# Contramodules

Leonid Positselski  Sector of Algebra and Number Theory, Institute for
Information Transmission Problems, Moscow 127994; and
Laboratory of Algebraic Geometry, National Research University Higher School
of Economics, Moscow 117312, Russia; and
Mathematics Department, Technion — Israel Institute of Technology, Haifa
32000, Israel<EMAIL_ADDRESS>
###### Abstract.

Contramodules are module-like algebraic structures endowed with infinite
summation (or, occasionally, integration) operations satisfying natural
axioms. Introduced originally by Eilenberg and Moore in 1965 in the case of
coalgebras over commutative rings, contramodules experience a small
renaissance now after being all but forgotten for three decades between
1970–2000. Here we present a review of various definitions and the most basic
results related to contramodules (drawing mostly from our monographs and
preprints [52, 54, 56, 57])—including contramodules over corings, topological
associative rings, topological Lie algebras and topological groups,
semicontramodules over semialgebras, and a “contra version” of the
Bernstein–Gelfand–Gelfand category ${\mathsf{O}}$. Several underived
manifestations of the comodule-contramodule correspondence phenomenon are
discussed.

###### Contents

  1. 0 Introduction
  2. 1 First Examples
    1. 1.1 Contramodules over coalgebras over fields
    2. 1.2 Basic properties of comodules and contramodules
    3. 1.3 Contramodules over the formal power series
    4. 1.4 Contramodules over the $l$-adic integers
    5. 1.5 Counterexamples
    6. 1.6 Recovering the contramodule structure
    7. 1.7 Contramodules over the Virasoro algebra
    8. 1.8 Contramodules over topological groups
  3. 2 Comodule and Contramodule Categories
    1. 2.1 Contramodules over topological rings
    2. 2.2 Contramodules over the adic completions of Noetherian rings
    3. 2.3 Contramodules over topological algebras over fields
    4. 2.4 Contramodules over topological Lie algebras
    5. 2.5 Contramodules over corings
    6. 2.6 Semicontramodules over semialgebras
    7. 2.7 The category ${\mathsf{O}}^{\mathsf{ctr}}$
    8. 2.8 Tate Harish-Chandra pairs
  4. 3 Tensor Operations and Adjusted Objects
    1. 3.1 Comodules and contramodules over coalgebras over fields
    2. 3.2 Contramodules over pro-Artinian local rings
    3. 3.3 Flat contramodules over topological rings
    4. 3.4 Underived co-contra correspondence over corings
    5. 3.5 Underived semico-semicontra correspondence
    6. 3.6 Co-contra correspondence over topological rings

##  0\. Introduction

0.0. Comodules over coalgebras or corings are familiar to many algebraists.
Being asked about the natural ways to assign an abelian category to a
coalgebra over a field, most people would probably mention the left comodules
and the right comodules. This is indeed a good answer in the case of module
categories over a _ring_ , where considering the left modules or the right
modules exhausts the basic possibilities. But the “left or right comodules”
answer is strikingly _incomplete_ , for in fact there are four such abelian
categories. In addition to the left and right _comodules_ , there are the left
and right _contramodules_ , which are no less basic, and very much analogous,
or rather _dual-analogous_ to, though different from, the comodules.

Contramodules were introduced, _on par with_ comodules, in the classical 1965
AMS Memoir of Eilenberg and Moore [24], but little attention was paid. The
2003 monograph [14], which was supposed to contain state of the art on corings
and comodules at the time, never mentioned contramodules. As it was noticed in
the presentation [16], by the end of 2000’s decade there still existed only
_three_ papers featuring contramodules that a MathSciNet search would bring:
in addition to the Eilenberg and Moore’s original memoir, there were the 1965
paper [63] by Vázquez García (in Spanish) and the rather remarkable 1970 paper
of Barr [2]. The next mention of contramodules in any kind of mathematical
literature that the present author is aware of comes in his own letters [51],
written (in transliterated Russian) in 2000 and 2002.

The 2000–2002 letters were eventually noticed by two groups of authors [31,
15] and one of them got interested specifically in contramodules, so the
number of relevant MathSciNet search hits grew a little by now (see, e. g.,
[17] and [66]). In the meantime, the present author’s ideas about
contramodules and the co-contra correspondence materialized in a sequence of
long books, papers, and preprints [52, 54, 55, 56, 57]; there is also a
presentation [58] and two later and shorter preprints [59, 60]. Still we feel
that it may be difficult for a researcher or a student to navigate this corps
of work without additional guidance. The present overview is intended to
provide such guidance (the introductions to [57, 59, 60] may be also,
hopefully, of some help).

0.1. A _coring_ may be informally defined as a “coalgebra over a
noncommutative ring” (or more precisely, a coalgebra object in the tensor
category of bimodules over a ring). Eilenberg and Moore’s original definition
of contramodules [24] was formulated in the generality of coalgebras over
commutative rings (i. e., coalgebra objects in the tensor category of
modules), but the generalization to corings is straightforward. So a comodule
over a coring can be described as “a comodule along the coalgebra variables in
the coring and a module along the ring variables”; a contramodule over a
coring is “a contramodule along the coalgebra and a module along the ring”.

Another option is to consider “algebras over coalgebras” (or more precisely,
algebra objects in the tensor categories of bicomodules); these are what we
call _semialgebras_. The corresponding module objects are called the
_semimodules_ and the _semicontramodules_. Once again, a semimodule is “a
module along the algebra variables in the semialgebra and a comodule along the
coalgebra variables”; a semicontramodule is “a module along the algebra and a
contramodule along the coalgebra”.

In the maximal natural generality achieved in the monograph [52], one
considers three-story towers of “algebras over coalgebras over rings”, or
_semialgebras over corings_. These still have four module categories attached
to them, namely, the left and right semimodules and the left and right
semicontramodules. That is the generality level in which the principal results
of the main body of the book [52] are obtained.

There are many more “comodule-like” abelian categories in algebra than just
comodules over corings or semimodules over semialgebras, though. Generally,
just about every class of “discrete”, “smooth”, or “torsion” modules can be
viewed as that of comodules “along a part of the variables” in one sense or
another. Every such module category is typically accompanied by a much less
familiar, but no less interesting, abelian category of contramodules. Hence
one comes to the definitions of contramodules over topological rings and
topological Lie algebras.

0.2. Generally, contramodules are modules with _infinite summation operations_
, understood algebraically as operations of infinite arity subjected to
natural axioms. Contramodules feel like being in some sense “complete”, but
carry no underlying topologies on them. Indeed, simple counterexamples [52,
61, 67] show that contramodule infinite summation operations _cannot_ be
interpreted as any kind of limit of finite partial sums (for all the finite
partial sums of a particular series can vanish in a contramodule while the
infinite sum does not).

Comodule categories typically have exact functors of filtered inductive limits
and enough injective objects, but nonexact functors of infinite product and no
projectives. Contramodule categories have exact functors of infinite product,
and typically enough projective objects, but nonexact functors of infinite
direct sum and no injectives. The historical obscurity/neglect of
contramodules seems to be the reason why many people believe that projective
objects are much less common than injective ones in “naturally appearing”
abelian categories.

0.3. On the other hand, there is a remarkably simple case of contramodules
over the adic completion of a Noetherian ring, where the forgetful functor
from contramodules to modules is fully faithful, so the contramodule infinite
summation operation can be _recovered_ from the conventional module structure.
In this setting, there is a different stream of literature, going back to the
1959 paper by Harrison [34], where contramodules were known and studied under
different names (and neither the connection with the Eilenberg–Moore
definition, _nor_ the existence of the infinite summation operations were
apparently ever noticed). The key modern term in this connection is the _MGM_
(Matlis–Greenlees–May) _duality_ [46, 23, 68, 59].

So (what we would call) _projective contramodules_ over the ring of $l$-adic
integers ${\mathbb{Z}}_{l}$ were studied in [34] in connection with the
classification of (what Harrison called) _co-torsion abelian groups_. A
definitive result in this direction was obtained by Enochs in [26], where
(what are since known as) _flat cotorsion modules_ over a Noetherian
commutative ring were classified in terms of (what we call) projective
contramodules over complete Noetherian local rings (see also [57, Theorem
1.3.8]). The argument in [26] was based on Matlis’ classification of injective
modules [45]. An equivalence between the categories of (what we would call)
injective discrete modules and projective contramodules over
${\mathbb{Z}}_{l}$ was also noticed in [34].

As to the arbitrary (not necessarily projective) contramodules over
${\mathbb{Z}}_{l}$, these were studied under the name of _Ext- $l$-complete_
abelian groups by Bousfield and Kan [12] and as _weakly $l$-complete_ abelian
groups by Jannsen [36]. Finally, contramodules over the adic completions of
Noetherian (and certain other) rings became known as _cohomologically
complete_ modules in the papers of Yekutieli et al. [68, 69, 70]. All these
names are derived from reflection over the basic fact that contramodules over
${\mathbb{Z}}_{l}$ and other adic completions are actually _always_ adically
complete, but _not_ necessarily adically separated (as the above-mentioned
counterexamples show).

0.4. In the author’s own research, contramodules appear, first of all, in
connection with the phenomenon of _comodule-contramodule correspondence_ [58],
which means _covariant_ equivalences between appropriate categories of
comodules and contramodules. The simplest example is the natural equivalence
between the additive categories of injective left comodules and projective
left contramodules over a coalgebra ${\mathcal{C}}$ over a field $k$.
Attempting to extend this equivalence to _complexes_ of left
${\mathcal{C}}$-comodules and left ${\mathcal{C}}$-contramodules using
complexes of injective comodules and projective contramodules as resolutions,
one discovers that unbounded acyclic complexes of contramodules are sometimes
assigned to irreducible comodules and vice versa.

The same problem occurs in the more complicated situation of the
correspondence between complexes of left semimodules and left
semicontramodules over a semialgebra ${\boldsymbol{\mathcal{S}}}$ over
${\mathcal{C}}$ [27, 28, 64, 52]. Hence the _derived_ co-contra correspondence
is, generally speaking, an equivalence between _exotic_ , rather than
conventional, derived categories. The _coderived category_ of
${\mathcal{C}}$-comodules is equivalent to the homotopy category of complexes
of injective comodules, and similarly, the _contraderived category_ of
${\mathcal{C}}$-contramodules is equivalent to the homotopy category of
projective contramodules [54]. So the coderived category of left
${\mathcal{C}}$-comodules and the contraderived category of left
${\mathcal{C}}$-contamodules are naturally equivalent to each other,
${\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}})$
[52, Sections 0.2.6–7].

This phenomenon of equivalence between “derived categories of the second kind”
is reproduced in a situation not involving comodules or contramodules in the
papers [37, 42, 43, 60], where the homotopy categories of unbounded complexes
of projective or injective modules over a ring are studied and an equivalence
between them is sometimes obtained. An extension of this theory to quasi-
coherent sheaves on nonaffine schemes was developed in the papers [49, 48,
55]; and an even more advanced version involving _contraherent cosheaves_ was
suggested in [57, Section 5.7].

0.5. In the relative situation of semimodules and semicontramodules over a
semialgebra ${\boldsymbol{\mathcal{S}}}$ over a coalgebra ${\mathcal{C}}$, the
_derived semimodule-semicontramodule correspondence_ is an equivalence between
the _semiderived category_ of left ${\boldsymbol{\mathcal{S}}}$-semimodules
and the _semi_(contra)_derived category_ of left
${\boldsymbol{\mathcal{S}}}$-semicontramodules,

${\mathsf{D}}^{\mathsf{si}}({\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}})\simeq{\mathsf{D}}^{\mathsf{si}}({\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}).$

The former is a “mixture of the coderived category along the variables from
${\mathcal{C}}$ and the conventional derived category along the variables from
${\boldsymbol{\mathcal{S}}}$ relative to ${\mathcal{C}}$” , while the latter
is a “mixture of the contraderived category in the direction of
${\mathcal{C}}$ and the derived category in the direction of
${\boldsymbol{\mathcal{S}}}$ relative to ${\mathcal{C}}$” [52, Corollary and
Remark D.3.1]. A version of the derived semico-semicontra correspondence
reproduced in a situation not involving contramodules can be found in [60,
Section 5].

On the other hand, the coderived category of left comodules and the
contraderived category of left contramodules over a _coring_ ${\mathcal{C}}$
over a ring $A$ are equivalent when the ring $A$ has finite homological
dimension (so the coderived and contraderived categories of $A$-modules are
indistinguishable from their derived category). In other words, the coderived
category of comodules and the contraderived category of contramodules are
equivalent in the relative situation provided that the homological dimension
“along the ring variables” is finite (when it is not, one needs a dualizing
complex along the ring variables to be chosen).

Similarly, the conventional derived categories of comodules and contramodules
may be equivalent in a relative situation mixing ring and coalgebra varables
when the homological dimension “along the coalgebra variables” is finite. This
includes, e. g., the case of quasi-compact semi-separated schemes, which are
glued from the affine pieces by “a gluing procedure of finite homological
dimension” (not exceeding the number of the pieces). The related version of
derived co-contra correspondence for quasi-coherent sheaves and contraherent
cosheaves was developed under the name of the “naïve co-contra correspondence”
in [57, Chapter 4].

Furthermore, an affine Noetherian formal scheme is cut out from its ambient
Noetherian scheme by a “cutting out procedure of finite homological dimension”
[68, Corollaries 4.28 and 5.27]. This can be roughly explained by noticing
that the formal completion of a scheme $X$ along its closed subscheme $Z$
consists in “subtracting from $X$ the open complement $U$ to $Z$ in $X$”, and
$U$ is a quasi-compact scheme whenever, say, $X$ is affine and $Z$ is defined
by a finitely generated ideal. So the Matlis–Greenlees–May duality is, in
fact, an _equivalence between the conventional derived categories of torsion
modules and contramodules_ over certain formal schemes [59].

0.6. We refrain from elaborating further upon the various derived categories
and the derived co-contra correspondence in this paper, restricting ourselves
mostly to the foregoing short discussion in the introduction. Indeed, it
appears that the coderived and contraderived categories have attracted already
some attention in the recent years, and a number of people have mastered the
beginnings of the related techniques in one form or another. Besides, there is
the presentation [58] discussing the philosophy of the derived co-contra
correspondence.

Instead, we concentrate on the even more basic, and at the same time perhaps
presently more counterintuitive, concepts of the abelian categories of
contramodules. The simplest examples of the categories of contramodules over
coalgebras over fields, the $l$-adic integers, the Virasoro algebra, and
locally compact totally disconnected topological groups are discussed in
Section 1. The key definitions of the categories of contramodules over
topological rings, topological associative and Lie algebras, corings and
semialgebras, and the category ${\mathsf{O}}^{\mathsf{ctr}}$ are introduced in
Section 2. Tensor and Hom-like operations on the categories of contramodules
and comodules and relations between various classes of objects adjusted to
these operations (analogues and dual versions of the classes of flat,
projective, and injective modules) are briefly considered in (the first half
of) Section 3. Several _underived_ co-contra correspondence constructions are
discussed in the final Subsections 3.4–3.6.

Acknowledgement. I learned the definition of a contramodule from the hard copy
of the Eilenberg–Moore 1965 AMS Memoir stored in the library of the Institute
for Advanced Study, where I went in the Spring of 1999 to look for relevant
literature after the discovery of (what are now known as) the coderived and
contraderived categories shortly before. I was supported by an NSF grant at
the time. Subsequently, almost all of my research on contramodules was done in
Moscow, starting from the Summers of 2000 and 2002 and then in 2006–2014. I
was partially supported by grants from EPDI, CRDF, INTAS, Pierre Deligne’s
Balzan prize, Simons Foundation, and several RFBR grants over the years. This
paper was written when I was visiting the Technion in October 2014–March 2015,
where I was supported in part by a fellowship from the Lady Davis Foundation.
I am grateful to Dmitry Kaledin who suggested the idea of writing an overview
on contramodules to me several years ago. I would like to thank Joseph
Bernstein and Amnon Yekutieli for helpful discussions.

##  1\. First Examples

###  1.1. Contramodules over coalgebras over fields

We start with recalling the largely familiar definitions. A coassociative
_coalgebra_ ${\mathcal{C}}$ with counit over a field $k$ is a $k$-vector space
endowed with a _comultiplication_ map $\mu_{\mathcal{C}}\colon
C\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{C}}$ and a _counit_ map
$\varepsilon_{\mathcal{C}}\colon{\mathcal{C}}\longrightarrow k$ satisfying the
equations dual to the equations on the multiplication and unit maps of an
associative algebra with unit. Explicitly, the two compositions of the
comultiplication map $\mu$ with the two maps
$\mu\otimes{\mathrm{id}}_{\mathcal{C}}$ and
${\mathrm{id}}_{\mathcal{C}}\otimes\mu\colon{\mathcal{C}}\otimes_{k}{\mathcal{C}}\rightrightarrows{\mathcal{C}}\otimes_{k}{\mathcal{C}}\otimes_{k}{\mathcal{C}}$
induced by the comultiplication map

${\mathcal{C}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{C}}\rightrightarrows{\mathcal{C}}\otimes_{k}{\mathcal{C}}\otimes_{k}{\mathcal{C}}$

should be equal to each other,
$(\mu\otimes{\mathrm{id}}_{\mathcal{C}})\circ\mu=({\mathrm{id}}_{\mathcal{C}}\otimes\mu)\circ\mu$,
and both the compositions of the comultiplication map with the two maps
$\varepsilon\otimes{\mathrm{id}}_{\mathcal{C}}$ and
${\mathrm{id}}_{\mathcal{C}}\otimes\varepsilon\colon{\mathcal{C}}\otimes_{k}{\mathcal{C}}\rightrightarrows{\mathcal{C}}$
induced by the counit map $\varepsilon$

${\mathcal{C}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{C}}\rightrightarrows{\mathcal{C}}$

should be equal to the identity map,
$(\varepsilon\otimes{\mathrm{id}}_{\mathcal{C}})\circ\mu={\mathrm{id}}_{\mathcal{C}}=({\mathrm{id}}_{\mathcal{C}}\otimes\varepsilon)\circ\mu$.

