_{\mathcal{C}}{\mathfrak{P}},\medspace
l)\colon\operatorname{Hom}_{k}({\mathcal{C}}\odot_{\mathcal{C}}{\mathfrak{P}},\medspace{\mathfrak{E}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}({\mathcal{C}}\odot_{\mathcal{C}}{\mathfrak{P}},\medspace{\mathfrak{P}}),$

is surjective; so the rightmost vertical map is surjective, too. It follows
that the morphism
${\mathrm{id}}\colon{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$ can be lifted
to a ${\mathcal{C}}$-contramodule morphism
${\mathfrak{P}}\longrightarrow{\mathfrak{E}}$, i. e., our surjective morphism
of ${\mathcal{C}}$-contramodules
$l\colon{\mathfrak{E}}\longrightarrow{\mathfrak{P}}$ splits. ∎

###  3.2. Contramodules over pro-Artinian local rings

A _pro-Artinian commutative ring_ ${\mathfrak{R}}$ is the projective limit of
a filtered projective system of Artinian commutative rings and surjective
morphisms between them. Equivalently, ${\mathfrak{R}}$ is a complete and
separated topological commutative ring where open ideals form a base of
neighborhoods of zero and all the discrete quotient rings are Artinian [56,
Section A.2].

A _pro-Artinian local ring_ is the projective limit of a filtered projective
system of Artinian commutative local rings and surjective morphisms between
them. E. g., any complete Noetherian local ring can be viewed as a pro-
Artinian local ring.

Contramodules over pro-Artinian local rings are suggested in [56] for use in
the role of coefficients in homological theories involving tensor operations,
infinite direct sums and products, and reductions to the residue field. The
point is that the use of contramodules with their well-behaved reductions
(satisfying Nakayama’s lemma irrespectively of any finite generatedness
assumptions) allows to extend to pro-Artinian local rings many results
originally provable over a field only.

Let ${\mathfrak{R}}$ be a pro-Artinian local ring with the maximal ideal
${\mathfrak{m}}$ and the residue field $k$. The category of
${\mathfrak{R}}$-contramodules
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ and the reduction functor
${\mathfrak{P}}\longmapsto{\mathfrak{P}}/{\mathfrak{m}}{\mathfrak{P}}$ acting
from it to the category of $k$-vector spaces have the following formal
properties:

  1. (i)

${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ is an abelian category with
infinite direct sums and products; the infinite product functors in
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ are exact and preserved by
the forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$;

  2. (ii)

${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ is a tensor category with a
right exact tensor product functor $\otimes^{\mathfrak{R}}$ and an internal
Hom functor $\operatorname{Hom}_{\mathfrak{R}}$; the forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$
takes the unit object of the tensor structure
${\mathfrak{R}}\in{\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ to the
unit object ${\mathfrak{R}}\in{\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$
and commutes with the internal Hom functors;

  3. (iii)

there are enough projective objects in
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$; these are called the _free
${\mathfrak{R}}$-contramodules_, as they are precisely the direct sums of
copies of the object ${\mathfrak{R}}$; the class of free
${\mathfrak{R}}$-contramodules is preserved by infinite direct sums, infinite
products, and the operations $\otimes^{\mathfrak{R}}$ and
$\operatorname{Hom}_{\mathfrak{R}}$;

  4. (iv)

the reduction functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ preserves infinite direct sums, infinite
products, commutes with the tensor products, and commutes with the internal
Hom from free ${\mathfrak{R}}$-contramodules;

  5. (v)

the reduction functor does not annihilate any objects:
${\mathfrak{P}}/{\mathfrak{m}}{\mathfrak{P}}=0$ implies
${\mathfrak{P}}=\nobreak 0$ for any
${\mathfrak{P}}\in{\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$; in other
words, a morphism of free ${\mathfrak{R}}$-contramodules
${\mathfrak{F}}\longrightarrow{\mathfrak{G}}$ is an isomorphism whenever its
reduction
${\mathfrak{F}}/{\mathfrak{m}}{\mathfrak{F}}\longrightarrow{\mathfrak{G}}/{\mathfrak{m}}{\mathfrak{G}}$
is an isomorphism of $k$-vector spaces.

The constructions of these structures and the proofs of the listed assertions
can be found in [56, Sections 1.1–1.6]. Let us point out two caveats, which
are in fact two ways to formulate one and the same observation. Firstly, the
infinite direct sums of ${\mathfrak{R}}$-contramodules are _not_ always exact
functors. Secondly, the functors of tensor product
${\mathfrak{F}}\otimes^{\mathfrak{R}}{-}$ with a free
${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is _not_ always exact in
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$.

Both problems do not occur in the homological dimension $1$ case. E. g., in
the case of the ring of $l$-adic integers ${\mathfrak{R}}={\mathbb{Z}}_{l}$ or
the ring of formal power series in one variable ${\mathfrak{R}}=k[[z]]$ the
infinite direct sums in ${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ are
still exact, as are the tensor products with free
${\mathfrak{R}}$-contramodules. Of course, even in these cases the infinite
direct sums and tensor products of ${\mathfrak{R}}$-contramodules are not
preserved by the forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$
(and one would not expect them to be).

The definition of a left contramodule over a complete and separated
topological ring ${\mathfrak{R}}$ with open right ideals forming a base of
neighborhoods of zero was explained in Section 2.1. Let us now define the
operation of tensor product $\otimes^{\mathfrak{R}}$ of contramodules over a
commutative topological ring ${\mathfrak{R}}$ with open ideals forming a base
of neighborhoods of zero.

The following definition of a contrabilinear map for conramodules over a
commutative topological ring was suggested to the author by Deligne. Let
${\mathfrak{P}}$, ${\mathfrak{Q}}$ and ${\mathfrak{K}}$ be three contramodules
over ${\mathfrak{R}}$. A map
$b\colon{\mathfrak{P}}\times{\mathfrak{Q}}\longrightarrow{\mathfrak{K}}$ is
called _contrabilinear_ over ${\mathfrak{R}}$ if for any two families of
coefficients $r_{\alpha}$ and $s_{\beta}\in{\mathfrak{R}}$ converging to zero
in the topology of ${\mathfrak{R}}$ and any two families of elements
$p_{\alpha}\in{\mathfrak{P}}$ and $q_{\beta}\in{\mathfrak{Q}}$ the equation

$b\Big{(}\sum_{\alpha}r_{\alpha}p_{\alpha},\medspace\sum_{\beta}s_{\beta}q_{\beta}\Big{)}=\sum_{\alpha,\beta}(r_{\alpha}s_{\beta})\,b(p_{\alpha},q_{\beta})$

holds in ${\mathfrak{K}}$. An ${\mathfrak{R}}$-contramodule ${\mathfrak{L}}$
endowed with an ${\mathfrak{R}}$-contrabilinear map
${\mathfrak{P}}\times{\mathfrak{Q}}\longrightarrow{\mathfrak{L}}$ is called
the _contramodule tensor product_ of the ${\mathfrak{R}}$-contramodules
${\mathfrak{P}}$ and ${\mathfrak{Q}}$ if for any ${\mathfrak{R}}$-contramodule
${\mathfrak{K}}$ and any ${\mathfrak{R}}$-contrabilinear map
${\mathfrak{P}}\times{\mathfrak{Q}}\longrightarrow{\mathfrak{K}}$ there exists
a unique ${\mathfrak{R}}$-contramodule morphism
${\mathfrak{L}}\longrightarrow{\mathfrak{K}}$ making the triangle diagram
${\mathfrak{P}}\times{\mathfrak{Q}}\longrightarrow{\mathfrak{L}}\longrightarrow{\mathfrak{K}}$
commutative [56, Section 1.6].

One has to check that for any ${\mathfrak{R}}$-subcontramodules
${\mathfrak{P}}^{\prime}\subset{\mathfrak{P}}$ and
${\mathfrak{Q}}^{\prime}\subset{\mathfrak{Q}}$ any
${\mathfrak{R}}$-contrabilinear map
${\mathfrak{P}}\times{\mathfrak{Q}}\longrightarrow{\mathfrak{K}}$ annihilating
${\mathfrak{P}}^{\prime}\times{\mathfrak{Q}}$ and
${\mathfrak{P}}\times{\mathfrak{Q}}^{\prime}$ factorizes uniquely through a
contrabilinear map
${\mathfrak{P}}/{\mathfrak{P}}^{\prime}\times{\mathfrak{Q}}/{\mathfrak{Q}}^{\prime}\longrightarrow{\mathfrak{K}}$.
Then it follows that the contramodule tensor product
${\mathfrak{P}}/{\mathfrak{P}}^{\prime}\otimes^{\mathfrak{R}}{\mathfrak{Q}}/{\mathfrak{Q}}^{\prime}$
can be obtained as the cokernel of the natural morphism of contramodule tensor
products

${\mathfrak{P}}^{\prime}\otimes^{\mathfrak{R}}{\mathfrak{Q}}\,\oplus\,{\mathfrak{P}}\otimes^{\mathfrak{R}}{\mathfrak{Q}}^{\prime}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{P}}\otimes^{\mathfrak{R}}{\mathfrak{Q}},$

so in order to produce the contramodule tensor products of arbitrary pairs of
${\mathfrak{R}}$-contramodules it remains to explain what the tensor products
of free ${\mathfrak{R}}$-contramodules are. Here one notices that setting
${\mathfrak{R}}[[X]]\otimes^{\mathfrak{R}}{\mathfrak{R}}[[Y]]\simeq{\mathfrak{R}}[[X\times
Y]]$ for any two sets $X$ and $Y$ does the job. Moreover, one has
${\mathfrak{R}}[[X]]\otimes^{\mathfrak{R}}{\mathfrak{P}}\simeq\bigoplus_{X}{\mathfrak{P}}$
for any set $X$ and any ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, and
generally the contramodule tensor product is a right exact functor preserving
infinite direct sums in the category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$.