A _left comodule_ ${\mathcal{M}}$ over a coalgebra ${\mathcal{C}}$ is a
$k$-vector space endowed with a _left coaction_ map
$\nu_{\mathcal{M}}\colon{\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{M}}$
satisfying the coassociativity and counitality equations. Explicitly, the two
compositions of the coaction map $\nu$ with the two maps
$\mu\otimes{\mathrm{id}}_{\mathcal{M}}$ and
${\mathrm{id}}_{\mathcal{C}}\otimes\nu\colon{\mathcal{C}}\otimes_{k}{\mathcal{M}}\rightrightarrows{\mathcal{C}}\otimes_{k}{\mathcal{C}}\otimes_{k}{\mathcal{M}}$
induced by the comultiplication and coaction maps

${\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{M}}\rightrightarrows{\mathcal{C}}\otimes_{k}{\mathcal{C}}\otimes_{k}{\mathcal{M}}$

should be equal to each other,
$(\mu\otimes{\mathrm{id}}_{\mathcal{M}})\circ\nu=({\mathrm{id}}_{\mathcal{C}}\otimes\nu)\circ\nu$,
and the composition of the coaction map with the map
$\varepsilon\otimes{\mathrm{id}}_{\mathcal{M}}\colon{\mathcal{C}}\otimes_{k}{\mathcal{M}}\longrightarrow{\mathcal{M}}$
induced by the counit map $\varepsilon_{\mathcal{C}}$

${\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{M}}\longrightarrow{\mathcal{M}}$

should be equal to the identity map,
$(\varepsilon\otimes{\mathrm{id}}_{\mathcal{M}})\circ\nu={\mathrm{id}}_{\mathcal{M}}$.
A _right comodule_ ${\mathcal{N}}$ over ${\mathcal{C}}$ is a $k$-vector space
endowed with a right coaction map
$\nu=\nu_{\mathcal{N}}\colon{\mathcal{N}}\longrightarrow{\mathcal{N}}\otimes_{k}{\mathcal{C}}$
satisfying the similar equations,
$(\nu\otimes{\mathrm{id}}_{\mathcal{C}})\circ\nu=({\mathrm{id}}_{\mathcal{N}}\otimes\mu)\circ\nu$

${\mathcal{N}}\longrightarrow{\mathcal{N}}\otimes_{k}{\mathcal{C}}\rightrightarrows{\mathcal{N}}\otimes_{k}{\mathcal{C}}\otimes_{k}{\mathcal{C}},$

and
$({\mathrm{id}}_{\mathcal{N}}\otimes\varepsilon)\circ\nu={\mathrm{id}}_{\mathcal{N}}$

${\mathcal{N}}\longrightarrow{\mathcal{N}}\otimes_{k}{\mathcal{C}}\longrightarrow{\mathcal{N}}.$

In order to arrive to the definition of a contramodule over ${\mathcal{C}}$,
one only has to rewrite the most familiar definition of a module over an
associative algebra in a slightly different form before quite formally
dualizing it. Given an associative algebra $A$ over $k$ with the
multiplication map $m\colon A\otimes_{k}A\longrightarrow A$ and the unit map
$e\colon k\longrightarrow A$, one would usually define a left $A$-module $M$
as a $k$-vector space endowed with a left action map $n\colon
A\otimes_{k}M\longrightarrow M$ satisfying the associativity and unitality
equations $n\circ(m\otimes{\mathrm{id}}_{M})=n\circ({\mathrm{id}}_{A}\otimes
n)$

$A\otimes_{k}A\otimes_{k}M\rightrightarrows A\otimes_{k}M\longrightarrow M$

and $n\circ(e\otimes_{k}{\mathrm{id}}_{M})={\mathrm{id}}_{M}$

$M\longrightarrow A\otimes_{k}M\longrightarrow M.$

However, having a map $n$ is the same thing as having a map

$p\colon M\longrightarrow\operatorname{Hom}_{k}(A,M),$

which then has to satisfy the associativity and unitality equations written in
the form $\operatorname{Hom}(m,{\mathrm{id}}_{M})\circ
p=\operatorname{Hom}({\mathrm{id}}_{A},p)\circ p$

$M\longrightarrow\operatorname{Hom}_{k}(A,M)\rightrightarrows\operatorname{Hom}_{k}(A\otimes_{k}A,\medspace
M)\simeq\operatorname{Hom}_{k}(A,\operatorname{Hom}_{k}(A,M))$

and $\operatorname{Hom}(e,{\mathrm{id}}_{M})\circ p={\mathrm{id}}_{M}$

$M\longrightarrow\operatorname{Hom}_{k}(A,M)\longrightarrow M.$

In this approach, the difference between the left and right modules lies in
the way one identifies the the Hom from the tensor product
$\operatorname{Hom}_{k}(A\otimes_{k}A,\medspace M)$ with the double Hom space
$\operatorname{Hom}_{k}(A,\operatorname{Hom}_{k}(A,M))$: presuming the
identification

(1) $\operatorname{Hom}_{k}(U\otimes_{k}V,\medspace
W)\simeq\operatorname{Hom}_{k}(V,\operatorname{Hom}_{k}(U,W))$

leads to the definition of a _left_ $A$-module, while identifying
$\operatorname{Hom}_{k}(A\otimes_{k}A,\medspace N)$ with
$\operatorname{Hom}_{k}(A,\operatorname{Hom}_{k}(A,N))$ by the rule

(2) $\operatorname{Hom}_{k}(U\otimes_{k}V,\medspace
W)\simeq\operatorname{Hom}_{k}(U,\operatorname{Hom}_{k}(V,W))$

and writing the same equations produces the definition of a _right_ $A$-module
$N$.

Now we can formulate our main definition. A _left contramodule_
${\mathfrak{P}}$ over a coalgebra ${\mathcal{C}}$ is a $k$-vector space
endowed with a _left contraaction_ map

$\pi_{\mathfrak{P}}\colon\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$

satisfying the following _contraassociativity_ and _contraunitality_
equations. Firstly, the two compositions of the two maps
$\operatorname{Hom}(\mu,{\mathfrak{P}})\colon\operatorname{Hom}_{k}({\mathcal{C}}\otimes_{k}{\mathcal{C}},\medspace{\mathfrak{P}})\longrightarrow\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})$
and
$\operatorname{Hom}({\mathcal{C}},\pi)\colon\operatorname{Hom}_{k}({\mathcal{C}},\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}}))\longrightarrow\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})$
induced by the comultiplication map $\mu=\mu_{\mathcal{C}}$ and the
contraaction map $\pi=\pi_{\mathfrak{P}}$ with the contraaction map $\pi$

$\operatorname{Hom}_{k}({\mathcal{C}},\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}}))\simeq\operatorname{Hom}_{k}({\mathcal{C}}\otimes_{k}{\mathcal{C}},\medspace{\mathfrak{P}})\rightrightarrows\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$

should be equal to each other,
$\pi\circ\operatorname{Hom}(\mu,{\mathfrak{P}})=\pi\circ\operatorname{Hom}({\mathcal{C}},\pi)$,
presuming the identification of
$\operatorname{Hom}_{k}({\mathcal{C}}\otimes_{k}{\mathcal{C}},\medspace{\mathfrak{P}})\simeq\operatorname{Hom}_{k}({\mathcal{C}},\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}}))$
by the left rule (1). Secondly, the composition of the map
$\operatorname{Hom}(\varepsilon,{\mathfrak{P}})\colon{\mathfrak{P}}\longrightarrow\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})$
induced by the counit map $\varepsilon=\varepsilon_{\mathcal{C}}$ with the
contraaction map

${\mathfrak{P}}\longrightarrow\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$

should be equal to the identity map,
$\pi\circ\operatorname{Hom}(\varepsilon,{\mathfrak{P}})={\mathrm{id}}_{\mathfrak{P}}$.

This definition can be found in [52, Section 0.2.4]; see also [54, Section
2.2] (the classical source is [24, Section III.5]). Using the identification
by the right rule (2) instead of (1) produces the definition of a _right
contramodule_ over ${\mathcal{C}}$. The way to understand why (1) is the
“left” rule and (2) is the “right” one lies in replacing a basic field $k$
with a noncommutative ring; see Section 2.5 below.

###  1.2. Basic properties of comodules and contramodules

The simplest way to produce examples of contramodules is by applying the Hom
functor to comodules in the first argument. Specifically, let ${\mathcal{N}}$
be a right comodule over a coalgebra ${\mathcal{C}}$ over $k$ and $V$ be a
$k$-vector space. Then the vector space
${\mathfrak{P}}=\operatorname{Hom}_{k}({\mathcal{N}},V)$ has a natural
structure of left contramodule over ${\mathcal{C}}$. The left contraaction map
$\pi_{\mathfrak{P}}$ is constructed by applying the functor
$\operatorname{Hom}_{k}({-},V)$ to the right coaction map $\nu_{\mathcal{N}}$

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

Let us denote by $k{\operatorname{\mathsf{--vect}}}$ the category of
$k$-vector spaces, by ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ the
category of left ${\mathcal{C}}$-comodules, by
${\operatorname{\mathsf{comod--}}}{\mathcal{C}}$ the category of right
${\mathcal{C}}$-comodules, and by
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ the category of left
${\mathcal{C}}$-contramodules. The $k$-vector space of morphisms between left
${\mathcal{C}}$-comodules ${\mathcal{L}}$ and ${\mathcal{M}}$ will be denoted
by $\operatorname{Hom}_{\mathcal{C}}({\mathcal{L}},{\mathcal{M}})$, and the
vector space of morphisms between left ${\mathcal{C}}$-contramodules
${\mathfrak{P}}$ and ${\mathfrak{Q}}$ by
$\operatorname{Hom}^{\mathcal{C}}({\mathfrak{P}},{\mathfrak{Q}})$.

The category ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ is abelian and
the forgetful functor
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ is exact. To prove as much, one has to use
the observation that the tensor product functor ${\mathcal{C}}\otimes_{k}{-}$
is exact, or more specifically, _left exact_. The forgetful functor also
preserves inductive limits, so filtered inductive limits are exact functors in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$. The infinite products in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ are not preserved by the
forgetful functor (unless ${\mathcal{C}}$ is finite-dimensional) and are _not_
exact in ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ in general.

In other words, the abelian category of ${\mathcal{C}}$-comodules satisfies
Grothendieck’s axioms Ab5 and Ab3*, but not in general Ab4* [33, No 1.5]. It
also admits a set of generators (for which one can take the finite-dimensional
comodules), so it has enough injective objects [33, No 1.10]. These can be
explicitly described as follows.

A _cofree_ left ${\mathcal{C}}$-comodule is a ${\mathcal{C}}$-comodule of the
form ${\mathcal{C}}\otimes_{k}V$, where $V$ is a $k$-vector space, with the
left ${\mathcal{C}}$-coaction induced by the comultiplication in
${\mathcal{C}}$. For any left ${\mathcal{C}}$-comodule ${\mathcal{L}}$, there
is a natural isomorphism of $k$-vector spaces

$\operatorname{Hom}_{\mathcal{C}}({\mathcal{L}},\medspace{\mathcal{C}}\otimes_{k}V)\simeq\operatorname{Hom}_{k}({\mathcal{L}},V),$

so cofree ${\mathcal{C}}$-comodules are injective. The coaction map
$\nu\colon{\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{M}}$
embeds any left ${\mathcal{C}}$-comodule into a cofree one, so there are
enough cofree ${\mathcal{C}}$-comodules. It follows that a
${\mathcal{C}}$-comodule is injective if and only if it is a direct summand of
a cofree one [52, Sections 0.2.1, 1.1.2, and 5.1.5].

The category ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ is abelian and
the forgetful functor
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ is exact (here one has to observe that the
functor $\operatorname{Hom}_{k}({\mathcal{C}},{-})$ is exact, or more
specifically, _right exact_). The forgetful functor also preserves infinite
products, so infinite products are exact functors in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$. The infinite direct sums
are _not_ preserved by the forgetful functor (unless ${\mathcal{C}}$ is
finite-dimensional) and are not exact in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ in general. (However, _un_
like the infinite products of ${\mathcal{C}}$-comodules, the infinite direct
sums of ${\mathcal{C}}$-contramodules remain exact when the homological
dimension of the category ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$
does not exceed $1$ [56, Remark 1.2.1].)

In other words, the abelian category of ${\mathcal{C}}$-contramodules
satisfies Grothendieck’s axioms Ab3 and Ab4*, but not in general Ab4 or Ab5*.
It also has enough projective objects, which can be explicitly described as
follows.

A _free_ left ${\mathcal{C}}$-contramodule is a ${\mathcal{C}}$-contramodule
of the form $\operatorname{Hom}_{k}({\mathcal{C}},V)$, where $V$ is a
$k$-vector space, with the left ${\mathcal{C}}$-contraaction constructed as
explained in the beginning of this section. For any left
${\mathcal{C}}$-contramodule ${\mathfrak{Q}}$, there is a natural isomorphism
of $k$-vector spaces

$\operatorname{Hom}^{\mathcal{C}}(\operatorname{Hom}_{k}({\mathcal{C}},V),{\mathfrak{Q}})\simeq\operatorname{Hom}_{k}(V,{\mathfrak{Q}}),$

so free ${\mathcal{C}}$-contramodules are projective. The contraaction map
$\pi\colon\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
presents any ${\mathcal{C}}$-contramodule as the quotient contramodule of a
free one, so there are enough free contramodules. It follows that a
${\mathcal{C}}$-contramodule is projective if and only if it is a direct
summand of a free one [52, Sections 0.2.4, 3.1.2, and 5.1.5].

Notice that the class of injective ${\mathcal{C}}$-comodules is not only
closed under infinite products in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ (which holds in any abelian
category), but also under infinite direct sums. Similarly, the class of
projective ${\mathcal{C}}$-contramodules is not only closed under infinite
direct sums in ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ (as in any
abelian category), but also under infinite products. These observations are
important for the theory of coderived and contraderived categories [54,
Section 4.4, cf. Sections 3.7–3.8].

The correspondence assigning the free ${\mathcal{C}}$-contramodule
$\operatorname{Hom}_{k}({\mathcal{C}},V)$ to the cofree
${\mathcal{C}}$-comodule ${\mathcal{C}}\otimes_{k}V$ is an equivalence between
the additive categories of cofree left ${\mathcal{C}}$-comodules and free left
${\mathcal{C}}$-contramodules. Hence the additive categories of injective left
${\mathcal{C}}$-comodules and projective left ${\mathcal{C}}$-contramodules
are equivalent, too [52, Sections 0.2.6 and 5.1.3] (see also [17] and [54,
Sections 5.1–5.2]).

###  1.3. Contramodules over the formal power series

The linear duality functor identifies the category opposite to the category of
conventional infinite-dimensional (otherwise known as discrete, or ind-finite-
dimensional) vector spaces with the category of _linearly compact_ , or pro-
finite-dimensional, vector spaces. In particular, a coassociative coalgebra
with counit is the same thing (up to inverting the arrows) as a linearly
compact or pro-finite-dimensional topological associative algebra with unit.
Notice that any coassociative coalgebra is the union of its finite-dimensional
subcoalgebras [62, Section 2.2], so any topological associative algebra with a
pro-finite-dimensional underlying topological vector space is a projective
limit of finite-dimensional associative algebras.

In particular, one can idenfity coalgebras by the names of their dual linearly
compact topological algebras. In this section we consider the simplest example
of an infinite-dimensional coassociative coalgebra—the coalgebra
${\mathcal{C}}$ for which the dual topological algebra ${\mathcal{C}}^{*}$ is
isomorphic to the algebra $k[[z]]$ of formal Taylor power series in one
variable over a field $k$. Explicitly, ${\mathcal{C}}$ is the $k$-vector space
with a countable basis consisting of the formal symbols $1^{*}$, $z^{*}$,
$z^{2}{}^{*}$, …, $z^{n}{}^{*}$, …, $n\in{\mathbb{Z}}_{\geqslant 0}$, with the
comultiplication map given by the rule

$\mu(z^{n}{}^{*})=\sum_{i+j=n}z^{i}{}^{*}\otimes z^{j}{}^{*}$

and the counit map $\varepsilon(1^{*})=1$, $\varepsilon(z^{n}{}^{*})=0$ for
$n>0$.

Then a (left or right) ${\mathcal{C}}$-comodule ${\mathcal{M}}$ is the same
thing as a $k$-vector space endowed with a _locally nilpotent_ linear operator
$z\colon{\mathcal{M}}\longrightarrow{\mathcal{M}}$. In other words, for any
vector $m\in{\mathcal{M}}$ there must exist an integer $n\geqslant 1$ such
that $z^{n}(m)=0$ in ${\mathcal{M}}$. Indeed, given a linear operator $z$ on
${\mathcal{M}}$ one would define the contraaction map
$\nu\colon{\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{M}}$
by the formula

$\nu(m)=\sum_{n=0}^{\infty}z^{n}{}^{*}\otimes z^{n}(m),$

and the local nilpotence condition is needed for the sum to be well-defined
(i. e., finite) for every vector $m\in{\mathcal{M}}$.

A ${\mathcal{C}}$-contramodule structure on a $k$-vector space
${\mathfrak{P}}$ is, by the definition, the datum of a $k$-linear map
$\pi\colon\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
satisfying the contraassociativity and contraunitality axioms. Having such a
map is the same thing as the following _infinite summation operation_ being
defined in ${\mathfrak{P}}$. For every sequence of vectors $p_{0}$, $p_{1}$,
$p_{2}$ … $\in{\mathfrak{P}}$ there should be given a vector denoted
figuratively by

$\sum_{n=0}^{\infty}z^{n}p_{n}\in{\mathfrak{P}}.$

This infinitary operation in ${\mathfrak{P}}$ should satisfy the equations of
linearity

$\sum_{n=0}^{\infty}z^{n}(ap_{n}+bq_{n})=a\sum_{n=0}^{\infty}z^{n}p_{n}+b\sum_{n=0}^{\infty}z^{n}q_{n},$

contraassociativity

$\sum_{i=0}^{\infty}z^{i}\left(\sum_{j=0}^{\infty}z^{j}p_{ij}\right)=\sum_{n=0}^{\infty}z^{n}\left(\sum_{i+j=n}p_{ij}\right),$

and unitality

$\sum_{n=0}^{\infty}z^{n}p_{n}=p_{0}\quad\text{when
$p_{1}=p_{2}=p_{3}=\dotsb=0$ in ${\mathfrak{P}}$}$

for any $p_{n}$, $q_{n}$, $p_{ij}\in{\mathfrak{P}}$ and $a$, $b\in k$. Notice
that in the main (middle) equation the first three summation signs denote the
contramodule infinite summation operation, while the fourth one is the
conventional finite sum of elements of a vector space [52, Section A.1.1].

As we will see below in Section 1.6, the ${\mathcal{C}}$-contramodule
structure on a vector space ${\mathfrak{P}}$ is in fact determined by a single
linear operator $z\colon{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$,

$z(p)=1\cdot 0+z\cdot p+z^{2}\cdot 0+z^{3}\cdot 0+\dotsb$

However, _un_ like for the comodules, for contramodules over more complicated
coalgebras the similar statement is, of course, no longer true.

###  1.4. Contramodules over the $l$-adic integers

A left module ${\mathcal{M}}$ over a topological ring ${\mathfrak{R}}$ is
called _discrete_ if the action map
${\mathfrak{R}}\times{\mathcal{M}}\longrightarrow{\mathcal{M}}$ is continuous
in the discrete topology of ${\mathcal{M}}$ and the given topology of
${\mathfrak{R}}$. In other words, this means that the annihilator of every
element of ${\mathcal{M}}$ must be an open left ideal in ${\mathfrak{R}}$.
Discrete left ${\mathfrak{R}}$-modules form an abelian category, which we
denote by ${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$.

The discussion of topological algebras dual to coalgebras in the previous
section ignored one point which we now have to clarify. Given a coassociative
coalgebra ${\mathcal{C}}$ over $k$, one can define the multiplication on the
dual vector space ${\mathcal{C}}^{*}$ in two approximately equally natural
ways which differ by the passage to the opposite algebra, i. e., switching the
left and right arguments of the product map. Let us make the choice of
defining the multiplication on ${\mathcal{C}}^{*}$ in in such a way that left
${\mathcal{C}}$-comodules acquire natural structures of left
${\mathcal{C}}^{*}$-modules and right ${\mathcal{C}}$-comodules become right
${\mathcal{C}}^{*}$-modules. Explicitly, this means applying the formula

$\langle fg,c\rangle=\langle f,c_{(2)}\rangle\langle g,c_{(1)}\rangle$

where $\langle\,,\,\rangle$ denotes the natural pairing
${\mathcal{C}}^{*}\times{\mathcal{C}}\longrightarrow k$ and $c\longmapsto
c_{(1)}\otimes c_{(2)}$ is Sweedler’s symbolic notation for the
comultiplication map $\mu$ [62, Section 1.2].

Then the category of left ${\mathcal{C}}$-comodules can be described as the
full subcategory in the category of left ${\mathcal{C}}^{*}$-modules
${\mathcal{C}}^{*}{\operatorname{\mathsf{--mod}}}$ consisting precisely of
those ${\mathcal{C}}^{*}$-modules that are discrete with respect to the
topology of ${\mathcal{C}}^{*}$. Similarly, a right ${\mathcal{C}}$-comodule
is the same thing as a discrete right ${\mathcal{C}}^{*}$-module [62, Section
2.1].

Now the explicit description of contramodules over the coalgebra
${\mathcal{C}}$ with ${\mathcal{C}}^{*}=k[[t]]$ given in the previous section
raises the question about defining contramodules over topological rings other
than pro-finite-dimensional algebras over fields. The most close analogues of
the rings $k[[t]]$ being the rings of $l$-adic integers ${\mathbb{Z}}_{l}$,
they are the natural starting point of the desired generalization (whose full
development we postpone until Sections 2.1–2.3).

So let $l$ be a prime number. Let us start with mentioning that a discrete
module over the topological ring of $l$-adic integers ${\mathbb{Z}}_{l}$ is
the same thing as an $l$-primary abelian group, i. e., an abelian group where
the order of every element is a power of $l$. The category
${\mathbb{Z}}_{l}{\operatorname{\mathsf{--discr}}}$ is abelian with exact
functors of filtered inductive limits, which are also preserved by the
embedding functor
${\mathbb{Z}}_{l}{\operatorname{\mathsf{--discr}}}\longrightarrow{\mathsf{Ab}}$
into the category of abelian groups. The infinite products in
${\mathbb{Z}}_{l}{\operatorname{\mathsf{--discr}}}$ are not preserved by the
forgetful functor and _not_ exact. In other words, the category
${\mathbb{Z}}_{l}{\operatorname{\mathsf{--discr}}}$ satisfies Ab5 and Ab3*,
but not Ab4*. It has enough injective objects, but no nonzero projectives. The
injective discrete ${\mathbb{Z}}_{l}$-modules are precisely the direct sums of
copies of the group $\mathbb{Q}_{l}/{\mathbb{Z}}_{l}$.

A _${\mathbb{Z}}_{l}$ -contramodule_ ${\mathfrak{P}}$ is an abelian group
endowed with the following infinite summation operation. For any sequence of
elements $p_{0}$, $p_{1}$, $p_{2}$ … $\in{\mathfrak{P}}$ an element denoted
symbolically by

$\sum_{n=0}^{\infty}l^{n}p_{n}\in{\mathfrak{P}}$

should be defined. This infinitary operation should satisfy the equations of
additivity

$\sum_{n=0}^{\infty}l^{n}(p_{n}+q_{n})=\sum_{n=0}^{\infty}l^{n}p_{n}+\sum_{n=0}^{\infty}l^{n}q_{n},$

contraassociativity

$\sum_{i=0}^{\infty}l^{i}\left(\sum_{j=0}^{\infty}l^{j}p_{ij}\right)=\sum_{n=0}^{\infty}l^{n}\left(\sum_{i+j=n}p_{ij}\right),$

and unitality + compatibility with the abelian group structure

$\sum_{n=0}^{\infty}l^{n}p_{n}=p_{0}+p_{1}+\dotsb+p_{1}\text{ ($l$ summands
$p_{1}$) \ when $p_{2}=p_{2}=p_{3}=\dotsb=0$}$

for any elements $p_{n}$, $q_{n}$, and $p_{ij}\in{\mathfrak{P}}$.

For any $l$-primary abelian group ${\mathcal{M}}$ and abelian group $V$, the
abelian group $\operatorname{Hom}_{\mathbb{Z}}({\mathcal{M}},V)$ has a natural
${\mathbb{Z}}_{l}$-contramodule structure provided by the rule

$\left(\sum_{n=0}^{\infty}l^{n}p_{n}\right)(m)=\sum_{n=0}^{\infty}p_{n}(l^{n}m)$

for any $p_{n}\in\operatorname{Hom}_{\mathbb{Z}}({\mathcal{M}},V)$ and
$m\in{\mathcal{M}}$. The category
${\mathbb{Z}}_{l}{\operatorname{\mathsf{--contra}}}$ of
${\mathbb{Z}}_{l}$-contramodules is abelian and the forgetful functor
${\mathbb{Z}}_{l}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathsf{Ab}}$
is exact. As we will see in Section 1.6, the forgetful functor is fully
faithful. It preserves infinite products, but _not_ infinite direct sums. Both
the infinite direct sums and infinite products are exact functors in
${\mathbb{Z}}_{l}{\operatorname{\mathsf{--contra}}}$. In other words, the
category ${\mathbb{Z}}_{l}{\operatorname{\mathsf{--discr}}}$ satisfies Ab4 and
Ab4* (but _not_ Ab5 or Ab5*). It has enough projective objects, but no
injectives.

The _free ${\mathbb{Z}}_{l}$-contramodule_ generated by a set $X$ is the set
${\mathbb{Z}}_{l}[[X]]$ of all infinite formal linear combinations $\sum_{x\in
X}a_{x}x$ of elements of $X$ with the coefficients $a_{x}\in{\mathbb{Z}}_{l}$
such that for every $n\geqslant 1$ all but a finite number of $a_{x}$ are
divisible by $l^{n}$ in ${\mathbb{Z}}_{l}$. Notice that any formal linear
combination satisfying this condition is, in fact, supported in an at most
countable subset in $X$. As we will see below in Sections 2.1–2.2, for any
${\mathbb{Z}}_{l}$-contramodule ${\mathfrak{P}}$ the group of all
${\mathbb{Z}}_{l}$-contramodule morphisms
${\mathbb{Z}}_{l}[[X]]\longrightarrow{\mathfrak{P}}$ is isomorphic to the
group ${\mathfrak{P}}^{X}$ of arbitrary maps of sets
$X\longrightarrow{\mathfrak{P}}$. The classes of free and projective
${\mathbb{Z}}_{l}$-contramodules coincide.

The additive categories of injective discrete ${\mathbb{Z}}_{l}$-modules and
projective ${\mathbb{Z}}_{l}$-contramodules are equivalent; the equivalence is
provided by the functors
${\mathcal{M}}\longmapsto\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Q}_{l}/{\mathbb{Z}}_{l},{\mathcal{M}})$
and
${\mathfrak{P}}\longmapsto\mathbb{Q}_{l}/{\mathbb{Z}}_{l}\otimes_{\mathbb{Z}}{\mathfrak{P}}$
[34, Proposition 2.1]. In particular, one has

$\textstyle\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Q}_{l}/{\mathbb{Z}}_{l},\medspace\bigoplus_{X}\mathbb{Q}_{l}/{\mathbb{Z}}_{l})\simeq{\mathbb{Z}}_{l}[[X]]\quad\text{and}\quad\mathbb{Q}_{l}/{\mathbb{Z}}_{l}\otimes_{Z}{\mathbb{Z}}_{l}[[X]]\simeq\bigoplus_{X}\mathbb{Q}_{l}/{\mathbb{Z}}_{l}.$

###  1.5. Counterexamples

For any topological ring ${\mathfrak{R}}$, one can compute infinite products
in the abelian category ${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$ in
the following way. Let ${\mathcal{M}}_{\alpha}$ be a family of discrete left
${\mathfrak{R}}$-modules; denote by $M$ their product in the abelian category
of arbitrary ${\mathfrak{R}}$-modules. Then the product of the family of
objects ${\mathcal{M}}_{\alpha}$ in the category
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$ can be obtained as an
${\mathfrak{R}}$-submodule ${\mathcal{M}}\subset M$ consisting precisely of
all the elements $m\in M$ whose annihilators in ${\mathfrak{R}}$ are open left
ideals.

In particular, this provides a rule for computing infinite products in the
abelian categories ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ of
comodules over coalgebras over fields. Another way to formulate such a rule is
as follows. In any abelian category, infinite products are left exact
functors; in other words, they preserve kernels of morphisms. Since any
${\mathcal{C}}$-comodule can be presented as the kernel of a morphism of
cofree ${\mathcal{C}}$-comodules, it suffices to know what the products of
families of cofree ${\mathcal{C}}$-comodules are. The latter are easily seen
to be given by the formula
$\prod_{\alpha}{\mathcal{C}}\otimes_{k}V_{\alpha}={\mathcal{C}}\otimes_{k}\prod_{\alpha}V_{\alpha}$.

Similarly, in order to compute the infinite direct sum of a family of objects
in ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$, one can present these as
the cokernels of morphisms of free ${\mathcal{C}}$-contramodules. Since any
${\mathcal{C}}$-contramodule can be obtained as such a cokernel and the
infinite direct sums preserve cokernels, it remains to use the formula
$\bigoplus_{\alpha}\operatorname{Hom}_{k}({\mathcal{C}},V_{\alpha})=\operatorname{Hom}_{k}({\mathcal{C}},\medspace\bigoplus_{\alpha}V_{\alpha})$
for the direct sum of a family of free ${\mathcal{C}}$-contramodules.

Let us return to the example of the coalgebra ${\mathcal{C}}$ dual to the
topological algebra of formal power series $k[[z]]$ considered in Section 1.3.
Viewed as a discrete module over the algebra ${\mathcal{C}}^{*}=k[[z]]$, the
coalgebra ${\mathcal{C}}$ can be identified with the quotient module
$k((z))/k[[z]]$ of the $k[[z]]$-module of Laurent series $k((z))$ by its
submodule $k[[z]]$. Consider the family of discrete $k[[z]]$-modules
$z^{-n}k[[z]]/k[[z]]$, $n=1$, $2$, … They can be included into short exact
sequences of discrete $k[[z]]$-modules

$0\text{$\mskip 1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}z^{-n}k[[z]]/k[[z]]\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}k((z))/k[[z]]\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}k((z))/z^{-n}k[[z]]\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip 1.5mu$}0.$

Passing to the infinite product of these short exact sequences in the category
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ over all $n\geqslant 1$, one
discovers that the map
$k((z))/k[[z]]\otimes_{k}\prod_{n}k=\prod_{n}k((z))/k[[z]]\longrightarrow\prod_{n}k((z))/z^{-n}k[[z]]=k((z))/k[[z]]\otimes_{k}\prod_{n}kz^{-n}$
is not surjective, as, e. g., the vector
$(z^{-n-1})_{n}\in\prod_{n}k((z))/z^{-n}k[[z]]$ does not belong to its image.
One also computes the infinite product $\prod_{n}z^{-n}k[[z]]/k[[z]]$ in the
category ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ as isomorphic to the
inductive limit
$\varinjlim_{m}\big{(}\prod_{n=1}^{m}z^{-n}k[[z]]/k[[z]]\times\prod_{n=m+1}^{\infty}z^{-m}k[[z]]/k[[z]]\big{)}$.

Now consider the family of ${\mathcal{C}}$-contramodules $k[[z]]/z^{n}k[[z]]$,
$n=1$, $2$ … They can be viewed as parts of the short exact sequences of
${\mathcal{C}}$-contramodules

$0\text{$\mskip 1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}z^{n}k[[z]]\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}k[[z]]\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}k[[z]/z^{n}k[[z]]\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip 1.5mu$}0.$

Passing to the infinite direct sum of these short exact sequences in the
category ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ over all
$n\geqslant 1$, one finds out that the map
$\operatorname{Hom}_{k}({\mathcal{C}},\medspace\bigoplus_{n}kz^{n})=\bigoplus_{n}z^{n}k[[z]]\allowbreak\longrightarrow\bigoplus_{n}k[[z]]=\operatorname{Hom}_{k}({\mathcal{C}},\medspace\bigoplus_{n}k)$
is injective. Its cokernel
${\mathfrak{P}}=\bigoplus_{n}k[[z]]/z^{n}k[[z]]\in{\mathcal{C}}{\operatorname{\mathsf{--contra}}}$
is the ${\mathcal{C}}$-contramodule that we are interested in.