The set $\operatorname{Hom}_{R}({\mathfrak{P}},{\mathfrak{Q}})$ of all
${\mathfrak{R}}$-contramodule morphisms between two
${\mathfrak{R}}$-contramodules ${\mathfrak{P}}$ and ${\mathfrak{Q}}$ is
endowed with the ${\mathfrak{R}}$-contramodule structure provided by the
pointwise infinite summation operations

$\Big{(}\sum_{\alpha}r_{\alpha}f_{\alpha}\Big{)}(p)=\sum_{\alpha}r_{\alpha}f_{\alpha}(p)$

for any family of morphisms
$f_{\alpha}\colon{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}$, any element
$p\in{\mathfrak{P}}$, and any family of coefficients
$r_{\alpha}\in{\mathfrak{R}}$ converging to zero in the topology of
${\mathfrak{R}}$ [56, Section 1.5]. One easily checks from the definitions
that

$\operatorname{Hom}_{\mathfrak{R}}({\mathfrak{P}}\otimes^{\mathfrak{R}}{\mathfrak{Q}},\medspace{\mathfrak{K}})\simeq\operatorname{Hom}_{\mathfrak{R}}({\mathfrak{P}},\operatorname{Hom}_{\mathfrak{R}}({\mathfrak{Q}},{\mathfrak{K}}))$

for any ${\mathfrak{R}}$-contramodules ${\mathfrak{P}}$, ${\mathfrak{Q}}$, and
${\mathfrak{K}}$, as is required of the internal Hom functor in a tensor
category.

Given an ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, one denotes by
${\mathfrak{m}}{\mathfrak{P}}\subset{\mathfrak{P}}$ the image of the
contraaction map
${\mathfrak{R}}[[{\mathfrak{P}}]]\supset{\mathfrak{m}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$.
The passage to the quotient $k$-vector space
${\mathfrak{P}}/{\mathfrak{m}}{\mathfrak{P}}$ provides the construction of the
reduction functor
${\mathfrak{P}}\longmapsto{\mathfrak{P}}/{\mathfrak{m}}{\mathfrak{P}}$.

We have explained the constructions of all the structures mentioned in (i-v).
The proof of the assertion that the class of projective
${\mathfrak{R}}$-contramodules is closed under the operations of infinite
product and the internal Hom functor depends on the assumption that the ring
${\mathfrak{R}}$ is pro-Artinian [56, Lemma 1.3.7], as does the proof of the
fact that the reduction functor preserves infinite products [56, Lemma 1.3.6].
The proof of the Nakayama lemma (v) is based on the assumption of topological
nilpotence of the ideal ${\mathfrak{m}}$ implied by the definition of a pro-
Artinial local ring (see Section 2.1), and the assertion that all projective
${\mathfrak{R}}$-contramodules are free follows from the Nakayama lemma [56,
Lemma 1.3.2].

In addition to ${\mathfrak{R}}$-contramodules, the coefficient formalism
developed in [56, Section 1] also includes _discrete ${\mathfrak{R}}$-modules_
or _${\mathfrak{R}}$ -comodules_; see Section 3.6 for a short discussion and
[56, Sections 1.4 and 1.9] for the details.

###  3.3. Flat contramodules over topological rings

The aim of this section is to describe a certain class of adjusted
contramodules over topological associative rings with a countable base of
neighborhoods of zero formed by open two-sided ideals that plays a crucial
role in [57, Appendix D] and will probably prove to be important and useful in
other contexts as well. Before passing to the general setting, let us briefly
return to the discussion of contramodules over the adic completions of
Noetherian rings by centrally generated ideals from Section 2.2.

Let $m$ be an ideal generated by central elements in a right Noetherian ring
$R$; denote by ${\mathfrak{R}}=\varprojlim_{n}R/m^{n}$ the $m$-adic completion
of the ring $R$, viewed as a complete and separated topological ring in its
natural topology. Let ${\mathfrak{m}}=\varprojlim_{n}m/m^{n}$ denote the ideal
generated by the image of $m\subset R$ in the ring ${\mathfrak{R}}$.

The following result is [57, Proposition C.5.4]; in the commutative case, its
proof can be found in [56, Lemma B.9.2].

######  Lemma 1.

A left ${\mathfrak{R}}$ contramodule ${\mathfrak{P}}$ is a flat $R$-module if
and only if the $R/m^{n}$-module
${\mathfrak{P}}/{\mathfrak{m}}^{n}{\mathfrak{P}}$ is flat for every
$n\geqslant 1$. The natural map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{m}}^{n}{\mathfrak{P}}$
is an isomorphism if this is the case. ∎

In other words, the nonseparatedness phenomenon demonstrated by the
counterexamples from [61, 67, 52] described in Section 1.5 above does not
occur for ${\mathfrak{R}}$-contramodules satisfying either of the two
equivalent flatness conditions from the first sentence of Lemma. In the more
general setting below, what was an assertion of a lemma becomes a well-behaved
definition.

In the rest of the section we follow [57, Section D.1]. Let
$R_{0}\longleftarrow R_{1}\longleftarrow R_{2}\longleftarrow
R_{3}\longleftarrow\dotsb$ be a projective system of associative rings and
surjective morphisms between them. Consider the projective limit
${\mathfrak{R}}=\varprojlim_{n}R_{n}$ and endow it with the topology of
projective limit of discrete rings $R_{n}$. Let
${\mathfrak{I}}_{n}\subset{\mathfrak{R}}$ denote the kernels of the natural
surjective morphisms of rings ${\mathfrak{R}}\longrightarrow R_{n}$; then the
open ideals ${\mathfrak{I}}_{n}$ form a base of neighborhoods of zero in the
topological ring ${\mathfrak{R}}$. To keep our notation in line with that of
[57, Appendix D], we switch a bit away from our previous notation pattern and
denote by ${\mathfrak{J}}{\times}{\mathfrak{P}}\subset{\mathfrak{P}}$ the
image of the contraaction map
${\mathfrak{J}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$ for any closed
ideal ${\mathfrak{J}}\subset{\mathfrak{R}}$ and any
${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$.

We recall from Section 2.1 that the category of left
${\mathfrak{R}}$-contramodules
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ is an abelian category with
infinite direct sums, exact infinite products, and enough projective objects.
The next result is [57, Lemma D.1.1]; see also [52, Lemma A.2.3].

######  Lemma 2.

For any left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, the natural map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
is surjective. ∎

Here is the promised definition. A left ${\mathfrak{R}}$-contramodule
${\mathfrak{F}}$ is called _flat_ if the map
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
is surjective and the $R_{n}$-module
${\mathfrak{F}}/{\mathfrak{J}}_{n}{\times}{\mathfrak{F}}$ is flat for every
$n$. The following proposition is the main related result.

######  Proposition 1.

The class of flat left ${\mathfrak{R}}$-contramodules is closed under
extensions and the passages to the kernels of surjective morphisms. For any
short exact sequence of flat ${\mathfrak{R}}$-contramodules
$0\longrightarrow{\mathfrak{H}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$, the short sequences of $R_{n}$-modules
$0\longrightarrow{\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}\longrightarrow{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}\longrightarrow
0$ are exact.

###### Proof.

This is [57, Lemmas D.1.3 and D.1.4]. ∎

The next result provides a characterization of projective
${\mathfrak{R}}$-contramodules, generalizing the results of [67, Corollary
4.5] and [56, Corollary B.8.2] (see also [69, Theorem 1.10] and [57, Corollary
C.5.6]).

######  Proposition 2.

A left ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is a projective object
in ${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ if and only if it is
flat and the left ${\mathfrak{R}}_{n}$-modules
${\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ are projective for
all $n\geqslant 0$. In other words, a left ${\mathfrak{R}}$-contramodule
${\mathfrak{F}}$ is projective if and only if the map
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
is an isomorphism and the $R_{n}$-modules
${\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ are projective.

###### Proof.

This is [57, Corollary D.1.8(a)]. ∎

###  3.4. Underived co-contra correspondence over corings

It was already mentioned in the end of Section 1.2 that the categories of
_injective_ left _comodules_ and _projective_ left _contramodules_ over a
coalgebra ${\mathcal{C}}$ over a field $k$ are naturally equivalent.
Similarly, at the end of Section 1.4 we pointed out the equivalence between
the categories of injective discrete modules and projective contramodules over
the ring of $l$-adic integers ${\mathbb{Z}}_{l}$. These are the simplest
instances of a very general homological phenomenon called the _comodule-
contramodule correspondence_ , which has many manifestations in algebra [46,
23, 68, 37, 42, 43, 54, 56, 59, 60], algebraic geometry [49, 48, 55, 57], and
representation theory [27, 28, 64, 52]. In the remaining three sections we
restrict ourselves to an overview of those versions of the co-contra
correspondence that can be readily formulated on the level of _additive_ or
_exact_ categories, while referring to the presentation [58] and the
introduction to the paper [59] for a discussion of the derived comodule-
contramodule correspondence.

Let ${\mathcal{C}}$ be a coassociative coring over an associative ring $A$
(see Section 2.5). Then the assignment of the left
${\mathcal{C}}$-contramodule $\operatorname{Hom}_{A}({\mathcal{C}},V)$ to the
left ${\mathcal{C}}$-comodule ${\mathcal{C}}\otimes_{A}V$ and vice versa
establishes an equivalence between the full additive subcategories of
coinduced ${\mathcal{C}}$-comodules in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ and induced
${\mathcal{C}}$-contramodules in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$. Indeed, one has

$\operatorname{Hom}_{\mathcal{C}}({\mathcal{C}}\otimes_{A}U,\medspace{\mathcal{C}}\otimes_{A}V)\simeq\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}U,\medspace
V)\\\
\simeq\operatorname{Hom}_{A}(U,\operatorname{Hom}_{A}({\mathcal{C}},V))\simeq\operatorname{Hom}^{\mathcal{C}}(\operatorname{Hom}_{A}({\mathcal{C}},U),\operatorname{Hom}_{A}({\mathcal{C}},V))$

for any left $A$-modules $U$ and $V$ [52, Section 0.2.6]. This is a particular
case of the isomorphism of _Kleisli categories_ for a pair (left adjoint
comonad, right adjoint monad) in any base category [39, 17].