Let us start with introducing a more careful notation. Let ${\mathfrak{E}}$
denote the free ${\mathcal{C}}$-contramodule generated by a $k$-vector space
$E$ with a countable basis $e_{1}$, $e_{2}$, $e_{3}$, … Explicitly,
${\mathfrak{E}}$ is the set of all formal linear combinations
$\sum_{n=1}^{\infty}a_{n}(z)e_{n}$, where the sequence of formal power series
$a_{n}(z)\in k[[z]]$ converges to zero in the topology of $k[[z]]$. The
$k[[z]]$-contramodule infinite summation operations on ${\mathfrak{E}}$ are
defined in the obvious way. Let ${\mathfrak{F}}$ be the similar free
${\mathcal{C}}$-contramodule generated by a $k$-vector space $F$ with a basis
$f_{1}$, $f_{2}$, $f_{3}$, … Define a morphism of
${\mathcal{C}}$-contramodules ${\mathfrak{F}}\longrightarrow{\mathfrak{E}}$ by
the rule
$\sum_{n=1}^{\infty}b_{n}(z)f_{n}\longmapsto\sum_{n=1}^{\infty}z^{n}b_{n}(z)e_{n}$.
Clearly, this morphism is injective; denote its cokernel by
${\mathfrak{P}}={\mathfrak{E}}/{\mathfrak{F}}$.

Set $p_{n}\in{\mathfrak{P}}$ to be the images of the elements
$e_{n}\in{\mathfrak{E}}$ under the surjective morphism
${\mathfrak{E}}\longrightarrow{\mathfrak{P}}$. Then the infinite sum
$p=\sum_{n=1}^{\infty}z^{n}p_{n}$ is a nonzero vector in ${\mathfrak{P}}$,
since the element $\sum_{n=1}^{\infty}z^{n}e_{n}$ does not belong to
${\mathfrak{F}}\subset{\mathfrak{E}}$ (there being no element
$\sum_{n=1}^{\infty}f_{n}$ in ${\mathfrak{F}}$). On the other hand, every
finite partial sum
$zp_{1}+z^{2}p_{2}+\dotsb+z^{n}p_{n}=zp_{1}+\dotsb+z^{n}p_{n}+z^{n+1}\cdot
0+\dotsb$ vanishes in ${\mathfrak{P}}$, the finite sum
$ze_{1}+\dotsb+z^{n}e_{n}$ being the image of the vector
$f_{1}+\dotsb+f_{n}\in{\mathfrak{F}}$ in ${\mathfrak{E}}$. It follows that our
vector $p=z^{n}(p_{n}+zp_{n+1}+\dotsb)$ belongs to $z^{n}{\mathfrak{P}}$ for
every $n\geqslant 1$, so the $z$-adic topology on ${\mathfrak{P}}$ is not
separated. This counterexample can be found in [52, Section A.1.1]; it also
occured, under slightly different guises, in [61, Example 2.5] and [67,
Example 3.20].

Among other things, ${\mathfrak{P}}$ provides an example of a
${\mathcal{C}}$-contramodule that does not have the form
$\operatorname{Hom}_{k}({\mathcal{N}},V)$ for any ${\mathcal{C}}$-comodule
${\mathcal{N}}$. An example of a finite-dimensional contramodule not of this
form (over a more complicated coalgebra ${\mathcal{C}}$) can be found in [52,
Section A.1.2]. Concerning the above coalgebra ${\mathcal{C}}$ with
${\mathcal{C}}^{*}=k[[z]]$, let us point out that the natural map
${\mathfrak{Q}}\longrightarrow\varprojlim_{n}{\mathfrak{Q}}/z^{n}{\mathfrak{Q}}$,
though not necessarily injective, is always _surjective_ for a
${\mathcal{C}}$-contramodule ${\mathfrak{Q}}$ [52, Lemma A.2.3]. Indeed, let
$q_{n}\in{\mathfrak{Q}}$ be a sequence of vectors such that $q_{n+1}-q_{n}\in
z^{n}{\mathfrak{Q}}$ for every $n=1$, $2$, … Suppose
$q_{n+1}-q_{n}=z^{n}p_{n}$; then the infinite sum
$q=q_{0}+\sum_{n=1}^{\infty}z^{n}p_{n}$ provides an element
$q\in{\mathfrak{Q}}$ for which $q-q_{n}\in z^{n}{\mathfrak{Q}}$ for every
$n\geqslant 1$.

Similarly, consider the family of $l$-primary abelian groups
$l^{-n}{\mathbb{Z}}/{\mathbb{Z}}$, $n=1$, $2$, … They can be included into
short exact sequences of $l$-primary abelian groups

$0\text{$\mskip 1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}l^{-n}{\mathbb{Z}}/{\mathbb{Z}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\mathbb{Q}_{l}/{\mathbb{Z}}_{l}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\mathbb{Q}_{l}/l^{-n}{\mathbb{Z}}_{l}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip 1.5mu$}0.$

Passing to the infinite product of these short exact sequences in the category
${\mathbb{Z}}_{l}{\operatorname{\mathsf{--discr}}}$ over all $n\geqslant 1$,
one discovers that the map
$\prod_{n}\mathbb{Q}_{l}/{\mathbb{Z}}_{l}\longrightarrow\prod_{n}\mathbb{Q}_{l}/l^{-n}{\mathbb{Z}}_{l}$
is not surjective, as, e. g., the element
$(l^{-n-1})_{n}\in\prod_{n}\mathbb{Q}_{l}/l^{-n}{\mathbb{Z}}_{l}$ does not
belong to its image. One also computes the infinite product
$\prod_{n}l^{-n}{\mathbb{Z}}/{\mathbb{Z}}$ in the category
${\mathbb{Z}}_{l}{\operatorname{\mathsf{--discr}}}$ as isomorphic to the
inductive limit
$\varinjlim_{m}\big{(}\prod_{n=1}^{m}l^{-n}{\mathbb{Z}}/{\mathbb{Z}}\times\prod_{n=m+1}^{\infty}l^{-m}{\mathbb{Z}}/{\mathbb{Z}}\big{)}$.

Consider the family of ${\mathbb{Z}}_{l}$-contramodules
${\mathbb{Z}}/l^{n}{\mathbb{Z}}$, $n=1$, $2$, … They can be viewed as parts of
the short exact sequences of ${\mathbb{Z}}_{l}$-contramodules

$0\text{$\mskip 1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}l^{n}{\mathbb{Z}}_{l}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathbb{Z}}_{l}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathbb{Z}}/l^{n}{\mathbb{Z}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip 1.5mu$}0.$

Passing to the infinite direct sum of these short exact sequences in the
category ${\mathbb{Z}}_{l}{\operatorname{\mathsf{--contra}}}$ over all
$n\geqslant 1$, one finds out that the map
$\bigoplus_{n}l^{n}{\mathbb{Z}}_{l}\longrightarrow\bigoplus_{n}{\mathbb{Z}}_{l}$
is injective. Its cokernel
${\mathfrak{P}}=\bigoplus_{n}{\mathbb{Z}}/l^{n}{\mathbb{Z}}\in{\mathcal{C}}{\operatorname{\mathsf{--contra}}}$
can be described as follows.

Let ${\mathfrak{E}}$ denote the free ${\mathbb{Z}}_{l}$-contramodule generated
by a sequence of symbols $e_{1}$, $e_{2}$, $e_{3}$, … Explicitly,
${\mathfrak{E}}$ is the set of all formal linear combinations
$\sum_{n=1}^{\infty}a_{n}e_{n}$, where the sequence of $l$-adic integers
$a_{n}\in{\mathbb{Z}}_{l}$ converges to zero in the topology of
${\mathbb{Z}}_{l}$. Let ${\mathfrak{F}}$ be the similar
${\mathbb{Z}}_{l}$-contramodule generated by a sequence of symbols $f_{1}$,
$f_{2}$, $f_{3}$, … Define a morphism of ${\mathcal{C}}$-contramodules
${\mathfrak{F}}\longrightarrow{\mathfrak{E}}$ by the rule
$\sum_{n=1}^{\infty}b_{n}f_{n}\longmapsto\sum_{n=1}^{\infty}l^{n}b_{n}e_{n}$.
Clearly, this morphism is injective; its cokernel
${\mathfrak{E}}/{\mathfrak{F}}$ is our ${\mathbb{Z}}_{l}$-contramodule
${\mathfrak{P}}$.

Set $p_{n}=e_{n}\bmod{\mathfrak{F}}\in{\mathfrak{P}}$. Then the infinite sum
$p=\sum_{n=1}^{\infty}l^{n}p_{n}$ is a nonzero element in ${\mathfrak{P}}$,
since the element $\sum_{n=1}^{\infty}l^{n}e_{n}$ does not belong to
${\mathfrak{F}}\subset{\mathfrak{E}}$. On the other hand, every summand
$l^{n}p_{n}$ vanishes in ${\mathfrak{P}}$, the element $l^{n}e_{n}$ being the
image of the element $f_{n}\in{\mathfrak{F}}$ in ${\mathfrak{E}}$. It follows
that the element $p$ belongs to $l^{n}{\mathfrak{P}}$ for every $n\geqslant
1$, so the $l$-adic topology on ${\mathfrak{P}}$ is not separated. Notice that
the natural map
${\mathfrak{Q}}\longrightarrow\varprojlim_{n}{\mathfrak{Q}}/l^{n}{\mathfrak{Q}}$,
though not necessarily injective, is always surjective for a
${\mathbb{Z}}_{l}$-contramodule ${\mathfrak{P}}$ [57, Lemma D.1.1]. The proof
is similar to the above argument for $k[[z]]$-contramodules.

###  1.6. Recovering the contramodule structure

We have seen in the previous section that a $k[[z]]$-contramodule can contain
infinitely $z$-divisible _vectors_ , i. e., vectors $p\in{\mathfrak{P}}$ for
which there exists a sequence of vectors $p_{n}\in{\mathfrak{P}}$ such that
$p=z^{n}p_{n}$ for every $n\geqslant 1$. Let us now show that _no_
$k[[z]]$-contramodule can contain infinitely $z$-divisible _$k[z]$
-submodules_. In other words, one can never choose the sequence of vectors
$p_{n}\in{\mathfrak{P}}$ in a compatible way, i. e., any sequence of vectors
$p_{n}\in{\mathfrak{P}}$ such that $p_{n}=zp_{n+1}$ for all $n\geqslant 0$ is
the sequence of zero vectors.

Indeed, consider the expression
$q=\sum_{n=0}^{\infty}z^{n}p_{n}\in{\mathfrak{P}}$. By assumption, we have

$\sum_{n=0}^{\infty}z^{n}p_{n}=\sum_{n=0}^{\infty}z^{n}\cdot
zp_{n+1}=\sum_{n=0}^{\infty}z^{n+1}p_{n+1}=\sum_{n=1}^{\infty}z^{n}p_{n},$

that is $q=q-p_{0}$ and $p_{0}=0$. Here the last two equations conceal the use
of the contraassociativity axiom from Section 1.3, which is being applied to
the double sequence of vectors $p_{ij}=p_{i+1}$ when $j=1$ and $p_{ij}=0$
otherwise. The assertion we have proven is essentially a particular case of
_Nakayama’s lemma for contramodules_ (see Section 2.1 below).

Now we are in the position to show that the forgetful functor
$k[[z]]{\operatorname{\mathsf{--contra}}}\longrightarrow
k[z]{\operatorname{\mathsf{--mod}}}$ (where we denote by
$k[[z]]{\operatorname{\mathsf{--contra}}}$ the category
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ of contramodules over the
coalgebra ${\mathcal{C}}$ with ${\mathcal{C}}^{*}=k[[z]]$) is _fully faithful_
, i. e., the ${\mathcal{C}}$-contramodule structure on a $k$-vector space
${\mathfrak{P}}$ can be uniquely recovered from the single linear operator
$z\colon{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$. Indeed, suppose that we
want to “compute” the value of the infinite sum
$\sum_{n=0}^{\infty}z^{n}p_{n}$ in ${\mathfrak{P}}$. Consider the infinite
system of linear equations

(3) $q_{n}=p_{n}+zq_{n+1},\quad\text{$n=0$, $1$, $2$,~{}\ldots}$

in the indeterminates $q_{n}\in{\mathfrak{P}}$. We have just shown that the
related system of homogeneous linear equations $q_{n}=zq_{n+1}$ has no nonzero
solutions in ${\mathfrak{P}}$. Hence a solution of the system (3) is unique if
it exists. Given a $k[[z]]$-contramodule structure in ${\mathfrak{P}}$, one
produces such a solution by setting

$q_{n}=\sum_{i=0}^{\infty}z^{i}p_{n+i}.$

The value of $\sum_{n=0}^{\infty}z^{n}p_{n}$ can be recovered as the vector
$q_{0}\in{\mathfrak{P}}$.

We have essentially shown that a $k[z]$-module $P$ admits an (always unique)
$k[[z]]$-contramodule structure if and only if the system of nonhomogeneous
linear equations (3) has a unique solution in $q_{n}$ for every sequence of
vectors $p_{n}\in P$. The latter condition is equivalent to the vanishing of
the two Ext spaces $\operatorname{Ext}^{*}_{k[z]}(k[z,z^{-1}],\medspace P)$
(see [52, Remark A.1.1] and [56, Lemmas B.5.1 and B.7.1]).

Similarly, we have seen that a ${\mathbb{Z}}_{l}$-contramodule
${\mathfrak{P}}$ can contain infinitely $l$-divisible _elements_ , i. e.,
there can be nonzero elements $p\in{\mathfrak{P}}$ for which there exists a
sequence of elements $p_{n}\in{\mathfrak{P}}$ such that $p=l^{n}p_{n}$ for
every $n\geqslant 1$. Let us show that no ${\mathbb{Z}}_{l}$-contramodule can
contain infinitely $l$-divisible _subgroups_. In other words, one can never
choose the sequence $p_{n}\in{\mathfrak{P}}$ in a compatible way, i. .e, any
sequence of elements $p_{n}\in{\mathfrak{P}}$ such that $p_{n}=lp_{n+1}$ for
all $n\geqslant 0$ is the zero sequence.

Indeed, consider the expression
$\sum_{n=0}^{\infty}l^{n}p_{n}\in{\mathfrak{P}}$. By assumption, we have

$\sum_{n=0}^{\infty}l^{n}p_{n}=\sum_{n=0}^{\infty}l^{n}\cdot
lp_{n+1}=\sum_{n=0}^{\infty}l^{n+1}p_{n+1}=\sum_{n=1}^{\infty}l^{n}p_{n},$

that is $q=q-p_{0}$ and $p_{0}=0$. Here the first equation signifies the use
of the “compatibility with the abelian group structure” axiom from Section
1.4, while the last two equations presume an application of the
contraassociativity axiom.

Let us show that the forgetful functor
${\mathbb{Z}}_{l}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathsf{Ab}}$
is fully faithful, i. e., a ${\mathbb{Z}}_{l}$-contramodule structure on an
abelian group ${\mathfrak{P}}$ is uniquely determined by the abelian group
structure. Suppose that we want to “compute” the value of the infinite sum
$\sum_{n=0}^{\infty}l^{n}p_{n}$ in ${\mathfrak{P}}$. Consider the infinite
system of linear equations

(4) $q_{n}=p_{n}+lq_{n+1},\quad\text{$n=0$, $1$, $2$~{}\ldots}$

in the indeterminates $q_{n}\in{\mathfrak{P}}$. We have just shown that the
related system of homogeneous linear equations $q_{n}=lq_{n+1}$ has no nonzero
solutions in ${\mathfrak{P}}$. Hence a solution of the system (4) is unique if
it exists. Assuming a ${\mathbb{Z}}_{l}$-contramodule structure in
${\mathfrak{P}}$, one produces such a solution by setting

$q_{n}=\sum_{i=0}^{\infty}l^{i}p_{n+i}.$

The value of $\sum_{n=0}^{\infty}l^{n}p_{n}$ can be recovered as the vector
$q_{0}\in{\mathfrak{P}}$.

We have essentially shown that an abelian group $P$ admits an (always unique)
${\mathbb{Z}}_{l}$-contramodule structure if and only if the system of
nonhomogeneous linear equations (4) has a unique solution in $q_{n}$ for every
sequence of elements $p_{n}\in P$. The latter condition is equivalent to the
vanishing of the two Ext groups
$\operatorname{Ext}^{*}_{\mathbb{Z}}({\mathbb{Z}}[l^{-1}],P)$ (see [52, Remark
A.3] and [56, Theorem B.1.1 and Lemma B.7.1]; cf. [12, Sections VI.3–4] and
[36, Definition 4.6 and Remark 4.7]).

###  1.7. Contramodules over the Virasoro algebra

The definition of contramodules over the formal power series algebra in terms
of infinite summation operations, as stated in Section 1.3, opens the door to
generalizations of the notion of a contramodule to various topological
algebraic structures, including not only associative rings, but also
topological Lie algebras. In this section we demonstrate the possibility of
such a definition in the simple example of the Virasoro Lie algebra.

The _punctured formal disk_ , otherwise known as the _formal circle_ over a
field $k$ is defined as a “space” such that the ring of functions on it is the
ring of formal Laurent power series $k((z))$. The Lie algebra of vector fields
on the formal circle $k[[z]]d/dz$ is the set of all expressions of the form
$f(z)d/dz$ with $f(z)\in k((z))$, endowed with the obvious $k$-vector space
structure and the Lie bracket $[f\,d/dz,\medspace
g\,d/dz]=(f\,dg/dz-g\,df/dz)\,d/dz$. The vector fields $L_{i}=z^{i+1}\,d/dz$
form a topological basis in the vector space $k((z))d/dz$, in which the Lie
bracket takes the form $[L_{i},L_{j}]=(j-i)L_{i+j}$.

The _Virasoro algebra_ $\mathbb{V}\mathrm{ir}$ is a central extension of the
Lie algebra $k((z))d/dz$ with a one-dimensional kernel spanned by an element
denoted by $C$. The $k$-vector space $\mathbb{V}\mathrm{ir}=k((z))d/dz\oplus
kC$ has a topological basis formed by the vectors $L_{i}$, $i\in{\mathbb{Z}}$,
and $C$, in which the Lie bracket is given by the rules

$[L_{i},C]=0,\qquad[L_{i},L_{j}]=(j-i)L_{i+j}+\delta_{i+j,\text{$\mskip
1.5mu$}0}\frac{i^{3}-i}{12}\,C,$

where $\delta$ is the Kronecker symbol, for all $i$, $j\in{\mathbb{Z}}$ [28,
64, 38].

A _discrete module_ ${\mathcal{M}}$ over the Virasoro algebra is a module over
the Lie algebra $\mathbb{V}\mathrm{ir}$ for which the action map
$\mathbb{V}\mathrm{ir}\times{\mathcal{M}}\longrightarrow{\mathcal{M}}$ is
continuous in the $z$-adic topology of $\mathbb{V}\mathrm{ir}$ and the
discrete topology of ${\mathcal{M}}$. In other words, ${\mathcal{M}}$ is a
vector space endowed with linear operators $L_{i}$ and
$C\colon{\mathcal{M}}\longrightarrow{\mathcal{M}}$ satisfying the above
commutation relations and the discreteness condition, according to which for
any vector $x\in{\mathcal{M}}$ there should exist an integer $n$ such that
$L_{i}x=0$ for all $i>n$.

A _contramodule_ ${\mathfrak{P}}$ over the Virasoro algebra
$\mathbb{V}\mathrm{ir}$ is a $k$-vector space endowed with a linear operator
$C\colon{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$ and an infinite summation
operation assigning to every sequence of vectors $p_{-n}$, $p_{-n+1}$,
$p_{-n+2}$ … $\in{\mathfrak{P}}$, $n\in{\mathbb{Z}}$, a vector denoted
symbolically by $\sum_{i=-n}^{\infty}L_{i}p_{i}\in{\mathfrak{P}}$. This
infinitary operation, or rather, sequence of infinitary operations indexed by
the integers $n$, should satisfy the equations of agreement

$\sum_{i=-n}^{\infty}L_{i}p_{i}=\sum_{i=-m}^{\infty}L_{i}p_{i}\quad\text{when
$-n<-m$ and $p_{-n}=\dotsb=p_{-m-1}=0$},$

linearity

$\sum_{i=-n}^{\infty}L_{i}(ap_{i}+bq_{i})=a\sum_{i=-n}^{\infty}L_{i}p_{i}+b\sum_{i=-n}^{\infty}L_{i}q_{i},$

and the contra-Jacobi identity

$\sum_{i=-n}^{\infty}L_{i}(Cp_{i})\text{$\mskip 1.5mu$}=\text{$\mskip
1.5mu$}C\sum_{i=-n}^{\infty}L_{i}p_{i}$

and

$\sum_{i=-n}^{\infty}L_{i}\left(\sum_{j=-m}^{\infty}L_{j}p_{ij}\right)-\sum_{j=-m}^{\infty}L_{j}\left(\sum_{i=-n}^{\infty}L_{i}p_{ij}\right)\\\
=\,\sum_{h=-n-m}^{\infty}L_{h}\left(\sum_{i+j=h}^{i\geqslant-n,\medspace
j\geqslant-m}(j-i)p_{ij}\right)+C\sum_{i+j=0}^{i\geqslant-n,\medspace
j\geqslant-m}\frac{i^{3}-i}{12}p_{ij}.$

for any $p_{i}$, $q_{i}$, $p_{ij}\in{\mathfrak{P}}$ and $a$, $b\in k$. This
definition (for the Lie algebra $k((z))d/dz$ without the central extension)
can be found in [52, Section D.2.7].

For any discrete module ${\mathcal{M}}$ over the Virasoro algebra and any
$k$-vector space $V$, the vector space
${\mathfrak{P}}=\operatorname{Hom}_{k}({\mathcal{M}},V)$ has a natural
structure of $\mathbb{V}\mathrm{ir}$-contramodule. The central element $C$
acts in ${\mathfrak{P}}$ by the usual formula $(Cp)(x)=-p(Cx)$ for
$p\in{\mathfrak{P}}$ and $x\in{\mathcal{M}}$, while the infinite summation
operations are provided by the rule

$\big{(}\sum_{i=-n}^{\infty}L_{i}p_{i}\big{)}(x)=-\sum_{i=-n}^{\infty}p_{i}(L_{i}x),$

for $p_{i}\in{\mathfrak{P}}$ and $x\in{\mathcal{M}}$, where the second
summation sign stands for the conventional sum of an eventually vanishing
sequence of vectors in $V$.

The category $\mathbb{V}\mathrm{ir}{\operatorname{\mathsf{--discr}}}$ of
discrete modules over the Virasoro algebra is abelian and the forgetful
functor $\mathbb{V}\mathrm{ir}{\operatorname{\mathsf{--discr}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ is exact. Both the infinite direct sums and
infinite products exist in
$\mathbb{V}\mathrm{ir}{\operatorname{\mathsf{--discr}}}$. The forgetful
functor preserves infinite direct sums (but not infinite products), so
filtered inductive limits are exact in
$\mathbb{V}\mathrm{ir}{\operatorname{\mathsf{--discr}}}$. In other words, the
abelian category $\mathbb{V}\mathrm{ir}{\operatorname{\mathsf{--discr}}}$
satisfies the axioms Ab5 and Ab3*. It also admits a set of generators, so it
has enough injectives.

The category $\mathbb{V}\mathrm{ir}{\operatorname{\mathsf{--contra}}}$ of
contramodules over the Virasoro algebra is abelian and the forgetful functor
$\mathbb{V}\mathrm{ir}{\operatorname{\mathsf{--contra}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ is exact. Both the infinite direct sums and
the infinite products exist in
$\mathbb{V}\mathrm{ir}{\operatorname{\mathsf{--contra}}}$. The forgetful
functor preserves infinite products, which are therefore exact functors in
$\mathbb{V}\mathrm{ir}{\operatorname{\mathsf{--contra}}}$; so this category
satisfies Ab4 and Ab3*. There are also enough projective objects in
$\mathbb{V}\mathrm{ir}{\operatorname{\mathsf{--contra}}}$. We will explain
their construction in Section 2.4 below.

###  1.8. Contramodules over topological groups

The aim of this section is to demonstrate the definition of contramodules over
a locally compact, totally disconnected topological group. A typical example
of such a group is the group $GL_{n}(\mathbb{Q}_{l})$ of invertible square
matrices over the rational $l$-adic numbers (endowed with the topology induced
by the topology of $\mathbb{Q}_{l}$).

In this section, all the _topological spaces_ are presumed to be Hausdorff,
locally compact, and totally disconnected. Open-closed subsets in such a
topological space $X$ form a topology base [10, Corollaire II.4.4]. A
_topological group_ is a topological space with a group structure given by
continuous multiplication and inverse element maps. Open subgroups in such a
topological group $G$ form a base of neighborhoods of zero [10, Corollaire
III.4.6.1]. When $G$ is compact, the same can be said about its open normal
subgroups; so $G$ is profinite.

A _discrete module_ ${\mathcal{M}}$ over a topological group $G$ is an abelian
group endowed with an action of $G$ provided by a continuous map
$G\times{\mathcal{M}}\longrightarrow{\mathcal{M}}$ in the given topology of
$G$ and the discrete topology of ${\mathcal{M}}$. In other words, an action of
$G$ in ${\mathcal{M}}$ is discrete if and only if the stabilizer of any
element of ${\mathcal{M}}$ is an open subgroup in $G$. A discrete action can
be also viewed as a map ${\mathcal{M}}\longrightarrow{\mathcal{M}}\\{G\\}$,
where for any topological space $X$ and abelian group $A$ we denote by
$A\\{X\\}$ the group of all locally constant $A$-valued functions
$X\longrightarrow A$ on $X$. Denoting by $G{\operatorname{\mathsf{--mod}}}$
the category of nontopological $G$-modules, i. e., abelian groups $M$ endowed
with an arbitrary action of $G$ viewed as an abstract group, and by
$G{\operatorname{\mathsf{--discr}}}$ the category of discrete $G$-modules,
there is a natural fully faithful functor
$G{\operatorname{\mathsf{--discr}}}\longrightarrow
G{\operatorname{\mathsf{--mod}}}$.

Let us introduce a bit more notation. Given a topological space $X$ and an
abelian group $A$, we denote by $A(X)$ the group of all locally constant
compactly supported $A$-valued functions on $X$. For any topological spaces
$X$ and $Y$, there is a natural isomorphism $A(X\times Y)\simeq A(X)(Y)$.
Furthermore, denote by $A[[X]]$ the abelian group of finitely additive
compactly supported $A$-valued measures defined on the open-closed subsets of
$X$. For any continuous map of topological spaces $X\longrightarrow Y$, the
push-forward map $A[[X]]\longrightarrow A[[Y]]$ is defined [52, Section
E.1.1].

For any topological spaces $X$, $Y$ and an abelian group $A$, there is a
natural map $A[[X\times Y]]\longrightarrow A[[X]][[Y]]$ assigning to an
$A$-valued measure $\nu$ on $X\times Y$ the measure taking an open-closed
subset $V\subset Y$ to the measure taking an open-closed subset $U\subset X$
to the element $\nu(U\times V)\in A$. This map is an isomorphism when _both_
the spaces $X$ and $Y$ are discrete or _both_ of them are compact, but not
otherwise.

A _contramodule_ over a topological group $G$ is an abelian group
${\mathfrak{P}}$ endowed with a _$G$ -contraaction_ map
$\pi\colon{\mathfrak{P}}[[G]]\longrightarrow{\mathfrak{P}}$, which can be
viewed as an _integration operation_ and denoted symbolically by

$\pi(\mu)=\int_{G}g^{-1}(d\mu_{g}),$

where $d\mu_{g}\in{\mathfrak{P}}$ denotes the value of a measure
$\mu\in{\mathfrak{P}}[[G]]$ on a small piece of the group $G$ containing an
element $g\in G$, while $g^{-1}(d\mu_{g})\in{\mathfrak{P}}$ is a small element
in ${\mathfrak{P}}$ obtained by applying to $d\mu_{g}$ the presumed
generalized action of $g^{-1}\in G$ in ${\mathfrak{P}}$.

The map $\pi$ must satisfy the following _contraassociativity_ and
_contraunitality_ equations. Firstly, the composition ${\mathfrak{P}}[[G\times
G]]\longrightarrow{\mathfrak{P}}[[G]][[G]]\longrightarrow{\mathfrak{P}}[[G]]$
of the above-described map ${\mathfrak{P}}[[G\times
G]]\longrightarrow{\mathfrak{P}}[[G]][[G]]$ with the iterated contraaction map
${\mathfrak{P}}[[G]][[G]]\longrightarrow{\mathfrak{P}}[[G]]\longrightarrow{\mathfrak{P}}$
should be equal to the composition ${\mathfrak{P}}[[G\times
G]]\longrightarrow{\mathfrak{P}}[[G]]\longrightarrow{\mathfrak{P}}$ of the
push-forward map ${\mathfrak{P}}[[G\times
G]]\longrightarrow{\mathfrak{P}}[[G]]$ with respect to the multiplication map
$G\times G\longrightarrow G$ with the contraaction map
${\mathfrak{P}}[[G]]\longrightarrow{\mathfrak{P}}$,

${\mathfrak{P}}[[G\times
G]]\rightrightarrows{\mathfrak{P}}[[G]]\longrightarrow{\mathfrak{P}}.$

Secondly, the point measure supported at the unit element $e\in G$ and taking
a prescribed value $p\in{\mathfrak{P}}$ on the neighborhoods of $e$ should be
taken to the element $p$ by the contraaction map,

${\mathfrak{P}}\longrightarrow{\mathfrak{P}}[[G]]\longrightarrow{\mathfrak{P}}.$

Given a point $g\in G$ and an element $p\in{\mathfrak{P}}$, denote by
$g^{-1}(p)\in{\mathfrak{P}}$ the element one obtains by applying the
contraaction map to the point measure supported at $g$ and taking the value
$p$ on its neighborhoods. This rule defines a natural action of $G$ (as an
abstract, nontopological group) on any $G$-contramodule ${\mathfrak{P}}$,
providing a forgetful functor
$G{\operatorname{\mathsf{--contra}}}\longrightarrow
G{\operatorname{\mathsf{--mod}}}$ [52, Section E.1.3].

For any discrete $G$-module ${\mathcal{M}}$ and an abelian group $V$, the
abelian group $\operatorname{Hom}_{\mathbb{Z}}({\mathcal{M}},V)$ has a natural
$G$-contramodule structure. The contraaction map
$\operatorname{Hom}_{\mathbb{Z}}({\mathcal{M}},V)[[G]]\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathcal{M}},V)$
assigns to a measure $\mu$ the additive map taking an element
$x\in{\mathcal{M}}$ to the value of the integral

$\pi(\mu)(x)=\int_{G}d\mu_{g}(gm)\text{$\mskip 1.5mu$}\in\text{$\mskip
1.5mu$}V.$

The ${\mathcal{M}}$-valued function $g\longmapsto gm$ being locally constant
on $G$ and the $\operatorname{Hom}_{\mathbb{Z}}({\mathcal{M}},V)$-valued
measure $\mu$ being compactly supported in $G$, the integral is well-defined
(cf. [52, Section E.1.4]).

The category of discrete $G$-modules is abelian and the forgetful functor
$G{\operatorname{\mathsf{--discr}}}\longrightarrow{\mathsf{Ab}}$ is exact.
Filtered inductive limits are exact functors in
$G{\operatorname{\mathsf{--discr}}}$; they are also preserved by the forgetful
functor. In other words, the category $G{\operatorname{\mathsf{--discr}}}$
satisfies the axioms Ab5 and Ab3*. It also admits a set of generators, so it
has enough injective objects. The category of $G$-contramodules is abelian and
the forgetful functor
$G{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathsf{Ab}}$ is exact.
Infinite products are exact functors in $G{\operatorname{\mathsf{--contra}}}$;
they are also preserved by the forgetful functor. So the category
$G{\operatorname{\mathsf{--contra}}}$ satisfies the axioms Ab3 and Ab4*. It
has enough projective objects.

The embedding functor $G{\operatorname{\mathsf{--discr}}}\longrightarrow
G{\operatorname{\mathsf{--mod}}}$ and the forgetful functor
$G{\operatorname{\mathsf{--contra}}}\longrightarrow
G{\operatorname{\mathsf{--mod}}}$ have the similar properies, the forgetful
functor $G{\operatorname{\mathsf{--mod}}}\longrightarrow{\mathsf{Ab}}$
preserving the inductive and projective limits of any diagrams. We will see
below in Section 2.6 how discrete $G$-modules and $G$-contramodules can be
interpreted as _semimodules_ and _semicontramodules_ over a certain
semialgebra ${\boldsymbol{\mathcal{S}}}$, opening the way to explicit
constructions of injective and projective objects in
$G{\operatorname{\mathsf{--discr}}}$ and
$G{\operatorname{\mathsf{--contra}}}$.

##  2\. Comodule and Contramodule Categories

###  2.1. Contramodules over topological rings

As we discussed in Sections 1.3–1.4, one would like to extend the definition
of a contramodule from the topological algebras dual to coalgebras over fields
to topological rings of more general nature. Before proceeding to present the
desired definition, let us start with reintroducing the conventional modules
over a ring.

Given a (nontopological) associative ring $R$ with unit, one can define left
$R$-modules in the following fancy way. For any set $X$, denote by $R[X]$ the
set of formal linear combinations of elements of $X$ with coefficients in $R$
(i. e., the underlying set of the free $R$-module generated by $X$). The
assignment $X\longmapsto R[X]$ is a covariant functor from the category of
sets to itself. The key observation is that this functor has a natural
structure of a _monad_ [44, Chapter VI] on the category of sets.

In other words, for any set $X$ there is a natural map of “opening the
parentheses” $\phi_{X}\colon R[R[X]]\longrightarrow R[X]$, assigning a formal
linear combination of elements of $X$ to a formal linear combination of formal
linear combinations. There is also a natural map $\epsilon_{X}\colon
X\longrightarrow R[X]$ defined in terms of the zero and unit elements of the
ring $R$. The associativity and unitality axioms of a monad [44, Section VI.1]
are satisfied by these two natural transformations.