Adjoining the direct summands to the full subcategories of coinduced
${\mathcal{C}}$-comodules and induced ${\mathcal{C}}$-contramodules, one
obtains what are called the full subcategories of relatively injective (or
${\mathcal{C}}/A$-injective) ${\mathcal{C}}$-comodules and relatively
projective ${\mathcal{C}}$-contramodules in the book [14, Sections 18.17–18]
and the paper [17, Sections 2.7–8]. This may be the standard terminology;
still in the monograph [52, Section 5.1.3] we chose to call these _quite
relatively injective_ (quite ${\mathcal{C}}/A$-injective)
${\mathcal{C}}$-comodules and _quite relatively projective_ (quite
${\mathcal{C}}/A$-projective) ${\mathcal{C}}$-contramodules, while preserving
the shorter terms with the single word “relatively” for the wider and more
important classes of comodules and contramodules discussed below.

A left ${\mathcal{C}}$-comodule ${\mathcal{J}}$ is said to be _quite
${\mathcal{C}}/A$-injective_ if the short sequence of abelian groups

(8)
$0\longrightarrow\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}},{\mathcal{J}})\longrightarrow\operatorname{Hom}_{\mathcal{C}}({\mathcal{L}},{\mathcal{J}})\longrightarrow\operatorname{Hom}_{\mathcal{C}}({\mathcal{M}},{\mathcal{J}})\longrightarrow
0$

is exact for every short exact sequence of left ${\mathcal{C}}$-comodules
$0\longrightarrow{\mathcal{K}}\longrightarrow{\mathcal{L}}\longrightarrow{\mathcal{M}}\longrightarrow
0$ that _splits_ as a short exact sequence of left _$A$ -modules_. We recall
that the category of left ${\mathcal{C}}$-comodules is not abelian in general,
there being a problem with the kernels of morphisms (see Section 2.5).
However, any $A$-split surjection of ${\mathcal{C}}$-comodules has a kernel
preserved by the forgetful functor
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$.

Moreover, the category of left ${\mathcal{C}}$-comodules with the class of all
$A$-split short exact sequences is an exact category (see [18] for the
definition, discussion, and references); the quite ${\mathcal{C}}/A$-injective
${\mathcal{C}}$-comodules are simply the injective objects of this exact
category structure. In particular, the coaction morphism
${\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{A}{\mathcal{M}}$ of any
left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ is split by the $A$-module map
${\mathcal{C}}\otimes_{A}{\mathcal{M}}\allowbreak\longrightarrow{\mathcal{M}}$
induced by the counit $\varepsilon$ of the coring ${\mathcal{C}}$. Considering
the corresponding $A$-split short exact sequence of ${\mathcal{C}}$-comodules,
one easily concludes that a ${\mathcal{C}}$-comodule is quite
${\mathcal{C}}/A$-injective if and only if it is a direct summand of a
coinduced ${\mathcal{C}}$-comodule.

Similarly, a left ${\mathcal{C}}$-contramodule ${\mathfrak{F}}$ is said to be
_quite ${\mathcal{C}}/A$-projective_ if the short sequence of abelian groups

(9)
$0\longrightarrow\operatorname{Hom}^{\mathcal{C}}({\mathfrak{F}},{\mathfrak{P}})\longrightarrow\operatorname{Hom}^{\mathcal{C}}({\mathfrak{F}},{\mathfrak{Q}})\longrightarrow\operatorname{Hom}^{\mathcal{C}}({\mathfrak{F}},{\mathfrak{R}})\longrightarrow
0$

is exact for every short exact sequence of left ${\mathcal{C}}$-contramodules
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ that _splits_ as a short exact sequence of left $A$-modules. Recall that
the category of left ${\mathcal{C}}$-contramodules is not abelian in general,
there being a problem with the cokernels of morphisms. However, any $A$-split
embedding of ${\mathcal{C}}$-contramodules has a cokernel preserved by the
forgetful functor
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$.

Moreover, the category of left ${\mathcal{C}}$-contramodules with the class of
all $A$-split short exact sequences is an exact category; the quite
${\mathcal{C}}/A$-projective ${\mathcal{C}}$-contramodules are simply the
projective objects of this exact category structure. In particular, the
contraaction morphism
$\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
of any ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ is split by the
$A$-module map
${\mathfrak{P}}\longrightarrow\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})$
induced by the counit of the coring ${\mathcal{C}}$. Considering the
corresponding $A$-split short exact sequence of ${\mathcal{C}}$-contramodules,
one easily concludes that a ${\mathcal{C}}$-contramodule is quite
${\mathcal{C}}/A$-injective if and only if it is a direct summand of an
induced ${\mathcal{C}}$-contramodule.

So the full additive subcategories of quite ${\mathcal{C}}/A$-injective
${\mathcal{C}}$-comodules in ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
and quite ${\mathcal{C}}/A$-projective ${\mathcal{C}}$-contramodules in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ are equivalent for any
coassociative coring ${\mathcal{C}}$. When the coring ${\mathcal{C}}$
coincides with the ring $A$ (i. e., the counit map
${\mathcal{C}}\longrightarrow A$ is bijective), this reduces to the identity
equivalence of the category of left $A$-modules with itself. However, the
category of $A$-modules is abelian and not only additive; viewing it just as
an additive category is rather unsatisfactory from our point of view. Still
the categories of quite relatively injective comodules and quite relatively
projective contramodules do not seem to carry any homological structures
beyond those of additive categories; in particular, they do _not_ have any
nontrivial exact category structures. The following definitions [52, Sections
5.1.4 and 5.3] are purported to overcome this drawback (cf. [53, Sections 4.1
and 4.3], where similarly defined relatively adjusted modules are called, more
in line with the traditional terminology, “weakly relatively projective” and
“weakly relatively injective”).

Assume that the coring ${\mathcal{C}}$ is a projective left and a flat right
$A$-module; then the categories of left ${\mathcal{C}}$-comodules and
${\mathcal{C}}$-contramodules are abelian. A left ${\mathcal{C}}$-comodule
${\mathcal{J}}$ is called _injective relative to $A$_
(${\mathcal{C}}/A$-injective) if the short sequence of Hom groups (8) is exact
for any short exact sequence of left ${\mathcal{C}}$-comodules
$0\longrightarrow{\mathcal{K}}\longrightarrow{\mathcal{L}}\longrightarrow{\mathcal{M}}\longrightarrow
0$ that are _projective as left $A$-modules_. Similarly, a left
${\mathcal{C}}$-contramodule ${\mathfrak{F}}$ is called _projective relative
to $A$_ (${\mathcal{C}}/A$-projective) if the short sequence of Hom groups (9)
is exact for any short exact sequence of left ${\mathcal{C}}$-contramodules
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ that are _injective as left $A$-modules_.

Assume further that the ring $A$ has finite left homological dimension (i. e.,
the category of left $A$-modules has finite homological dimension; cf. the
last sentence of Section 2.6). Then the full subcategory
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}_{{\mathcal{C}}/A{\operatorname{\mathsf{--inj}}}}$
of ${\mathcal{C}}/A$-injective ${\mathcal{C}}$-comodules is closed under
extensions and the passages to the cokernels of injective morphisms in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$. Similarly, the full
subcategory
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}_{{\mathcal{C}}/A{\operatorname{\mathsf{--proj}}}}$
of ${\mathcal{C}}/A$-projective ${\mathcal{C}}$-contramodules is closed under
extensions and the passages to the kernels of subjective morphisms in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ [52, Lemma 5.3.1]. Being
closed under extensions, the full subcategories
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}_{{\mathcal{C}}/A{\operatorname{\mathsf{--inj}}}}$
and
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}_{{\mathcal{C}}/A{\operatorname{\mathsf{--proj}}}}$
inherit the exact category structures of the abelian categories
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ and
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$. Moreover, the following
strong converse assertions hold.

######  Lemma.

(a) The subcategory of ${\mathcal{C}}/A$-injective ${\mathcal{C}}$-comodules
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}_{{\mathcal{C}}/A{\operatorname{\mathsf{--inj}}}}\subset{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
is the minimal full subcategory of the abelian category
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ containing the coinduced
${\mathcal{C}}$-comodules and closed under extensions and direct summands.

(b) The subcategory of ${\mathcal{C}}/A$-projective
${\mathcal{C}}$-contramodules
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}_{{\mathcal{C}}/A{\operatorname{\mathsf{--proj}}}}\subset{\mathcal{C}}{\operatorname{\mathsf{--contra}}}$
is the minimal full subcategory of the abelian category
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ containing the induced
${\mathcal{C}}$-contramodules and closed under extensions and direct summands.

###### Proof.

This is the strengthening of the result of [52, Remark 9.1] that one obtains
by replacing the resolution technique of [52, Lemma 9.1.2] with that of [25,
second half of the proof of Theorem 10]; see also [57, Corollary B.2.4].