Given the endofunctor
$R[{-}]\colon{\mathsf{Sets}}\longrightarrow{\mathsf{Sets}}$ endowed with the
natural transformations $\phi$ and $\epsilon$, one can define a left
$R$-module as an algebra/module over this monad on the category of sets. In
other words, a left ${\mathfrak{R}}$-module $M$ is a set endowed with a map of
sets $m\colon R[M]\longrightarrow M$ satisfying the associativity and
unitality axioms from [44, Section VI.2]. Specifically, the two maps
$\phi_{M}$ and $R[m]\colon R[R[M]]\longrightarrow R[M]$ should have equal
compositions with the map $m$,

$R[R[M]]\rightrightarrows R[M]\longrightarrow M,$

while the composition of the map $\epsilon_{M}\colon M\longrightarrow R[M]$
with the map $m$ should be equal to the identity map ${\mathrm{id}}_{M}$,

$M\longrightarrow R[M]\longrightarrow M.$

Now let ${\mathfrak{R}}$ be an associative topological ring with unit. We will
have to assume that ${\mathfrak{R}}$ is complete and separated, and open right
ideals form a base of neighborhoods of zero in ${\mathfrak{R}}$. In other
words, the natural map
${\mathfrak{R}}\longrightarrow\varprojlim{\mathfrak{R}}/{\mathfrak{J}}$, where
${\mathfrak{J}}$ runs over all the open right ideals, must be a topological
isomorphism. Notice that these are precisely the assumptions under which the
discrete right ${\mathfrak{R}}$-modules are a good category to be assigned to
${\mathfrak{R}}$ (see the beginning of Section 1.4; cf. [9, Section 1.4]);
otherwise, a discrete right ${\mathfrak{R}}$-module is the same thing as a
discrete right module over the completion of ${\mathfrak{R}}$ in the new
topology with a base of neighborhoods of zero consisting of the open right
ideals in the original one.

For any set $X$, denote by ${\mathfrak{R}}[[X]]$ the set of all infinite
formal linear combinations $\sum_{x}r_{x}x$ of elements of $X$ with the
coefficients in ${\mathfrak{R}}$ for which the family of coefficients
$r_{x}\in{\mathfrak{R}}$ converges to zero in the topology of
${\mathfrak{R}}$. The latter condition means that for any neighborhood of zero
$U\subset{\mathfrak{R}}$ the set of all $x\in X$ for which $r_{x}\notin U$
must be finite. We will endow the functor
${\mathfrak{R}}[[{-}]]\colon{\mathsf{Sets}}\longrightarrow{\mathsf{Sets}}$
with the structure of a monad on the category of sets by defining an “opening
of infinite parentheses” map
$\phi_{X}\colon{\mathfrak{R}}[[{\mathfrak{R}}[[X]]]]\longrightarrow{\mathfrak{R}}[[X]]$
and a unit map $\epsilon_{X}\colon X\longrightarrow{\mathfrak{R}}[[X]]$.

In order to define the map $\phi_{X}$, one essentially has to show that the
infinite sums of products in ${\mathfrak{R}}$ that one obtains after opening
the parentheses converge in the topology of ${\mathfrak{R}}$. That is where
our assumptions about the topological ring ${\mathfrak{R}}$ have to be used.
Indeed, one has
${\mathfrak{R}}[[X]]=\varprojlim_{\mathfrak{J}}{\mathfrak{R}}/{\mathfrak{J}}[X]$,
where ${\mathfrak{J}}$ runs over all the open right ideals in ${\mathfrak{R}}$
(and the notation $A[X]$ for a set $X$ and an abelian group $A$ stands for the
group of all finite formal linear combinations of the elements of $X$ with
coefficients in $A$).

Defining the “opening of parentheses” map
${\mathfrak{R}}/{\mathfrak{J}}[{\mathfrak{R}}[[X]]]\longrightarrow{\mathfrak{R}}/{\mathfrak{J}}[X]$
does not involve any actual infinite summation, since
${\mathfrak{J}}\subset{\mathfrak{R}}$ is an open right ideal. It remains to
consider the composition
${\mathfrak{R}}[[{\mathfrak{R}}[[X]]]]\longrightarrow{\mathfrak{R}}/{\mathfrak{J}}[{\mathfrak{R}}[[X]]]\longrightarrow{\mathfrak{R}}/{\mathfrak{J}}[X]$
and pass to the projective limit over ${\mathfrak{J}}$. The unit map
$\epsilon_{X}$ is easy to define; one can say that it is the composition
$X\longrightarrow{\mathfrak{R}}[X]\longrightarrow{\mathfrak{R}}[[X]]$.
Checking the monad equations for the natural transformations $\phi$ and
$\epsilon$ is straightforward.

A _left ${\mathfrak{R}}$-contramodule_ ${\mathfrak{P}}$ is an algebra/module
over this monad on the category of sets. In other words, it is a set endowed
with an _${\mathfrak{R}}$ -contraaction_ map
$\pi\colon{\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$
satisfying the (contra)associativity and unitality equations together with the
natural transformations $\phi$ and $\epsilon$. Specifically, the two maps
$\phi_{\mathfrak{P}}$ and
${\mathfrak{R}}[[\pi]]\colon{\mathfrak{R}}[[{\mathfrak{R}}[[{\mathfrak{P}}]]]]\longrightarrow{\mathfrak{R}}[[{\mathfrak{P}}]]$
should have equal compositions with the contraaction map $\pi$,

${\mathfrak{R}}[[{\mathfrak{R}}[[{\mathfrak{P}}]]]]\rightrightarrows{\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}},$

while the composition of the map
$\epsilon_{\mathfrak{P}}\colon{\mathfrak{P}}\longrightarrow{\mathfrak{R}}[[{\mathfrak{P}}]]$
with the contraaction map should be equal to the identity map
${\mathrm{id}}_{\mathfrak{P}}$,

${\mathfrak{P}}\longrightarrow{\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}.$

This definition can be found in [52, Remark A.3] and [56, Section 1.2].

Notice that a systematic study of a class of monads on the category of sets,
called the _algebraic_ monads and viewed as “generalized rings”, was
undertaken by Durov in [22]. The definition above was in part inspired by
Durov’s work. However, the monad $X\longmapsto{\mathfrak{R}}[[X]]$ is _not_
algebraic, as the functor ${\mathfrak{R}}[[{-}]]$ does not preserve filtered
inductive limits of sets.

For any set $X$, the “opening of parentheses” map
$\pi=\phi_{X}\colon{\mathfrak{R}}[[{\mathfrak{R}}[[X]]]]\longrightarrow{\mathfrak{R}}[[X]]$
provides the set ${\mathfrak{R}}[[X]]$ with a natural left
${\mathfrak{R}}$-contramodule structure. The ${\mathfrak{R}}$-contramodules
${\mathfrak{R}}[[X]]$ are called the _free_ ${\mathfrak{R}}$-contramodules.
For the reasons common to all monads [44, Section VI.5], for any set $X$ and
any left ${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ there is a natural
bijection/isomorphism of abelian groups
$\operatorname{Hom}^{\mathfrak{R}}({\mathfrak{R}}[[X]],{\mathfrak{Q}})\simeq\operatorname{Hom}_{\mathsf{Sets}}(X,{\mathfrak{Q}})$,
where we denote by
$\operatorname{Hom}^{\mathfrak{R}}({\mathfrak{P}},{\mathfrak{Q}})$ the group
of morphisms from an object ${\mathfrak{P}}$ to an object ${\mathfrak{Q}}$ in
the category of ${\mathfrak{R}}$-contramodules.

Equivalently, an ${\mathfrak{R}}$-contramodule can be defined as a set endowed
with the following _infinite summation operations_. For any set of indices
$\\{\alpha\\}$, any family of elements $p_{\alpha}\in{\mathfrak{P}}_{\alpha}$,
and any family of coefficients $r_{\alpha}\in{\mathfrak{R}}$ converging to
zero in the topology of ${\mathfrak{R}}$, the element denoted symbolically by
$\sum_{\alpha}r_{\alpha}p_{\alpha}\in{\mathfrak{P}}$ must be defined. This
series of infinitary operations in the set ${\mathfrak{P}}$ should satisfy the
equations of contraassociativity

$\sum_{\alpha}r_{\alpha}\sum_{\beta}r_{\alpha\beta}p_{\alpha\beta}=\sum_{\alpha,\beta}(r_{\alpha}r_{\alpha\beta})p_{\alpha\beta}\quad\text{
if $r_{\alpha}\to 0$ and $\forall\alpha$ $r_{\alpha\beta}\to 0$ in
${\mathfrak{R}}$,}$

unitality

$\sum_{\alpha}r_{\alpha}p_{\alpha}=p_{\alpha_{0}}\quad\text{ if the set
$\\{\alpha\\}$ consists of one element $\alpha_{0}$ and $r_{\alpha_{0}}=1$},$

and distributivity

$\sum_{\alpha,\beta}r_{\alpha\beta}p_{\alpha}=\sum_{\alpha}\left(\sum_{\beta}r_{\alpha\beta}\right)p_{\alpha}\quad\text{
if $r_{\alpha\beta}\to 0$ in ${\mathfrak{R}}$}.$

Here the summation over double indices $\alpha,\beta$ presumes a set of pairs
$\\{(\alpha,\beta)\\}$ mapping into another set $\\{\alpha\\}$ by a map
denoted symbolically by $(\alpha,\beta)\longmapsto\alpha$ (i. e., the range of
possible $\beta$’s may depend on a chosen $\alpha$). The summation sign in the
parentheses in the third equation denotes the convergent sum in
${\mathfrak{R}}$, while all the other summation signs stand for the infinite
summation operation in ${\mathfrak{P}}$. Our conditions on the topology of
${\mathfrak{R}}$ guarantee that the family $r_{\alpha}r_{\alpha\beta}$
converges to zero whenever both the family $r_{\alpha}$ does and the families
$r_{\alpha\beta}$ do for every fixed $\beta$.

Restricting the summation operations to finite sets of indices $\\{\alpha\\}$,
one discovers that every left ${\mathfrak{R}}$-contramodule has an underlying
left ${\mathfrak{R}}$-module structure. Equivalently, one composes the
contraaction map
${\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$ with the
natural embedding
${\mathfrak{R}}[{\mathfrak{P}}]\longrightarrow{\mathfrak{R}}[[{\mathfrak{P}}]]$
in order to endow the underlying set of an ${\mathfrak{R}}$-contramodule
${\mathfrak{P}}$ with the structure of an ${\mathfrak{R}}$-module. We have
constructed the forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\allowbreak\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$
from the category of left ${\mathfrak{R}}$-contramodules
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ to the category
${\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$ of left modules over the ring
${\mathfrak{R}}$ viewed as an abstract (nontopological) ring.

For any discrete right ${\mathfrak{R}}$-module ${\mathcal{N}}$ and any abelian
group $V$, the group of all additive maps
$\operatorname{Hom}_{\mathsf{Ab}}({\mathcal{N}},V)$ has a natural left
${\mathfrak{R}}$-contramodule structure with the infinite summation operations
defined by the rule

$\left(\sum_{\alpha}r_{\alpha}p_{\alpha}\right)(x)=\sum_{\alpha}p_{\alpha}(xr_{\alpha})$

for any $p_{\alpha}\in{\mathfrak{P}}$, $x\in{\mathcal{N}}$ and a family of
coefficients $r_{\alpha}$ converging to zero in the topology of
${\mathfrak{R}}$. Here the summation sign in the right-hand side denotes the
sum of a family of elements in $V$ all but a finite number of which vanish, as
$xr_{\alpha}=0$ for all but a finite subset of indices $\alpha$.

For any topological ring ${\mathfrak{R}}$ the category
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$ of discrete left
${\mathfrak{R}}$-modules is abelian and the forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}\longrightarrow{\mathsf{Ab}}$
is exact. Filtered inductive limits are exact functors in
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$; they are also preserved by
the forgetful functor. In other words, the category
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$ satisfies the axioms Ab5 and
Ab3* (but _not_ in general Ab4*). It also admits a set of generators, so it
has enough injectives.

For any complete and separated topological ring ${\mathfrak{R}}$ with a base
of neighborhoods of zero formed by the open right ideals the category of left
${\mathfrak{R}}$-contramodules is abelian and the forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathsf{Ab}}$
is exact. To convince oneself that this is so, one uses the definition of
${\mathfrak{R}}$-contramodules in terms of infinite summation operations in
order to define the ${\mathfrak{R}}$-contramodule structures on the kernel and
cokernel of any morphism of ${\mathfrak{R}}$-contramodules taken in the
category of abelian groups. It helps to start from writing down the equations
of compatibility of the contramodule infinite summation operations with the
conventional finite operations in an abelian group or an
${\mathfrak{R}}$-module [56, Section 1.2].

Infinite products are exact functors in
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$; they are also preserved by
the forgetful functor. There are enough projective objects in
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$; an
${\mathfrak{R}}$-contramodule is projective if and only if it is a direct
summand of a free one. Infinite direct sums of free
${\mathfrak{R}}$-contramodules are computed by the rule
$\bigoplus_{\alpha}{\mathfrak{R}}[[X_{\alpha}]]={\mathfrak{R}}[[\coprod_{\alpha}X_{\alpha}]]$;
to compute the infinite direct sum of a family of arbitrary
${\mathfrak{R}}$-contramodules, one can present them as the cokernels of
morphisms of free contramodules and use the fact that infinite direct sums
commute with cokernes. Hence the category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ satisfies the axioms Ab3
and Ab4* (but _not_ in general Ab4).

The infinite products of discrete ${\mathfrak{R}}$-modules and the infinite
direct sums of ${\mathfrak{R}}$-contramodules are not preserved by the
respective forgetful functors in general. The embedding functor
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$
and the forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$
have the similar properties, the forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--mod}}}\longrightarrow{\mathsf{Ab}}$
preserving the inductive and projective limits of any diagrams.

The following version of _Nakayama’s lemma for discrete modules and
contramodules_ over a topological ring is one of their most important
properties.

######  Lemma.

(a) Let ${\mathfrak{R}}$ be a topological ring and
${\mathfrak{m}}\subset{\mathfrak{R}}$ be a topologically nilpotent ideal, i.
e., for any neighborhood of zero $U\subset{\mathfrak{R}}$ there exists an
integer $n\geqslant 1$ such that ${\mathfrak{m}}^{n}\subset U$. Then for any
nonzero discrete left ${\mathfrak{R}}$-module ${\mathcal{M}}$ the submodule
${}_{\mathfrak{m}}{\mathcal{M}}\subset{\mathcal{M}}$ of elements annihilated
by ${\mathfrak{m}}$ is nonzero.

(b) Let ${\mathfrak{R}}$ be a complete, separated topological ring with a base
of neighborhoods of zero formed by the open right ideals, and let
${\mathfrak{m}}\subset{\mathfrak{R}}$ be a topologically nilpotent closed
ideal. Then for any nonzero left ${\mathfrak{R}}$-contramodule
${\mathfrak{P}}$ the quotient contramodule
${\mathfrak{P}}/{\mathfrak{m}}{\mathfrak{P}}$ of ${\mathfrak{P}}$ by the image
${\mathfrak{m}}{\mathfrak{P}}$ of the contraaction map
${\mathfrak{m}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$ is nonzero.

Here the map ${\mathfrak{m}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$
is simply the restriction of the contraaction map
$\pi\colon{\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$ to
the subset
${\mathfrak{m}}[[{\mathfrak{P}}]]\subset{\mathfrak{R}}[[{\mathfrak{P}}]]$ of
all formal linear combinations with (converging families of) coefficients in
${\mathfrak{m}}$. Notice that this version of Nakayama’s lemma presumes _no_
finite generatedness condition on either the discrete module _or_ the
contramodule; on the other hand, it requires a rather strong global
topological nilpotency condition on the ideal ${\mathfrak{m}}$.

###### Proof.

Part (a): let $x\in{\mathcal{M}}$ be a nonzero element and
$U\subset{\mathfrak{R}}$ be its annihilator in ${\mathfrak{R}}$. The
${\mathfrak{R}}$-module ${\mathcal{M}}$ being discrete, $U$ is an open
neighborhood of zero in ${\mathfrak{R}}$; hence there exists an integer
$n\geqslant 1$ such that ${\mathfrak{m}}^{n}\subset U$, so
${\mathfrak{m}}^{n}x=0$. It remains to consider the maximal integer
$i\geqslant 0$ for which ${\mathfrak{m}}^{i}x\neq 0$; then
${\mathfrak{m}}^{i}x\subset{}_{\mathfrak{m}}{\mathcal{M}}$. The proof of part
(b) is a bit more complicated; see [52, Lemma A.2.1] and [56, Lemma 1.3.1]. ∎

###  2.2. Contramodules over the adic completions of Noetherian rings

Let $R$ be a right Noetherian associative ring, and let $m\subset R$ be an
ideal generated by central elements in $R$. Denote by
${\mathfrak{R}}=\varprojlim_{n}R/m^{n}$ the $m$-adic completion of the ring
$R$. In this section we explain how to describe the abelian category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ of left contramodules over
the complete ring ${\mathfrak{R}}$ viewed as a topological ring in the
projective limit topology (or, which is the same, the $m$-adic topology) in
terms of conventional modules over the original ring $R$.

######  Theorem.

The composition of forgetful functors
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--mod}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ provides a fully faithful embedding of
abelian categories
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$. A left $R$-module $P$ belongs to the full
subcategory ${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ if and only if any of the following
equivalent conditions holds:

(a) for any element $s\in m$ belonging to the center of the ring $R$ and any
$R[s^{-1}]$-module $L$ one has $\operatorname{Ext}_{R}^{i}(L,P)=0$ for all
$i\geqslant 0$;

(b) for any element $s\in m$ belonging to the center of the ring $R$ one has
$\operatorname{Ext}_{R}^{i}(R[s^{-1}],P)=0$ for all $i\geqslant 0$;

(c) for any element $s\in m$ one has
$\operatorname{Ext}^{*}_{{\mathbb{Z}}[t]}({\mathbb{Z}}[t,t^{-1}],P)=0$, where
${\mathbb{Z}}[t]$ denotes the ring of polynomials in one variable with
integral coefficients, ${\mathbb{Z}}[t,t^{-1}]$ is the ring of Laurent
polynomials, and $t$ acts in $P$ by the multiplication with $s$;

(d) for any $j=1$, …, $n$ and any $i=0$ or $1$ one has
$\operatorname{Ext}_{R}^{i}(R[s_{j}^{-1}],P)=0$, where $s_{1}$, …, $s_{j}$ is
a fixed set of central generators of the ideal $m\subset R$;

(e) for any $j=1$, …, $n$ and any $i=0$ or $1$ one has
$\operatorname{Ext}^{i}_{{\mathbb{Z}}[t]}({\mathbb{Z}}[t,t^{-1}],P)=0$, where
$t$ acts in $P$ by the multiplication with $s_{j}$.

In other words, an ${\mathfrak{R}}$-contramodule structure on a given left
${\mathfrak{R}}$-module is always _unique_ , and the theorem lists equivalent
conditions telling when it _exists_. Of course, for contramodules over
topological rings more complicated then the adic completions no such
description is in general possible.

In the case of a commutative ring $R$, the theorem essentially says that a
contramodule over the $m$-adic completion of $R$ is the same thing as a
_cohomologically $m$-adically complete $R$-module_ of Porta–Shaul–Yekutieli
[68, 69, 70]. A very brief sketch of the proof of the above theorem is
presented below; a detailed exposition can be found in [56, Appendix B] and
[57, Section C.5] (see also [52, Remark A.1.1]).

###### Sketch of proof.

First let us explain why any left $R$-module $P$ admitting a left
${\mathfrak{R}}$-contramodule structure satisfies the conditions (a) and (c).
The choice of an element $s\in m$ defines a continuous homomorphism of
topological rings ${\mathbb{Z}}[[t]]\longrightarrow{\mathfrak{R}}$, thus
endowing any left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ with a left
${\mathbb{Z}}[[t]]$-contramodule structure. One checks that this structure is
inherited by the groups
$\operatorname{Ext}^{i}_{{\mathbb{Z}}[t]}(L,{\mathfrak{P}})$ for any
${\mathbb{Z}}[t]$-module $L$ and, when $s$ is central in ${\mathfrak{R}}$, by
the groups $\operatorname{Ext}^{i}_{R}(L,{\mathfrak{P}})$ for any $R$-module
$L$. Now when $t$ or $s$ acts invertibly in $L$, the Ext groups in question
turn out to be ${\mathbb{Z}}[[t]]$-contramodules with an invertible action of
$t$, which have to vanish by the Nakayama Lemma 2.1(b) above (cf. [56, Section
B.2]).

Furthermore, for any central element $s\in R$ and any left $R$-module $P$ one
has
$\operatorname{Ext}_{R}^{i}(R[s^{-1}],P)\simeq\operatorname{Ext}_{{\mathbb{Z}}[t]}^{i}({\mathbb{Z}}[t,t^{-1}],P)$
and, of course, both groups always vanish for $i>1$ [56, Lemma B.7.1]. It
remains to show that any left $R$-module $P$ satisfying the condition (e) can
be endowed with a left ${\mathfrak{R}}$-contramodule structure in a unique
way. This is accomplished by the following sequence of lemmas.

Consider the topological ring of formal power series
${\mathfrak{T}}=R[[t_{1},\dotsc,t_{n}]]$ in the central variables $t_{1}$, …,
$t_{n}$ with coefficients in $R$; then there is a natural continuous ring
homomorphism ${\mathfrak{T}}\longrightarrow{\mathfrak{R}}$ taking $t_{j}$ to
$s_{j}$. Consider also the ring of polynomials $T=R[t_{1},\dotsc,t_{n}]$ and
the similar ring homomorphism $T\longrightarrow R$.

######  Lemma 1.

The ring homomorphism ${\mathfrak{T}}\longrightarrow{\mathfrak{R}}$ is
surjective, and its kernel ${\mathfrak{J}}$ is generated by the central
elements $s_{j}-t_{j}$ as an ideal in an abstract (nontopological) ring
${\mathfrak{T}}$. Moreover, any family of elements converging to zero in
${\mathfrak{R}}$ can be lifted to a family of elements converging to zero in
${\mathfrak{T}}$, and any family of elements in ${\mathfrak{J}}$ converging to
zero in the topology of ${\mathfrak{T}}$ can be presented as a linear
combination of $n$ families of elements in ${\mathfrak{T}}$, each of them
converging to zero, with the coefficients $s_{j}-t_{j}$.

###### Proof.

This is where the Noetherianness condition on the ring $R$ is being used; see
[56, Sections B.3–B.4] and [57, Lemma C.5.2]. ∎

######  Lemma 2.

The (contra)restriction of scalars functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{T}}{\operatorname{\mathsf{--contra}}}$
identifies the category of left ${\mathfrak{R}}$-contramodules with the full
subcategory in ${\mathfrak{T}}{\operatorname{\mathsf{--contra}}}$ consisting
of those left ${\mathfrak{T}}$-contramodules in which the elements
$s_{j}-t_{j}\in{\mathfrak{T}}$ act by zero.

###### Proof.

Follows from Lemma 1; see [56, Lemma B.4.1]. ∎

It is easy to interpret left ${\mathfrak{T}}$-contramodules as left
$R$-modules endowed with infinite summation operations with the coefficients
$t_{1}^{m_{1}}\dotsm t_{n}^{m_{n}}$ (see [56, proof of Lemma B.5.1]; cf.
Section 1.3 above). Hence it follows from Lemma 2 that the definition of
contramodules over the $l$-adic integers given in Section 1.4 is equivalent to
the general definition from Section 2.1 specialized to the case of
${\mathfrak{R}}={\mathbb{Z}}_{l}$.

######  Lemma 3.

The forgetful functor
${\mathfrak{T}}{\operatorname{\mathsf{--contra}}}\longrightarrow
T{\operatorname{\mathsf{--mod}}}$ identifies the category of left
${\mathfrak{T}}$-contramodules with the full subcategory in the category of
left $T$-modules consisting of all those modules $Q$ for which
$\operatorname{Ext}_{T}^{*}(T[t_{j}^{-1}],Q)=0$ for every $j=1$, …, $n$.

###### Proof.

The “unique recovering” argument here is just a more elaborated version of the
reasoning from Section 1.6 above. See [56, Sections B.5–B.7] and [57, Lemma
C.5.3] for the details. ∎

To finish the proof of the theorem, it remains to combine together the results
of Lemmas 2 and 3. ∎

Denote by ${\mathfrak{m}}=\varprojlim_{n}m/m^{n}\subset{\mathfrak{R}}$ the
extension of the ideal $m$ in the ring ${\mathfrak{R}}$. The following result
explains the term “cohomologically complete module” for left $R$-modules
satisfying the equivalent conditions of Theorem.

######  Proposition 1.

For any left $R$-contramodule ${\mathfrak{P}}$, the natural map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{m}}^{n}{\mathfrak{P}}\allowbreak=\varprojlim_{n}{\mathfrak{P}}/m^{n}{\mathfrak{P}}$
is surjective.

###### Proof.

See [52, Lemma A.2.3] or [57, Lemma D.1.1]. ∎

Notice that the natural functor
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is fully faithful for any topological ring
$R$ with a base of neighborhoods of zero consisting of open left ideals $J$
and its completion ${\mathfrak{R}}=\varprojlim_{J}R/J$. Moreover, when
${\mathfrak{R}}=\varprojlim_{n}R/m^{n}$ is the completion of the ring $R$ in
the adic topology of an ideal $m\subset R$ generated by a finite set of
central elements $s_{j}$, an $R$-module $M$ belongs to the full subcategory
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}\subset
R{\operatorname{\mathsf{--mod}}}$ if and only if it is _$m$ -torsion_, i. e.,
one has $R[s_{j}^{-1}]\otimes_{R}M=0$ for all $j$ or, which is the same,
$\operatorname{Tor}^{R}_{*}(R[s_{j}^{-1}],M)=0$ for all $j$.

Let us point out that the class of $m$-adically complete and separated left
$R$-modules, i. e., left $R$-modules $P$ for which the map
$P\longrightarrow\varprojlim_{n}P/m^{n}P$ is an isomorphism, does _not_ have
good homological properties. Indeed, it is _not_ preserved not only by the
passages to the cokernels of injective morphisms (see Section 1.5), but _also_
by extensions in $R{\operatorname{\mathsf{--mod}}}$ or
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ [61, Example 2.5]. The full
subcategory of left ${\mathfrak{R}}$-contramodules
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$, on the other hand, not only contains all
the $m$-adically complete and separated left $R$-modules, but is also closed
under the kernels, cokernels, extensions, and projective limits in
$R{\operatorname{\mathsf{--mod}}}$.

In the respective assumptions on the topological ring ${\mathfrak{R}}$, let us
denote by $\operatorname{Ext}_{\mathfrak{R}}^{*}({\mathcal{L}},{\mathcal{M}})$
the Ext groups in the abelian category
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$ and by
$\operatorname{Ext}^{{\mathfrak{R}},\text{$\mskip
1.5mu$}*}({\mathfrak{P}},{\mathfrak{Q}})$ the Ext groups in the abelian
category ${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$. The next
proposition shows that the embeddings of abelian categories
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ and
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ have good homological properties.

######  Proposition 2.

(a) Let $R$ be a left Noetherian ring, $m\subset R$ be an ideal generated by
central elements, and ${\mathfrak{R}}=\varprojlim_{n}R/m^{n}$ be the $m$-adic
completion of $R$. Then the embedding functor
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces isomorphisms on all the Ext groups,
$\operatorname{Ext}^{i}_{\mathfrak{R}}({\mathcal{L}},{\mathcal{M}})\simeq\operatorname{Ext}^{i}_{R}({\mathcal{L}},{\mathcal{M}})$
for all ${\mathcal{L}}$,
${\mathcal{M}}\in{\mathfrak{R}}{\operatorname{\mathsf{--discr}}}$ and all
$i\geqslant 0$.

(b) Let $R$ be a right Noetherian ring, $m\subset R$ be an ideal generated by
central elements, and ${\mathfrak{R}}$ be the $m$-adic completion of $R$. Then
the embedding functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces isomorphisms on all the Ext groups,
$\operatorname{Ext}^{{\mathfrak{R}},\text{$\mskip
1.5mu$}i}({\mathfrak{P}},{\mathfrak{Q}})\simeq\operatorname{Ext}_{R}^{i}({\mathfrak{P}},{\mathfrak{Q}})$
for all ${\mathfrak{P}}$,
${\mathfrak{Q}}\in{\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ and all
$i\geqslant 0$.

###### Proof.

Part (a): it follows from the Artin–Rees lemma (see [47, Theorem 8.5] and [32,
Theorems 1.9 and 13.3]) that the functor
${\mathfrak{R}}{\operatorname{\mathsf{--discr}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ preserves injectivity of objects (cf. [55,
Section A.3]), which is clearly sufficient. Part (b): one shows that the
functor ${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ takes free ${\mathfrak{R}}$-contramodules to
flat $R$-modules and all ${\mathfrak{R}}$-contramodules to relatively
cotorsion $R$-modules (see [56, Sections B.8–B.10] and [57, Propositions
C.5.4–C.5.5]). ∎

###  2.3. Contramodules over topological algebras over fields

The “set-theoretical” definition of ${\mathfrak{R}}$-contramodules given in
Section 2.1 is intended to incorporate “arithmetical” examples such as that of
the ring ${\mathfrak{R}}={\mathbb{Z}}_{l}$ of $l$-adic integers. In the case
of a topological algebra ${\mathfrak{R}}$ over a field $k$, the definition can
be simplified, facilitating the comparision with the notion of a contramodule
over a coalgebra ${\mathcal{C}}$ over $k$.

A topological vector space $V$ over a field $k$ is said to have a _linear
topology_ if its open vector subspaces form a base of neighborhoods of zero in
it. In the sequel, we presume all our topological vector spaces to have linear
topologies and, unless otherwise mentioned, to be complete and separated. In
other words, the natural map $V\longrightarrow\varprojlim_{U}V/U$, where $U$
runs over all the open vector subspaces in $V$, should be a topological
isomorphism (see [9] or [52, Section D.1]). Given a topological vector space
$V$ and an abstract (nontopological) vector space $P$ over a field $k$, we
denote by $V\otimes^{^}\,P$ the projective limit
$\varprojlim_{U}V/U\otimes_{k}P$ taken over all the open vector subspaces
$U\subset V$, viewed as an abstract (nontopological) vector space.

For any associative algebra $R$ over a field $k$, one can define left
$R$-modules as modules over the monad $M\longmapsto R\otimes_{k}M$ on the
category of $k$-vector spaces $M\in k{\operatorname{\mathsf{--vect}}}$. We
would like to extend this definition of $R$-modules to the case of topological
algebras over $k$. Let ${\mathfrak{R}}$ be a complete and separated
topological associative algebra over a field $k$ where open right ideal form a
base of neighborhoods of zero. Then the functor
$P\longmapsto{\mathfrak{R}}\otimes^{^}\,P$ has a natural structure of a monad
on the category of (nontopological) $k$-vector spaces $P\in
k{\operatorname{\mathsf{--vect}}}$. Indeed, let us construct the natural
transformations of multiplication and unit in this monad.

For any open right ideal ${\mathfrak{J}}\subset{\mathfrak{R}}$ the
multiplication map
${\mathfrak{R}}/{\mathfrak{J}}\times{\mathfrak{R}}\longrightarrow{\mathfrak{R}}/{\mathfrak{J}}$,
being continuous in the discrete topology of ${\mathfrak{R}}/{\mathfrak{J}}$
and the given topology of ${\mathfrak{R}}$, defines a structure of discrete
right ${\mathfrak{R}}$-module on the quotient space
${\mathfrak{R}}/{\mathfrak{J}}$, so the annihilator of every element of
${\mathfrak{R}}/{\mathfrak{J}}$ is an open right ideal in ${\mathfrak{R}}$.
Hence the multiplication map
${\mathfrak{R}}/{\mathfrak{J}}\otimes_{k}{\mathfrak{R}}\longrightarrow{\mathfrak{R}}/{\mathfrak{J}}$
induces a natural linear map
${\mathfrak{R}}/{\mathfrak{J}}\otimes_{k}({\mathfrak{R}}\otimes^{^}\,P)\longrightarrow{\mathfrak{R}}/{\mathfrak{J}}\otimes_{k}P$.
Composing this map with the projection
${\mathfrak{R}}\otimes^{^}\,({\mathfrak{R}}\otimes^{^}\,P)\longrightarrow{\mathfrak{R}}/{\mathfrak{J}}\otimes_{k}({\mathfrak{R}}\otimes^{^}\,P)$
and passing to the projective limit over open right ideals ${\mathfrak{J}}$,
we obtain the desired monad multiplication map
$\phi_{P}\colon{\mathfrak{R}}\otimes^{^}\,({\mathfrak{R}}\otimes^{^}\,P)\longrightarrow{\mathfrak{R}}\otimes^{^}\,P$.

The unit map $\epsilon_{P}\colon P\longrightarrow{\mathfrak{R}}\otimes^{^}\,P$
of our monad is obtained as the composition of the map
$P\longrightarrow{\mathfrak{R}}\otimes_{k}P$ induced by the unit element in
${\mathfrak{R}}$ with the completion map
${\mathfrak{R}}\otimes_{k}P\longrightarrow{\mathfrak{R}}\otimes^{^}\,P$.
Verifying the associativity and unitality axioms of a monad for the functor
${\mathfrak{R}}\otimes^{^}\,{-}\colon\allowbreak
k{\operatorname{\mathsf{--vect}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ endowed with the natural transformations
$\phi$ and $\epsilon$ is straightforward.

A _left ${\mathfrak{R}}$-contramodule_ ${\mathfrak{P}}$ is an algebra/module
over this monad on the category of $k$-vector spaces. In other words, it is a
$k$-vector space endowed with a _contraaction_ map
$\pi\colon{\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$
satisfying the following _contraassociativity_ and contraunitality equations.
Firstly, the two maps $\phi_{\mathfrak{P}}$ and
${\mathfrak{R}}\otimes^{^}\,\pi\colon{\mathfrak{R}}\otimes^{^}\,({\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}})\longrightarrow{\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}$
should have equal compositions with the contraaction map $\pi$,

${\mathfrak{R}}\otimes^{^}\,({\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}})\rightrightarrows{\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}.$

Secondly, the composition of the map
$\epsilon_{\mathfrak{P}}\colon{\mathfrak{P}}\longrightarrow{\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}$
with the contraaction map should be equal to the identity map
${\mathrm{id}}_{\mathfrak{P}}$,

${\mathfrak{P}}\longrightarrow{\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}.$

A _free ${\mathfrak{R}}$-contramodule_ is an ${\mathfrak{R}}$-contramodule of
the form ${\mathfrak{P}}={\mathfrak{R}}\otimes^{^}\,P$, where $P$ is a
$k$-vector space, with the contraaction map
$\pi=\phi_{P}\colon{\mathfrak{R}}\otimes^{^}\,({\mathfrak{R}}\otimes^{^}\,P)\longrightarrow{\mathfrak{R}}\otimes^{^}\,P$.
This definition of ${\mathfrak{R}}$-contramodules can be found in [52, Section
D.5.2].