To be more specific, let us prove part (b). Let ${\mathfrak{P}}$ be a left
${\mathcal{C}}$-contramodule. The ${\mathcal{C}}$-contraaction map
${\mathfrak{G}}=\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
is a surjective morphism of ${\mathcal{C}}$-contramodules; let us denote its
kernel by ${\mathfrak{K}}$. According to [52, Lemma 3.1.3(b)], there exists an
injective ${\mathcal{C}}$-contramodule morphism
${\mathfrak{K}}\longrightarrow{\mathfrak{E}}$ from ${\mathfrak{K}}$ into an
$A$-injective left ${\mathcal{C}}$-contramodule ${\mathfrak{E}}$ such that the
quotient contramodule ${\mathfrak{E}}/{\mathfrak{K}}$ is a finitely iterated
extension of induced ${\mathcal{C}}$-contramodules. Denote by ${\mathfrak{F}}$
the fibered coproduct $({\mathfrak{E}}\oplus{\mathfrak{G}})/{\mathfrak{K}}$ of
the ${\mathcal{C}}$-contramodules ${\mathfrak{E}}$ and ${\mathfrak{G}}$ over
${\mathfrak{K}}$; then ${\mathfrak{F}}$ is an extension of the
${\mathcal{C}}$-contramodules ${\mathfrak{E}}/{\mathfrak{K}}$ and
${\mathfrak{G}}$, and there is an surjective morphism
${\mathfrak{F}}\longrightarrow{\mathfrak{P}}$ with the kernel
${\mathfrak{E}}$.

Now suppose that the ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ is
${\mathcal{C}}/A$-projective. Then the Ext group
$\operatorname{Ext}^{{\mathcal{C}},\text{$\mskip
1.5mu$}1}({\mathfrak{P}},{\mathfrak{E}})$ in the abelian category
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ vanishes by [52, Lemma
5.3.1(b)], hence the ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ is a direct
summand of a finitely iterated extension of induced
${\mathcal{C}}$-contramodules ${\mathfrak{F}}$. Notice that the length of the
iterated extension in this construction is bounded by the left homological
dimension of the ring $A$. ∎

According to [52, Theorem 5.3], the exact categories of
${\mathcal{C}}/A$-injective left ${\mathcal{C}}$-comodules and
${\mathcal{C}}/A$-projective left ${\mathcal{C}}$-contramodules are naturally
equivalent

${\mathcal{C}}{\operatorname{\mathsf{--comod}}}_{{\mathcal{C}}/A{\operatorname{\mathsf{--inj}}}}\simeq{\mathcal{C}}{\operatorname{\mathsf{--contra}}}_{{\mathcal{C}}/A{\operatorname{\mathsf{--proj}}}}.$

The equivalence is provided by the functor
$\Psi_{\mathcal{C}}\colon{\mathcal{M}}\longmapsto\operatorname{Hom}_{\mathcal{C}}({\mathcal{C}},{\mathcal{M}})$
of ${\mathcal{C}}$-comodule homomorphisms from the left
${\mathcal{C}}$-comodule ${\mathcal{C}}$ and the functor
$\Phi_{\mathcal{C}}\colon{\mathfrak{P}}\longmapsto{\mathcal{C}}\odot_{\mathcal{C}}{\mathfrak{P}}$
of _contratensor product_ of left ${\mathcal{C}}$-contramodules with the right
${\mathcal{C}}$-comodule ${\mathcal{C}}$ [52, Sections 0.2.6–7 and 5.1.1] (cf.
Section 3.1; see also [17, Section 5]).

###  3.5. Underived semico-semicontra correspondence

Let ${\boldsymbol{\mathcal{S}}}$ be a semialgebra over a coalgebra
${\mathcal{C}}$ over a field $k$ (see Section 2.6). Assume that
${\boldsymbol{\mathcal{S}}}$ is an injective left ${\mathcal{C}}$-comodule and
an injective right ${\mathcal{C}}$-comodule, so the categories of left
${\boldsymbol{\mathcal{S}}}$-semimodules and left
${\boldsymbol{\mathcal{S}}}$-semicontramodules
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$ and
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$ are abelian.
The full subcategory
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}$
of left ${\boldsymbol{\mathcal{S}}}$-semimodules that are _injective as left
${\mathcal{C}}$-comodules_ is obviously closed under extensions (and cokernels
of injective morphisms) in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$, while the full
subcategory
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}$
of left ${\boldsymbol{\mathcal{S}}}$-semicontramodules that are _projective as
left ${\mathcal{C}}$-contramodules_ is closed under extensions (and kernels of
surjective morphisms) in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$. Hence the full
subcategories
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}$
and
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}$
inherit the exact category structures of the abelian categories
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$ and
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$.

According to [52, Sections 0.3.7 and 6.2] (see also [51]), the exact
categories of ${\mathcal{C}}$-injective left
${\boldsymbol{\mathcal{S}}}$-semimodules and ${\mathcal{C}}$-projective left
${\boldsymbol{\mathcal{S}}}$-semicontramodules are naturally equivalent. The
equivalence is provided by the functor
$\Psi_{\boldsymbol{\mathcal{S}}}\colon{\boldsymbol{\mathcal{M}}}\longmapsto\operatorname{Hom}_{\boldsymbol{\mathcal{S}}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathcal{M}}})$
of ${\boldsymbol{\mathcal{S}}}$-semimodule homomorphisms from the left
${\boldsymbol{\mathcal{S}}}$-semimodule ${\boldsymbol{\mathcal{S}}}$ and the
functor
$\Phi_{\boldsymbol{\mathcal{S}}}\colon{\boldsymbol{\mathfrak{P}}}\longmapsto{\boldsymbol{\mathcal{S}}}\circledcirc_{\boldsymbol{\mathcal{S}}}{\boldsymbol{\mathfrak{P}}}$
of (_semi_)_contratensor product_ of left
${\boldsymbol{\mathcal{S}}}$-semicontramodules with the right
${\boldsymbol{\mathcal{S}}}$-semimodule ${\boldsymbol{\mathcal{S}}}$ (cf. the
definition of the contratensor product over a coalgebra in Section 3.1).
Furthermore, the equivalence of exact categories
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}\simeq{\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}$
forms a commutative diagram of functors with the equivalence
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}_{\mathsf{inj}}\simeq{\mathcal{C}}{\operatorname{\mathsf{--contra}}}_{\mathsf{proj}}$
between the additive categories of injective left ${\mathcal{C}}$-comodules
and projective left ${\mathcal{C}}$-contramodules, and the forgetful functors
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}_{\mathsf{inj}}$
and
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--contra}}}_{\mathsf{proj}}$,

(10) $\dgARROWLENGTH=4em\begin{diagram}$

The particular case considered in Section 2.8, with the semialgebra

${\boldsymbol{\mathcal{S}}}^{r}_{\varkappa+\varkappa_{0}}({\mathfrak{g}},{\mathcal{C}})\simeq{\boldsymbol{\mathcal{S}}}={\boldsymbol{\mathcal{S}}}^{l}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$

corresponding to a central extension
$\varkappa\colon({\mathfrak{g}}^{\prime},{\mathcal{C}})\longrightarrow({\mathfrak{g}},{\mathcal{C}})$
of Tate Harish-Chandra pairs satisfying the condition of Theorem 2.8 is
especially notable. In this situation we obtain the commutative diagram of
equivalences of exact/additive categories and forgetful functors

(11) $\dgARROWLENGTH=4em\begin{diagram}$

where
${\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}\subset{\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$
and
${\mathsf{O}}^{\mathsf{ctr}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}\subset{\mathsf{O}}^{\mathsf{ctr}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$
denote the full exact subcategories of ${\mathcal{C}}$-injective Harish-
Chandra modules and ${\mathcal{C}}$-projective Harish-Chandra contramodules
with the central charge $\varkappa$.

It was pointed out by Feigin–Fuchs [27], [28, Remark 2.4] and
Meurman–Frenkel–Rocha-Caridi–Wallach [64] back in the first half of 1980’s
that the categories of Verma modules over the Virasoro algebra on any pair of
complementary levels $C=c$ and $C=26-c$ are anti-isomorphic. The above result
extends this classical observation to the whole exact subcategories of
${\mathcal{C}}$-adjusted objects in the abelian categories ${\mathsf{O}}$ and
${\mathsf{O}}^{\mathsf{ctr}}$ over any Tate Harish-Chandra pair satisfying the
nondegeneracy condition.

Indeed, consider the Tate Harish-Chandra pair
$({\mathfrak{g}},{\mathcal{C}})=\big{(}k((z))d/dz,{\mathcal{C}}(H)\big{)}$
over a field $k$ of characteristic $0$ with the pro-algebraic subgroup $H$
corresponding to the compact open Lie subalgebra
${\mathfrak{h}}=zk[[z]]d/dz\subset k((z))d/dz$ as described in the beginning
of Section 2.8. The group $H$ acts in the Lie algebra ${\mathfrak{g}}$ by
changing the independent variable in the vector fields,
$a^{-1}\big{(}f(z)d/dz\big{)}=f(a(z))\,d/da(z)=f(a(z))/a^{\prime}(z)\,d/dz$
for all $a\in H(k)$, $f\in k((z))$.

A _Verma module_ over $\mathbb{V}\mathrm{ir}$ is an
$U(\mathbb{V}\mathrm{ir})$-module induced from a one-dimensional module
$kv_{0}$ over the compact open subalgebra ${\mathfrak{h}}\oplus
kC\subset\mathbb{V}\mathrm{ir}$. The subalgebra
$z^{2}k[[z]]d/dz\subset{\mathfrak{h}}$ topologically spanned by the basis
vectors $L_{n}$ with $n\geqslant 1$ acts by zero in $kv_{0}$, while the
generators $C$ and $L_{0}$ act by certain scalars $c$ and $h_{0}\in k$. These
modules belong to the categories
${\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$ with the respective
central charges $\varkappa=c$, but, as such, do _not_ play any noticeable role
in our theory. Indeed, they do _not_ belong to the subcategories
${\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}\subset{\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$,
being freely generated as modules over a Lie subalgebra _complementary_ to
${\mathfrak{h}}$ in $\mathbb{V}\mathrm{ir}$ and having no particular
adjustness properties as comodules over ${\mathcal{C}}={\mathcal{C}}(H)$.