For any discrete right ${\mathfrak{R}}$-module ${\mathcal{N}}$ and any
(nontopological) $k$-vector space $E$, the vector space
${\mathfrak{P}}=\operatorname{Hom}_{k}({\mathcal{N}},E)$ has a natural left
${\mathfrak{R}}$-contramodule structure provided by the contraaction map
$\pi\colon{\mathfrak{R}}\otimes^{^}\,\operatorname{Hom}_{k}({\mathcal{N}},E)\longrightarrow\operatorname{Hom}_{k}({\mathcal{N}},E)$
defined symbolically by the formula

$\pi(r\otimes^{^}\,f)(n)=f(nr),$

where $r\in R$, $n\in{\mathcal{N}}$,
$f\in\operatorname{Hom}_{k}({\mathcal{N}},E)$, and the expression in the
right-hand side makes sense, since the right action map
${\mathcal{N}}\otimes_{k}{\mathfrak{R}}\otimes_{k}\operatorname{Hom}_{k}({\mathcal{N}},E)\longrightarrow{\mathcal{N}}\otimes_{k}\operatorname{Hom}_{k}({\mathcal{N}},E)$
restricted to
$n\otimes{\mathfrak{R}}\otimes{\mathfrak{P}}\subset{\mathcal{N}}\otimes_{k}{\mathfrak{R}}\otimes_{k}{\mathfrak{P}}$
factorizes through the surjection
$n\otimes{\mathfrak{R}}\otimes{\mathfrak{P}}\longrightarrow
n\otimes{\mathfrak{R}}/{\mathfrak{J}}\otimes{\mathfrak{P}}$ for a certain open
right ideal ${\mathfrak{J}}\subset{\mathfrak{R}}$.

Let us show that our new definition of ${\mathfrak{R}}$-contramodules is
equivalent to the one from Section 2.1 in the case of a topological algebra
${\mathfrak{R}}$ over a field $k$. The following argument can be found in [56,
Section 1.10].

Recall that the category of $k$-vector spaces can be defined as the category
of algebras/modules over the monad $X\longmapsto k[X]$ on the category of
sets. Hence for any $k$-vector space $P$ there is a natural action map
$p\colon k[P]\longrightarrow P$. Moreover, this map is the coequalizer of the
pair of maps $k[[P]]\rightrightarrows k[P]$, one of which is the “opening of
parentheses” map $\phi_{P}$, while the other one is the map $k[p]$ induced by
the map $p$. Indeed, applying the forgetful functor
$k{\operatorname{\mathsf{--vect}}}\longrightarrow{\mathsf{Sets}}$ makes this
even a split coequalizer with an explicit splitting defined in terms of the
unit map of our monad [44, Sections VI.6–7]. Subtracting one of the maps in
the pair from the other one, we obtain an exact sequence in the category
$k{\operatorname{\mathsf{--vect}}}$

(5) $k[k[P]]\longrightarrow k[P]\longrightarrow P\longrightarrow 0$

for any $k$-vector space $P$.

Notice the natural isomorphism
${\mathfrak{R}}\otimes^{^}\,k[X]\simeq{\mathfrak{R}}[[X]]$ for any set $X$.
Furthermore, any additive functor on the category of $k$-vector spaces is
exact. Thus applying the functor ${\mathfrak{R}}\otimes^{^}\,{-}$ to the exact
sequence (5), we obtain an exact sequence

(6) ${\mathfrak{R}}[[k[P]]]\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{R}}[[P]]\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{R}}\otimes^{^}\,P\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip 1.5mu$}0$

for any $k$-vector space $P$. In particular, we have obtained a natural
surjective map
${\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}$
for any $k$-vector space ${\mathfrak{P}}$; composing it with the contraaction
map $R\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$ of a
contramodule ${\mathfrak{P}}$ over the topological $k$-algebra
${\mathfrak{R}}$, we obtain a contraaction map
${\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$ defining the
structure of a contramodule over the topological ring ${\mathfrak{R}}$ on the
set ${\mathfrak{P}}$.

Conversely, starting from the contraaction map
${\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$ of a
contramodule ${\mathfrak{P}}$ over the topological ring ${\mathfrak{R}}$, one
can first of all compose it with the natural embedding
$k[{\mathfrak{P}}]\longrightarrow{\mathfrak{R}}[[{\mathfrak{P}}]]$, defining a
$k$-vector space structure $k[{\mathfrak{P}}]\longrightarrow{\mathfrak{P}}$ on
the set ${\mathfrak{P}}$. Furthermore, restricting the contraassociativity
equation
${\mathfrak{R}}[[{\mathfrak{R}}[[{\mathfrak{P}}]]]]\rightrightarrows{\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$
to the subset
${\mathfrak{R}}[[k[{\mathfrak{P}}]]]\subset{\mathfrak{R}}[[{\mathfrak{R}}[[{\mathfrak{P}}]]]]$,
one discovers that the two maps
${\mathfrak{R}}[[k[{\mathfrak{P}}]]]\rightrightarrows{\mathfrak{R}}[[{\mathfrak{P}}]]$
have equal compositions with the contraaction map
${\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$. So we see
from the exact sequence (6) that the contraaction map
${\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$ factorizes
through the surjective map
${\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}$,
providing ${\mathfrak{P}}$ with a contraaction map
${\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$ of a
contramodule over the topological $k$-algebra ${\mathfrak{R}}$.

Of course, one still has to check that the map
${\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$
satisfies the contraassociativity and contraunitality equations if and only if
the corresponding map
${\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$ does. Here it
helps to notice that the natural map

(7) ${\mathfrak{R}}[[{\mathfrak{R}}[[{\mathfrak{P}}]]]]\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{R}}\otimes^{^}\,({\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}})$

is surjective, so any two maps
${\mathfrak{R}}\otimes^{^}\,({\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
are equal to each other whenever their compositions with the map (7) are. We
have also seen that the class of free ${\mathfrak{R}}$-contramodules as
defined in this section coincides with the one introduced in Section 2.1 when
${\mathfrak{R}}$ is a topological algebra over a field.

Now we can finally compare our notion of a contramodule over a topological
ring/topological algebra over a field $k$ with the definition of a
contramodule over a (coassociative) coalgebra ${\mathcal{C}}$ over $k$ given
in Section 1.1. Let ${\mathfrak{R}}={\mathcal{C}}^{*}$ be the dual vector
space to the coalgebra ${\mathcal{C}}$ endowed with its pro-finite-dimensional
topological algebra structure (see Sections 1.3–1.4). Then there is a natural
isomorphism $R\otimes^{^}\,P\simeq\operatorname{Hom}_{k}({\mathcal{C}},P)$ for
any $k$-vector space $P$, making an ${\mathfrak{R}}$-contraaction map
${\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$ the
same thing as a ${\mathcal{C}}$-contraaction map
$\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$.

The vector spaces
${\mathfrak{R}}\otimes^{^}\,({\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}})$ and
$\operatorname{Hom}_{k}({\mathcal{C}},\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}}))$
parametrizing the systems of contraassociativity equations on the two kinds of
contraaction maps being also naturally isomorphic, one easily checks that a
map ${\mathfrak{R}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$
defines a left ${\mathfrak{R}}$-contramodule structure on a $k$-vector space
${\mathfrak{P}}$ if and only if the corresponding map
$\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
defines a left ${\mathcal{C}}$-contramodule structure on ${\mathfrak{P}}$.

###  2.4. Contramodules over topological Lie algebras

The definition of the category of contramodules over the Virasoro algebra
given in Section 1.7 calls for a generalization to a reasonably large class of
topological Lie algebras.

There are some naïve approaches: for example, it is easy to define comodules
and contramodules over Lie coalgebras in the way analogous to the definitions
for coassociative coalgebras explained in Section 1.1. This provides the
notion of a contramodule over a linearly compact topological Lie algebra.
Notice that the class of all Lie coalgebras is in some sense not as narrow as
that of coassociative coalgebras over fields: unlike in the coassociative case
(see Section 1.3), a Lie coalgebra does not have to be the union of its
finite-dimensional subcoalgebras.

Indeed, it suffices to consider the case of the Lie coalgebra ${\mathcal{L}}$
dual to the linearly compact Lie algebra $k[[z]]d/dz$ over a field $k$ of zero
characteristic, that is the Lie subalgebra topologically spanned by the
generators $L_{-1}$, $L_{0}$, $L_{1}$, $L_{2}$ … in the algebra $k((z))d/dz$.
The Lie algebra $k[[z]]d/dz$ having no nonzero proper closed ideals, the Lie
coalgebra ${\mathcal{L}}$ has no nonzero proper subcoalgebras at all.
Nevertheless, the class of linearly compact Lie algebras does not even contain
the Virasoro algebra.

So let us start with the class of locally linearly compact, or _Tate_ Lie
algebras. A topological vector space $V$ is said to be _locally linearly
compact_ , or a _Tate vector space_ if it has a linearly compact open
subspace, or equivalently, if (linearly) compact open subspaces form a base of
neighborhoods of zero in $V$. In other words, a topological vector space is a
Tate vector space if it is topologically isomorphic to the direct sum of a
compact vector space and a discrete vector space (see [9, Sections 1.1–1.2 and
the references therein] and [52, Section D.1.1]).

A _Tate Lie algebra_ ${\mathfrak{g}}$ is a Tate vector space endowed with a
continuous Lie algebra structure, i. e., a Lie bracket
${\mathfrak{g}}\times{\mathfrak{g}}\longrightarrow{\mathfrak{g}}$ that is
continuous as a function of two variables. Any Tate Lie algebra has a base of
neighborhoods of zero consisting of open Lie subalgebras (see footnotes in [7,
Section 3.8.17] or [9, Section 1.4], or a paragraph in [52, Section D.1.8]).
For example, the “Laurent totalization”
${\mathfrak{g}}=\bigoplus_{n<0}{\mathfrak{g}}_{n}\oplus\prod_{n\geqslant
0}{\mathfrak{g}}_{n}$ of any ${\mathbb{Z}}$-graded Lie algebra
$\bigoplus_{n\in{\mathbb{Z}}}{\mathfrak{g}}_{n}$ with finite-dimensional
components ${\mathfrak{g}}_{n}$ is a Tate Lie algebra with compact open
subalgebras $\prod_{i\geqslant n}{\mathfrak{g}}_{i}\subset{\mathfrak{g}}$,
$n\geqslant 0$, forming a base of neighborhoods of zero. This includes such
classical examples as the Virasoro and Kac–Moody Lie algebras.

A _contramodule_ ${\mathfrak{P}}$ over a Tate Lie algebra ${\mathfrak{g}}$
over a field $k$ is a $k$-vector space endowed with a _contraaction_ map
${\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$
satisfying the following (system of) contra-Jacobi equation(s). Given a
compact vector space $V$, denote by $V^{\vee}$ the discrete vector space to
dual to $V$, so that $V^{\vee}{}^{*}$ is isomorphic to $V$ as a topological
vector space. For any abstract (nontopological) vector space $P$, there is
then a natural isomorphism of (nontopological) vector spaces
$V\otimes^{^}\,P\simeq\operatorname{Hom}_{k}(V^{\vee},P)$. So in particular we
have
${\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}}\simeq\varinjlim_{V}\operatorname{Hom}_{k}(V^{\vee},{\mathfrak{P}})$,
where $V$ runs over all the compact vector subspaces $V\subset{\mathfrak{g}}$.

Now let $U$, $V$, and $W\subset{\mathfrak{g}}$ be three compact vector
subspaces for which $[U,V]\subset W$; then there is a natural cobracket map
$W^{\vee}\longrightarrow V^{\vee}\otimes_{k}U^{\vee}$. It required that the
composition

$\operatorname{Hom}_{k}(V\otimes_{k}U,\medspace{\mathfrak{P}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}(W,{\mathfrak{P}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{P}}$

of the map induced by the cobracket map with the contraaction map should be
equal to the difference of the interated contraaction maps

$\displaystyle\operatorname{Hom}_{k}(V\otimes_{k}U,\medspace{\mathfrak{P}})$
$\displaystyle\text{$\mskip 1.5mu$}\simeq\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}(U,\operatorname{Hom}_{k}(V,{\mathfrak{P}}))$
$\displaystyle\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}(U,{\mathfrak{P}})$ $\displaystyle\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{P}}$ and
$\displaystyle\operatorname{Hom}_{k}(V\otimes_{k}U,\medspace{\mathfrak{P}})$
$\displaystyle\text{$\mskip 1.5mu$}\simeq\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}(V,\operatorname{Hom}_{k}(U,{\mathfrak{P}}))$
$\displaystyle\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}(V,{\mathfrak{P}})$ $\displaystyle\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{P}}.$

This definition can be found in [52, Section D.2.7]. Contramodules over Tate
Lie algebras serve as the coefficients for the theory of _semi-infinite
cohomology_ of Lie algebras (as opposed to the semi-infinite _homology_ [29],
[7, Section 3.8]); see [52, Section D.5.6] for the definition and Section 2.8
below for a brief overview.

In order to extend the definition of a ${\mathfrak{g}}$-contramodule to
topological Lie algebras ${\mathfrak{g}}$ of more general nature, we need to
introduce a bit more topological linear algebra background. The following
three _topological tensor product_ operations were defined in [9, Section 1.1]
(see also [52, Section D.1.3]).

For any topological vector spaces $V$ and $W$, the $!$-tensor product
$V\otimes^{!}W$ is the completion of the tensor product $V\otimes_{k}W$ with
respect to the topology with a base of neighborhoods of zero formed by the
subspaces $V^{\prime}\otimes W+V\otimes W^{\prime}\subset V\otimes_{k}W$,
where $V^{\prime}\subset V$ and $W^{\prime}\subset W$ are open vector
subspaces. In other words, one has
$V\otimes^{!}W=\varprojlim_{V^{\prime},W^{\prime}}V/V^{\prime}\otimes_{k}\nobreak
W/W^{\prime}$, with the projective limit topology.

Furthermore, the $*$-tensor product $V\otimes^{*}W$ is the completion of the
tensor product $V\otimes_{k}W$ with respect to the topology in which a vector
subspace $T\subset V\otimes_{k}W$ is open if and only if it satisfies the
following three conditions:

  1. (i)

there exist open vector subspaces $V^{\prime}\subset V$, $W^{\prime}\subset W$
such that $V^{\prime}\otimes_{k}W^{\prime}\subset T$;

  2. (ii)

for any vector $v\in V$ there exists an open subspace $W^{\prime\prime}\subset
W$ such that $v\otimes W^{\prime\prime}\subset T$;

  3. (iii)

for any vector $w\in W$ there exists an open subspace $V^{\prime\prime}\subset
V$ such that $V^{\prime\prime}\otimes w\subset T$.

For any topological vector space $U$, a bilinear map $V\times W\longrightarrow
U$ is continuous (as a function of two variables) if and only if it can be
(always uniquely) extended to a continuous linear map
$V\otimes^{*}W\longrightarrow U$.

Finally, the $\overset{\leftarrow}{\ }$-tensor product
$V\mathbin{\overset{\leftarrow}{\vphantom{o}\smash{\otimes}}}W$ is the
completion of the tensor product $V\otimes_{k}W$ with respect to the topology
in which a vector subspace $T\subset V\otimes_{k}W$ is open if and only if it
satisfies the following two conditions:

  1. (i)

there exists an open vector subspace $V^{\prime}\subset V$ such that
$V^{\prime}\otimes_{k}W\subset T$;

  2. (ii)

for any vector $v\in V$ there exists an open subspace $W^{\prime\prime}\subset
W$ such that $v\otimes W^{\prime\prime}\subset T$.

The underlying abstract (nontopological) vector space of the topological
tensor product $V\mathbin{\overset{\leftarrow}{\vphantom{o}\smash{\otimes}}}W$
does not depend on the topology on $W$ and is naturally isomorphic to the
completed tensor product $V\otimes^{^}\,W$ introduced in Section 2.3. The
multiplication map
${\mathfrak{R}}\times{\mathfrak{R}}\longrightarrow{\mathfrak{R}}$ of a
(complete and separated) topological associative algebra ${\mathfrak{R}}$ can
be (uniquely) extended to a continuous linear map
${\mathfrak{R}}\mathbin{\overset{\leftarrow}{\vphantom{o}\smash{\otimes}}}{\mathfrak{R}}\longrightarrow{\mathfrak{R}}$
if and only if open right ideals form a base of neighborhoods of zero in
${\mathfrak{R}}$ [9, Section 1.4].