The _contragredient_ Verma modules are relevant for us instead. Notice first
of all that the pro-algebraic group $H$ contains a subgroup whose group of
points consists of the coordinate changes $z\longmapsto a_{1}z$ multiplying
the coordinate $z$ with a scalar factor $a_{1}\in k$. The category of
comodules (as well as contramodules) over (the coalgebra of) this algebraic
group, which is isomorphic to the multiplicative group $\mathbb{G}_{m}$, is
equivalent to the category of graded $k$-vector spaces. In particular, the
Verma modules ${\mathcal{M}}(c,h_{0})$ over $\mathbb{V}\mathrm{ir}$ carry the
grading by the weights of the semisimple operator $L_{0}$.

Furthermore, the discrete Lie subalgebra $\bigoplus_{n}kL_{n}\oplus
C\subset\mathbb{V}\mathrm{ir}$ spanned _nontopologically_ by the generators
$C$ and $L_{n}\in\mathbb{V}\mathrm{ir}$ has an involutive automorphism
$\sigma$ given by the rules $\sigma(L_{n})=-L_{-n}$ and $\sigma(C)=-C$. The
_contragredient Verma module_ ${\mathcal{M}}(c,h_{0})^{\vee}$ is the graded
dual vector space to ${\mathcal{M}}(c,h_{0})$ endowed by the induced action of
the Lie subalgebra $\bigoplus_{n}kL_{n}\oplus C$, twisted by the automorphism
$\sigma$ and then extended to the whole Lie algebra $\mathbb{V}\mathrm{ir}$ by
continuity. Both the passage to the dual module and the involution $\sigma$
change the sign of the central charge; hence ${\mathcal{M}}(c,h_{0})^{\vee}$
is again a $\mathbb{V}\mathrm{ir}$-module with the central charge $c$. So the
full subcategory of Verma modules on the level $\varkappa=c$ (with a varying
parameter $h_{0}\in k$) in
${\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$ is anti-isomorphic to
the full subcategory of contragredient Verma modules in the same category
${\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$.

Denoting by $H_{+}\subset H$ the pro-unipotent pro-algebraic subgroup whose
points are the power series $a(z)=1+a_{2}z^{2}+a_{3}z^{3}+\dotsb$ with
$a_{n}\in k$ for $n\geqslant 2$, the contragredient Verma modules can be
described as precisely those objects of the category
${\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$ whose structures of
${\mathcal{C}}(H_{+})$-comodules are those of cofree comodules with one
cogenerator. Alternatively, the contragredient Verma modules are distinguished
among all the objects of
${\mathsf{O}}_{\varkappa}({\mathfrak{g}},{\mathcal{C}})$ by the property that
their underlying ${\mathcal{C}}$-comodules are the “minimal possible”, i. e.,
_indecomposable_ injective comodules.

Finally, consider the category of Harish-Chandra _contramodules_ over
$({\mathfrak{g}},{\mathcal{C}})$ with the central charge
$\varkappa+\varkappa_{0}=-26+c$ whose ${\mathcal{C}}(H_{+})$-contramodule
structures are those of free contramodules with one generator, or
equivalently, whose underlying ${\mathcal{C}}$-contramodules are
indecomposable projective contramodules. This full subcategory in
${\mathsf{O}}_{\varkappa+\varkappa_{0}}^{\mathsf{ctr}}({\mathfrak{g}},{\mathcal{C}})$
is anti-equivalent to the category of contragredient Verma modules with the
central charge $26-c$ via the linear duality functor
${\mathfrak{P}}={\mathcal{N}}^{*}=\operatorname{Hom}_{k}({\mathcal{N}},k)$. It
is also _equivalent_ to the category of Verma modules with the central charge
$26-c$ via the functor assigning to a Verma module ${\mathcal{M}}$ the
_infinite product_ ${\mathfrak{P}}=\prod_{n\in{\mathbb{Z}}}{\mathcal{M}}_{n}$
of its grading components, endowed with the $\sigma$-twisted action of the
Virasoro Lie algebra and a natural ${\mathcal{C}}$-contramodule structure (cf.
the discussion of graded contramodules in [52, Section 11.1.1] and [54, Remark
2.2]).

So the classical duality between the categories of Verma modules on the
complementary levels appears in our setting, after the naïve twisting and
linear duality are taken into account, as the restriction of the equivalence
of exact categories (11) to the full subcategories of objects “of the minimal
possible size”. (Cf. [52, Corollary and Remark D.3.1], where a discussion of
these results in the _derived_ comodule-contramodule correspondence context
can be found; see also [52, Sections 0.2.6–7] for relevant counterexamples
demonstrating how exotic derived categories appear in the derived co-contra
correspondence.)

Now let us explain, as it was promised in Sections 2.6 and 2.8, how to use the
equivalence of exact categories (10) in order to construct injective objects
in the category ${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$
and projective objects in the category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$. As above, we
assume that the semialgebra ${\boldsymbol{\mathcal{S}}}$ is an injective left
and right ${\mathcal{C}}$-comodule.

######  Proposition.

(a) There are enough injective objects in the abelian category of left
${\boldsymbol{\mathcal{S}}}$-semimodules. A left
${\boldsymbol{\mathcal{S}}}$-semimodule is injective if and only if it is a
direct summand of an ${\boldsymbol{\mathcal{S}}}$-semimodule of the form
$\Phi_{\boldsymbol{\mathcal{S}}}(\operatorname{Hom}_{k}({\boldsymbol{\mathcal{S}}},V))$,
where $V$ is a $k$-vector space.

(b) There are enough projective objects in the abelian category of left
${\boldsymbol{\mathcal{S}}}$-semicontramodules. A left
${\boldsymbol{\mathcal{S}}}$-semicontramodule is projective if and only if it
is a direct summand of an ${\boldsymbol{\mathcal{S}}}$-semicontramodule of the
form
$\Psi_{\boldsymbol{\mathcal{S}}}({\boldsymbol{\mathcal{S}}}\otimes_{k}V)$,
where $V$ is a $k$-vector space.

###### Proof.

A left semimodule ${\boldsymbol{\mathcal{M}}}$ over a semialgebra
${\boldsymbol{\mathcal{S}}}$ over a coalgebra ${\mathcal{C}}$ over a field $k$
is called _semiprojective_ if it is a direct summand of the
${\boldsymbol{\mathcal{S}}}$-semimodule
${\boldsymbol{\mathcal{S}}}\otimes_{k}V$ for some $k$-vector space $V$.
Similarly, a left semicontramodule ${\boldsymbol{\mathfrak{P}}}$ over
${\boldsymbol{\mathcal{S}}}$ is called _semiinjective_ if it is a direct
summand of the ${\boldsymbol{\mathcal{S}}}$-semicontramodule
$\operatorname{Hom}_{k}({\boldsymbol{\mathcal{S}}},V)$ for some vector space
$V$ [52, Sections 3.4.3 and 9.2].

The semiprojective semimodules are projective objects in the exact category of
${\mathcal{C}}$-injective left ${\boldsymbol{\mathcal{S}}}$-semimodules, as
the functor

${\boldsymbol{\mathcal{M}}}\longmapsto\operatorname{Hom}_{\boldsymbol{\mathcal{S}}}({\boldsymbol{\mathcal{S}}}\otimes_{k}V,\medspace{\boldsymbol{\mathcal{M}}})\simeq\operatorname{Hom}_{k}(V,\Psi_{\boldsymbol{\mathcal{S}}}({\boldsymbol{\mathcal{M}}}))\\\
\simeq\operatorname{Hom}_{k}(V,\Psi_{\mathcal{C}}({\boldsymbol{\mathcal{M}}}))=\operatorname{Hom}_{k}(V,\operatorname{Hom}_{\mathcal{C}}({\mathcal{C}},{\boldsymbol{\mathcal{M}}}))\simeq\operatorname{Hom}_{\mathcal{C}}({\mathcal{C}}\otimes_{k}V,\medspace{\boldsymbol{\mathcal{M}}})$

is exact on
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}$.
For any ${\mathcal{C}}$-injective left ${\boldsymbol{\mathcal{S}}}$-semimodule
${\boldsymbol{\mathcal{M}}}$, the semiaction morphism
${\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{M}}}\longrightarrow{\boldsymbol{\mathcal{M}}}$
is an admissible epimorphism in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}$
from a semiprojective left ${\boldsymbol{\mathcal{S}}}$-semimodule
${\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{M}}}$
onto ${\boldsymbol{\mathcal{M}}}$; so the projective objects of the category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}$
are precisely the semiprojective semimodules.

Similarly, the semiinjective semicontramodules are injective objects in the
exact category of ${\mathcal{C}}$-projective left
${\boldsymbol{\mathcal{S}}}$-semicontramodules, as the functor

${\boldsymbol{\mathfrak{P}}}\longmapsto\operatorname{Hom}^{\boldsymbol{\mathcal{S}}}({\boldsymbol{\mathfrak{P}}},\medspace\operatorname{Hom}_{k}({\boldsymbol{\mathcal{S}}},V))\simeq\operatorname{Hom}_{k}(\Phi_{\boldsymbol{\mathcal{S}}}({\boldsymbol{\mathfrak{P}}}),V)\\\
\simeq\operatorname{Hom}_{k}(\Phi_{\mathcal{C}}({\boldsymbol{\mathfrak{P}}}),V)\simeq\operatorname{Hom}^{\mathcal{C}}({\boldsymbol{\mathfrak{P}}},\medspace\operatorname{Hom}_{k}({\mathcal{C}},V))$

is exact on
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}$.
For any ${\mathcal{C}}$-projective left
${\boldsymbol{\mathcal{S}}}$-semicontramodule ${\boldsymbol{\mathfrak{P}}}$,
the semicontraaction morphism
${\boldsymbol{\mathfrak{P}}}\longrightarrow\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}})$
is an admissible monomorphism in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}$
from ${\boldsymbol{\mathfrak{P}}}$ into a semiinjective left
${\boldsymbol{\mathcal{S}}}$-semicontramodule
$\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}})$.
So the injective objects of the category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}$
are precisely the semiinjective semicontramodules.

The functors $\Psi_{\boldsymbol{\mathcal{S}}}$ and
$\Phi_{\boldsymbol{\mathcal{S}}}$ being mutually inverse equivalences between
the exact categories
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}$
and
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}$,
it follows that the ${\boldsymbol{\mathcal{S}}}$-semimodules
$\Phi_{\boldsymbol{\mathcal{S}}}(\operatorname{Hom}_{k}({\boldsymbol{\mathcal{S}}},V))$
and their direct summands are the injective objects of the exact category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}$,
while the ${\boldsymbol{\mathcal{S}}}$-semicontramodules
$\Psi_{\boldsymbol{\mathcal{S}}}({\boldsymbol{\mathcal{S}}}\otimes_{k}V)$ and
their direct summands are the projective objects of the exact category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}$.
Furthermore, any left ${\boldsymbol{\mathcal{S}}}$-semimodule can be embedded
into a ${\mathcal{C}}$-injective ${\boldsymbol{\mathcal{S}}}$-semimodule. This
assertion is provided by the combination of the construction of [52, Lemma
1.3.3] (see also [51]) with the result of Lemma 3.1(a) above.

Similarly, any left ${\boldsymbol{\mathcal{S}}}$-semicontramodule is the
quotient contramodule of a ${\mathcal{C}}$-projective
${\boldsymbol{\mathcal{S}}}$-semicontramodule. To prove this fact, one has to
combine the construction of [52, Lemma 3.3.3] (which was present already in
[51]) with the assertion of [52, Lemma 5.2 or 5.3.2] whose proof we
reproduced, in our generality of coalgebras over fields, in Lemma 3.1(b)
above. Therefore, any left ${\boldsymbol{\mathcal{S}}}$-semimodule can be
embedded into an ${\boldsymbol{\mathcal{S}}}$-semimodule of the form
$\Phi_{\boldsymbol{\mathcal{S}}}(\operatorname{Hom}_{k}({\boldsymbol{\mathcal{S}}},V))$,
and any left ${\boldsymbol{\mathcal{S}}}$-semicontramodule is the quotient
contramodule of an ${\boldsymbol{\mathcal{S}}}$-semicontramodule of the form
$\Psi_{\boldsymbol{\mathcal{S}}}({\boldsymbol{\mathcal{S}}}\otimes_{k}V)$.

Finally, in order to show that any injective object
${\boldsymbol{\mathcal{J}}}$ of the exact category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}_{{\mathcal{C}}{\operatorname{\mathsf{--inj}}}}$
is also an injective object in the abelian category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$, suppose that we
are given an injective morphism
${\boldsymbol{\mathcal{J}}}\longrightarrow{\boldsymbol{\mathcal{L}}}$ from
${\boldsymbol{\mathcal{J}}}$ into a left
${\boldsymbol{\mathcal{S}}}$-semimodule ${\boldsymbol{\mathcal{L}}}$. Let
${\boldsymbol{\mathcal{L}}}\longrightarrow{\boldsymbol{\mathcal{M}}}$ be an
injective morphism from ${\boldsymbol{\mathcal{L}}}$ into a
${\mathcal{C}}$-injective left ${\boldsymbol{\mathcal{S}}}$-semimodule
${\boldsymbol{\mathcal{M}}}$. Then ${\boldsymbol{\mathcal{J}}}$ is a direct
summand in ${\boldsymbol{\mathcal{M}}}$, so there is a projection
${\boldsymbol{\mathcal{M}}}\longrightarrow{\boldsymbol{\mathcal{J}}}$
splitting the embedding
${\boldsymbol{\mathcal{J}}}\longrightarrow{\boldsymbol{\mathcal{M}}}$.
Restricting this projection to the subsemimodule
${\boldsymbol{\mathcal{L}}}\subset{\boldsymbol{\mathcal{M}}}$, we see that the
embedding
${\boldsymbol{\mathcal{J}}}\longrightarrow{\boldsymbol{\mathcal{L}}}$ also
splits.

Similarly, in order to prove that any projective object
${\boldsymbol{\mathfrak{F}}}$ of the exact category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}_{{\mathcal{C}}{\operatorname{\mathsf{--proj}}}}$
is also a projective object in the abelian category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$, suppose that
we are given a surjective morphism
${\boldsymbol{\mathfrak{Q}}}\longrightarrow{\boldsymbol{\mathfrak{F}}}$ onto
${\boldsymbol{\mathfrak{F}}}$ from a left
${\boldsymbol{\mathcal{S}}}$-semicontramodule ${\boldsymbol{\mathfrak{Q}}}$.
Let ${\boldsymbol{\mathfrak{P}}}\longrightarrow{\boldsymbol{\mathfrak{Q}}}$ be
a surjective morphism onto ${\boldsymbol{\mathfrak{Q}}}$ from a
${\mathcal{C}}$-projective left ${\boldsymbol{\mathcal{S}}}$-semicontramodule
${\boldsymbol{\mathfrak{P}}}$. Then ${\boldsymbol{\mathfrak{F}}}$ is a direct
summand in ${\boldsymbol{\mathfrak{P}}}$, so there is a section
${\boldsymbol{\mathfrak{F}}}\longrightarrow{\boldsymbol{\mathfrak{P}}}$
splitting the surjection
${\boldsymbol{\mathfrak{P}}}\longrightarrow{\boldsymbol{\mathfrak{F}}}$.
Composing this section with the projection
${\boldsymbol{\mathfrak{P}}}\longrightarrow{\boldsymbol{\mathfrak{Q}}}$, we
obtain a section
${\boldsymbol{\mathfrak{F}}}\longrightarrow{\boldsymbol{\mathfrak{Q}}}$
showing that the surjection
${\boldsymbol{\mathfrak{Q}}}\longrightarrow{\boldsymbol{\mathfrak{F}}}$ also
splits (cf. [52, proof of Lemma 9.2.1]). ∎

###  3.6. Co-contra correspondence over topological rings

In this last section we discuss generalizations of the equivalence between the
additive categories of injective comodules and projective contramodules over a
coalgebra over a field to topological rings ${\mathfrak{R}}$ more complicated
than the linearly compact topological algebras (which are dual to coalgebras
over fields). For examples of _derived_ co-contra correspondence over
topological rings the reader is referred to [57, Sections C.1, C.5, and D.2].

First let us suppose that ${\mathfrak{R}}$ is a pro-Artinian commutative ring
(see Section 3.2). By the definition, an _${\mathfrak{R}}$ -comodule_ is an
ind-object in the abelian category opposite to the category of discrete
${\mathfrak{R}}$-modules of finite length [56, Section 1.4]. There is a
natural contravariant functor
${\mathcal{M}}\longmapsto{\mathcal{M}}^{\mathrm{op}}$ assigning to every
${\mathfrak{R}}$-comodule a pro-object in the category of discrete
${\mathfrak{R}}$-modules of finite length. Furthermore, there is a
distinguished object ${\mathcal{C}}={\mathcal{C}}({\mathfrak{R}})$ in the
category ${\mathfrak{R}}{\operatorname{\mathsf{--comod}}}$ of
${\mathfrak{R}}$-comodules such that
${\mathcal{C}}^{\mathrm{op}}={\mathfrak{R}}$; the functor
${\mathcal{M}}\longmapsto{\mathcal{M}}^{\mathrm{op}}$, viewed as a
contravariant functor from ${\mathfrak{R}}{\operatorname{\mathsf{--comod}}}$
to the category of abelian groups, is represented by
${\mathcal{C}}({\mathfrak{R}})$. A _cofree_ ${\mathfrak{R}}$-comodule is a
direct sum of copies of the ${\mathfrak{R}}$-comodule ${\mathcal{C}}$; the
cofree ${\mathfrak{R}}$-comodules are injective, and any
${\mathfrak{R}}$-comodule can be embedded into a cofree one.

According to Matlis’ duality (see [45, Corollary 4.3] or [47, Theorem 18.6]),
choosing an injective hull of the irreducible module over an Artinian
commutative local ring $R$ fixes an anti-equivalence of the category of
$R$-modules of finite length with itself. Passing to the inductive limit of
such auto-anti-equivalences over all the discrete quotient rings of a pro-
Artinian commutative ring ${\mathfrak{R}}$, one obtains an auto-anti-
equivalence of the category of discrete ${\mathfrak{R}}$-modules of finite
length depending on the choice of a “minimal injective cogenerator” of the
abelian category ${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$, i. e., an
injective hull of the direct sum of the irreducible discrete
${\mathfrak{R}}$-modules. Hence choosing such an injective object
${\mathcal{E}}\in{\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$ identifies
the category of discrete ${\mathfrak{R}}$-modules with the category of
${\mathfrak{R}}$-comodules; the equivalence of categories
${\mathfrak{R}}{\operatorname{\mathsf{--comod}}}\simeq{\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$
takes the object
${\mathcal{C}}\in{\mathfrak{R}}{\operatorname{\mathsf{--comod}}}$ to the
object ${\mathcal{E}}$ [56, Section 1.9].

For any pro-Artinian commutative ring ${\mathfrak{R}}$, the categories of
injective ${\mathfrak{R}}$-comodules and projective
${\mathfrak{R}}$-contramodules are naturally equivalent,
${\mathfrak{R}}{\operatorname{\mathsf{--comod}}}_{\mathsf{inj}}\simeq{\mathfrak{R}}{\operatorname{\mathsf{--contra}}}_{\mathsf{proj}}$.
The equivalence is provided by the functor
$\Psi_{\mathfrak{R}}\colon{\mathcal{M}}\longmapsto\operatorname{Hom}_{\mathfrak{R}}({\mathcal{C}},{\mathcal{M}})$
of ${\mathfrak{R}}$-comodule homomorphisms from the ${\mathfrak{R}}$-comodule
${\mathcal{C}}({\mathfrak{R}})$ and the functor
$\Phi_{\mathfrak{R}}\colon{\mathfrak{P}}\longmapsto{\mathcal{C}}\odot_{\mathfrak{R}}{\mathfrak{P}}$
of _contratensor product_ of ${\mathfrak{R}}$-contramodules with the
${\mathfrak{R}}$-comodule ${\mathcal{C}}({\mathfrak{R}})$. It assigns the free
${\mathfrak{R}}$-contramodule ${\mathfrak{R}}[[X]]$ to the cofree
${\mathfrak{R}}$-comodule $\bigoplus_{X}{\mathcal{C}}({\mathfrak{R}})$ for any
set $X$. This result can be found in [52, Section 1.5].

More generally, let ${\mathfrak{R}}$ be a _right pseudo-compact_ topological
ring, i. e., a complete, separated topological ring with a base of
neighborhoods of zero formed by open right ideals ${\mathfrak{J}}$ for which
the right ${\mathfrak{R}}$-modules ${\mathfrak{R}}/{\mathfrak{J}}$ have finite
length [30, § IV.3]. We define _left ${\mathfrak{R}}$-comodules_ as the ind-
objects in the abelian category opposite to the category of discrete right
${\mathfrak{R}}$-modules of finite length. The category of left
${\mathfrak{R}}$-comodules ${\mathfrak{R}}{\operatorname{\mathsf{--comod}}}$
is anti-equivalent to the category of _pseudo-compact right
${\mathfrak{R}}$-modules_, i. e., pro-objects in the category of discrete
right ${\mathfrak{R}}$-modules of finite length or, which is the same, compact
and separated topological right ${\mathfrak{R}}$-modules with a base of
neighborhoods of zero formed by open submodules with discrete quotient modules
of finite length. As above, we denote this anti-equivalence by
${\mathcal{M}}\longmapsto{\mathcal{M}}^{\mathrm{op}}$.

There is a distinguished left ${\mathfrak{R}}$-comodule
${\mathcal{C}}={\mathcal{C}}({\mathfrak{R}})$ for which
${\mathcal{C}}^{\mathrm{op}}={\mathfrak{R}}$; the functor
${\mathcal{M}}\longmapsto{\mathcal{M}}^{\mathrm{op}}$, viewed as a
contravariant functor from ${\mathfrak{R}}{\operatorname{\mathsf{--comod}}}$
to the category of abelian groups, is represented by
${\mathcal{C}}({\mathfrak{R}})$. A _cofree_ left ${\mathfrak{R}}$-comodule is
a direct sum of copies of the ${\mathfrak{R}}$-comodule ${\mathcal{C}}$; the
cofree ${\mathfrak{R}}$-comodules are injective, and any left
${\mathfrak{R}}$-comodule can be embedded into a cofree one. The abelian
category ${\mathfrak{R}}{\operatorname{\mathsf{--comod}}}$ is locally finite
[30, § II.4]; and the choice of an injective cogenerator ${\mathcal{E}}$ in
any locally finite abelian category ${\mathsf{A}}$ fixes an equivalence
between ${\mathsf{A}}$ and the category of left comodules over the topological
ring
${\mathfrak{R}}=\operatorname{End}_{\mathsf{A}}({\mathcal{E}})^{\mathrm{op}}$
opposite to the ring of endomorphisms of the object
${\mathcal{E}}\in{\mathsf{A}}$. The topology on the ring ${\mathfrak{R}}$ is
defined to have a base of neighborhoods of zero consisting of (the right
ideals opposite to) the annihilators of submodules of finite length
${\mathcal{M}}\subset{\mathcal{E}}$. The equivalence of categories
${\mathsf{A}}\simeq{\mathfrak{R}}{\operatorname{\mathsf{--comod}}}$ assigns
the object ${\mathcal{C}}\in{\mathfrak{R}}{\operatorname{\mathsf{--comod}}}$
to the object ${\mathcal{E}}$ [30, Corollaire VI.4.1].

For any right pseudo-compact topological ring ${\mathfrak{R}}$, the categories
of injective left ${\mathfrak{R}}$-comodules and projective left
${\mathfrak{R}}$-contramodules are naturally equivalent,

${\mathfrak{R}}{\operatorname{\mathsf{--comod}}}_{\mathsf{inj}}\simeq{\mathfrak{R}}{\operatorname{\mathsf{--contra}}}_{\mathsf{proj}}.$

Indeed, the injective left ${\mathfrak{R}}$-comodules are the direct summands
of the cofree ${\mathfrak{R}}$-comodules $\bigoplus_{X}{\mathcal{C}}(R)$, and
the projective left ${\mathfrak{R}}$-contramodules are the direct summands of
the free ${\mathfrak{R}}$-contramodules ${\mathfrak{R}}[[X]]$, where $X$
denotes arbitrary sets. It remains to compute the groups of morphisms

$\textstyle\operatorname{Hom}_{\mathfrak{R}}\big{(}\bigoplus_{X}{\mathcal{C}},\medspace\bigoplus_{Y}{\mathcal{C}}\big{)}\simeq\prod_{X}\operatorname{Hom}_{\mathfrak{R}}\big{(}{\mathcal{C}},\medspace\bigoplus_{Y}{\mathcal{C}}\big{)}\simeq\prod_{X}\varprojlim_{{\mathcal{M}}\subset{\mathcal{C}}}\operatorname{Hom}_{\mathfrak{R}}\big{(}{\mathcal{M}},\medspace\bigoplus_{Y}{\mathcal{C}}\big{)}\\\
\textstyle\simeq\prod_{X}\varprojlim_{{\mathcal{M}}\subset{\mathcal{C}}}\bigoplus_{Y}\operatorname{Hom}_{\mathfrak{R}}({\mathcal{M}},{\mathcal{C}})\simeq\prod_{X}\varprojlim_{{\mathfrak{J}}\subset{\mathfrak{R}}}\bigoplus_{Y}{\mathfrak{R}}/{\mathfrak{J}}=\prod_{X}\varprojlim_{{\mathfrak{J}}\subset{\mathfrak{R}}}{\mathfrak{R}}/{\mathfrak{J}}[Y]\\\
\textstyle\simeq\prod_{X}{\mathfrak{R}}[[Y]]\simeq\prod_{X}\operatorname{Hom}^{\mathfrak{R}}\big{(}{\mathfrak{R}},{\mathfrak{R}}[[Y]]\big{)}\simeq\operatorname{Hom}^{\mathfrak{R}}\big{(}{\mathfrak{R}}[[X]],{\mathfrak{R}}[[Y]]\big{)}$

between the cofree ${\mathfrak{R}}$-comodules and the free
${\mathfrak{R}}$-contramodules in terms of projective limits over the
subcomodules of finite length ${\mathcal{M}}\subset{\mathcal{C}}$ and the open
right ideals ${\mathfrak{J}}\subset{\mathfrak{R}}$ in order obtain an
isomorphism of the categories they form. In other words, the additive
categories of projective left ${\mathfrak{R}}$-contramodules and projective
pseudo-compact right ${\mathfrak{R}}$-modules are _anti-equivalent_ (cf. [52,
the end of Remark A.3]).

One would like to generalize this equivalence from locally finite to locally
Noetherian abelian categories, i. e., abelian categories satisfying the axiom
Ab5 and admitting a set of generators consisting of Noetherian objects, or
equivalently, Ab5-categories where every object is the union of its Noetherian
subobjects and isomorphism classes of Noetherian objects form a set [30, §
II.4] (cf. [42]).

A remark at the end of [30, § IV.3] suggests considering topological rings
${\mathfrak{R}}$ with a base of neighborhoods of zero formed by (say, right)
ideals ${\mathfrak{J}}$ such that the quotient modules
${\mathfrak{R}}/{\mathfrak{J}}$ are Artinian, and topological right
${\mathfrak{R}}$-modules with a base of neighborhoods of zero formed by open
${\mathfrak{R}}$-submodules with Artinian quotient modules. Then the opposite
abelian category ${\mathsf{E}}({\mathfrak{R}})$ to such category of
${\mathfrak{R}}$-modules is locally Noetherian, and the object ${\mathcal{C}}$
opposite to the right ${\mathfrak{R}}$-module ${\mathfrak{R}}$ is injective in
it.

However, the direct summands of direct sums of copies of the object
${\mathcal{C}}$ do _not_ exhaust the class of injective objects in
${\mathsf{E}}({\mathfrak{R}})$, as one can see already in the example of the
discrete ring ${\mathfrak{R}}={\mathbb{Z}}$ with the Artinian right
${\mathfrak{R}}$-module $\mathbb{Q}_{l}/{\mathbb{Z}}_{l}$, which admits no
surjective continuous morphisms from topological products of copies of the
right ${\mathfrak{R}}$-module ${\mathfrak{R}}$. Furthermore, given a locally
Noetherian abelian category ${\mathsf{A}}$, choosing an injective object
${\mathcal{E}}\in{\mathsf{A}}$ such that all the injectives in ${\mathsf{A}}$
are direct summands of the direct sums of copies of ${\mathcal{E}}$ and leads
to a topological ring
${\mathfrak{R}}=\operatorname{End}_{\mathsf{A}}({\mathcal{E}})^{\mathrm{op}}$
which does _not_ satisfy the above Artinianness condition in general.

Indeed, it suffices to take ${\mathsf{A}}={\mathsf{Ab}}$ and
${\mathcal{E}}=\mathbb{R}/{\mathbb{Z}}$; then the Noetherian object
${\mathbb{Z}}\in{\mathsf{A}}$ is embeddable into ${\mathcal{E}}$ by means of
any irrational number in $\mathbb{R}/{\mathbb{Z}}$, hence the discrete right
${\mathfrak{R}}$-module
$\operatorname{Hom}_{\mathsf{Ab}}({\mathbb{Z}},\mathbb{R}/{\mathbb{Z}})=\mathbb{R}/{\mathbb{Z}}$
is the quotient module of
${\mathfrak{R}}=\operatorname{End}_{\mathsf{Ab}}(\mathbb{R}/{\mathbb{Z}})^{\mathrm{op}}$
by a certain open right ideal, and it is _not_ an Artinian module. _Nor_ does
the functor assigning the ${\mathfrak{R}}$-module
$\operatorname{Hom}_{\mathsf{Ab}}({\mathbb{Z}},\mathbb{R}/{\mathbb{Z}})$ to
the abelian group ${\mathbb{Z}}$ appear anywhere close to being an anti-
equivalence of abelian categories.

So we choose a slightly different path and start from an arbitrary locally
Noetherian abelian category ${\mathsf{A}}$. Recall that, being a Grothendieck
abelian category (an Ab5-category with a set of generators), any locally
Noetherian category has enough injectives [33, No 1.10].

######  Theorem.

For any locally Noetherian abelian category ${\mathsf{A}}$ there exists a
unique abelian category ${\mathsf{B}}$ with enough projectives such that the
full additive subcategories of injective objects in ${\mathsf{A}}$ and
projective objects in ${\mathsf{B}}$ are (covariantly) equivalent,

${\mathsf{A}}_{\mathsf{inj}}\simeq{\mathsf{B}}_{\mathsf{proj}}.$

All the infinite direct sums and products exist in the abelian categories
${\mathsf{A}}$ and ${\mathsf{B}}$, and both the subcategories of injective
objects in ${\mathsf{A}}$ and projective objects in ${\mathsf{B}}$ are closed
under both the infinite direct sums and the infinite products.

###### Proof.

To prove uniqueness, we notice that an abelian category with enough
projectives is determined by its full additive subcategory of projective
objects. Indeed, given an abelian category ${\mathsf{B}}^{\prime}$ with enough
projectives and an abelian category ${\mathsf{B}}^{\prime\prime}$, any
additive functor
${\mathsf{B}}^{\prime}_{\mathsf{proj}}\longrightarrow{\mathsf{B}}^{\prime\prime}$
can be uniquely extended to a right exact functor
${\mathsf{B}}^{\prime}\longrightarrow{\mathsf{B}}^{\prime\prime}$. In
particular, ${\mathsf{B}}^{\prime}$ and ${\mathsf{B}}^{\prime\prime}$ be two
abelian categories with equivalent full subcategories of projectives
${\mathsf{B}}^{\prime}_{\mathsf{proj}}\simeq{\mathsf{B}}^{\prime\prime}_{\mathsf{proj}}$.
Then the embedding functor
${\mathsf{B}}^{\prime}_{\mathsf{proj}}\longrightarrow{\mathsf{B}}^{\prime\prime}$
extends uniquely to a right exact functor
${\mathsf{B}}^{\prime}\longrightarrow{\mathsf{B}}^{\prime\prime}$, while the
embedding functor
${\mathsf{B}}^{\prime\prime}_{\mathsf{proj}}\longrightarrow{\mathsf{B}}^{\prime}$
extends uniquely to a right exact functor
${\mathsf{B}}^{\prime\prime}\longrightarrow{\mathsf{B}}^{\prime}$. The
compositions
${\mathsf{B}}^{\prime}\longrightarrow{\mathsf{B}}^{\prime\prime}\longrightarrow{\mathsf{B}}^{\prime}$
and
${\mathsf{B}}^{\prime\prime}\longrightarrow{\mathsf{B}}^{\prime}\longrightarrow{\mathsf{B}}^{\prime\prime}$
being right exact functors isomorphic to the identity functors on the full
subcategories of projective objects, they are also naturally isomorphic to
identity functors on the whole abelian categories ${\mathsf{B}}^{\prime}$ and
${\mathsf{B}}^{\prime\prime}$.

Now, given a locally Noetherian abelian category ${\mathsf{A}}$, choose an
injective object ${\mathcal{E}}\in{\mathsf{A}}$ such that all the injectives
in ${\mathsf{A}}$ are directs summands of the direct sums of copies of
${\mathcal{E}}$. E. g., one can take ${\mathcal{E}}$ to be the direct sum of
injective envelopes of all the quotient objects of Noetherian generators of
${\mathsf{A}}$. Consider the topological ring
${\mathfrak{R}}=\operatorname{End}_{\mathsf{A}}({\mathcal{E}})^{\mathrm{op}}$
with a base of neighborhoods of zero formed by the right ideals opposite to
the annihilators of Noetherian submodules ${\mathcal{M}}\subset{\mathcal{E}}$
in $\operatorname{End}_{\mathsf{A}}({\mathcal{E}})$. Set
${\mathsf{B}}={\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ to be the
abelian category of left ${\mathfrak{R}}$-contramodules.

To identify the full subcategory of direct sums of copies of the object
${\mathcal{E}}$ in ${\mathsf{A}}$ with the full subcategory of free
${\mathfrak{R}}$-contramodules in ${\mathsf{B}}$, one computes the Hom groups

$\textstyle\operatorname{Hom}_{\mathsf{A}}\big{(}\bigoplus_{X}{\mathcal{E}},\medspace\bigoplus_{Y}{\mathcal{E}}\big{)}\simeq\prod_{X}\operatorname{Hom}_{\mathsf{A}}\big{(}{\mathcal{E}},\medspace\bigoplus_{Y}{\mathcal{E}}\big{)}\simeq\prod_{X}\varprojlim_{{\mathcal{M}}\subset{\mathcal{E}}}\operatorname{Hom}_{\mathsf{A}}\big{(}{\mathcal{M}},\medspace\bigoplus_{Y}{\mathcal{E}}\big{)}\\\
\textstyle\simeq\prod_{X}\varprojlim_{{\mathcal{M}}\subset{\mathcal{E}}}\bigoplus_{Y}\operatorname{Hom}_{\mathsf{A}}({\mathcal{M}},{\mathcal{E}})\simeq\prod_{X}\varprojlim_{{\mathfrak{J}}\subset{\mathfrak{R}}}\bigoplus_{Y}{\mathfrak{R}}/{\mathfrak{J}}\\\
\textstyle\simeq\prod_{X}{\mathfrak{R}}[[Y]]\simeq\prod_{X}\operatorname{Hom}^{\mathfrak{R}}\big{(}{\mathfrak{R}},{\mathfrak{R}}[[Y]]\big{)}\simeq\operatorname{Hom}^{\mathfrak{R}}\big{(}{\mathfrak{R}}[[X]],{\mathfrak{R}}[[Y]]\big{)}$

in both subcategories in terms of projective limits over the Noetherian
subobjects ${\mathcal{M}}\subset{\mathcal{E}}$ and the open right ideals
${\mathfrak{J}}\subset{\mathfrak{R}}$. Adjoining the direct summands (the
images of idempotent endomorphisms) to both the full subcategories, one
obtains the desired equivalence between the full subcategories of injective
objects in ${\mathsf{A}}$ and projective objects in ${\mathsf{B}}$.

Any abelian category with infinite direct sums and a set of generators has
infinite products by Freyd’s Special Adjoint Functor existence Theorem [44,
Corollary V.8]; and the category of left contramodules over a topological ring
${\mathfrak{R}}$ with a base of neighborhoods of zero consisting of open right
ideals always has both the infinite direct sums and products, as we have seen
in Section 2.1. In any abelian category the infinite direct sums of projective
objects are projective and the infinite products of injective objects are
injective.

The infinite direct sums of injective objects in a locally Noetherian abelian
category ${\mathsf{A}}$ are injective [30, Corollaire II.4.1 and Proposition
IV.2.6]. Finally, to prove that the infinite products of projective objects in
our category ${\mathsf{B}}$ are projective, we notice that any family of
objects of the full subcategory
${\mathsf{B}}_{\mathsf{proj}}\subset{\mathsf{B}}$ has an infinite product in
${\mathsf{B}}_{\mathsf{proj}}$ (since any family of objects of
${\mathsf{A}}_{\mathsf{inj}}$ has an infinite product in
${\mathsf{A}}_{\mathsf{inj}}$). It is claimed that whenever an abelian
category ${\mathsf{B}}$ has enough projectives and the full subcategory of
projectives ${\mathsf{B}}_{\mathsf{proj}}\subset{\mathsf{B}}$ has infinite
products, these are also the infinite products of objects of
${\mathsf{B}}_{\mathsf{proj}}$ in the whole category ${\mathsf{B}}$, i. e.,
the embedding functor
${\mathsf{B}}_{\mathsf{proj}}\longrightarrow{\mathsf{B}}$ preserves infinite
products.

Indeed, let an object $X\in{\mathsf{B}}$ be presented as the cokernel of a
morphism of projective objects $Q\longrightarrow P$, so that there is an
initial fragment of projective resolution $Q\longrightarrow P\longrightarrow
X\longrightarrow 0$ in ${\mathsf{B}}$. Let $F_{\alpha}$ be a family of objects
in ${\mathsf{B}}_{\mathsf{proj}}$ and $F$ be their product in
${\mathsf{B}}_{\mathsf{proj}}$. Then one computes the group
$\operatorname{Hom}_{\mathsf{B}}(X,F)$ as the kernel of the map of abelian
groups
$\operatorname{Hom}_{\mathsf{B}}(P,F)\longrightarrow\operatorname{Hom}_{\mathsf{B}}(Q,F)$,
which is isomorphic to the product of the kernels of the maps
$\operatorname{Hom}_{\mathsf{B}}(P,F_{\alpha})\longrightarrow\operatorname{Hom}_{\mathsf{B}}(Q,F_{\alpha})$,
that is the product of the groups
$\operatorname{Hom}_{\mathsf{B}}(X,F_{\alpha})$ over the indices $\alpha$. ∎

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