# A bounded below, noncontractible,
acyclic complex of projective modules

Leonid Positselski  Institute of Mathematics, Czech Academy of Sciences
Žitná 25, 115 67 Prague 1
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

We construct examples of bounded below, noncontractible, acyclic complexes of
finitely generated projective modules over some rings $S$, as well as bounded
above, noncontractible, acyclic complexes of injective modules. The rings $S$
are certain rings of infinite matrices with entries in the rings of
commutative polynomials or formal power series in infinitely many variables.
In the world of comodules or contramodules over coalgebras over fields,
similar examples exist over the cocommutative symmetric coalgebra of an
infinite-dimensional vector space. A simpler, universal example of a bounded
below, noncontractible, acyclic complex of free modules with one generator,
communicated to the author by Canonaco, is included at the end of the paper.

###### Contents

  1. 1 Projective, Flat, and Injective Bounded Acyclicity Problems
  2. 2 The Injective Construction of Acyclic Complex of Projectives
  3. 3 Dual Rickard’s Acyclicity Theorem
  4. 4 The Projective Construction of Acyclic Complex of Projectives
  5. 5 Brief Preliminaries on Coalgebras
  6. 6 Comodule and Contramodule Acyclicity Theorems
  7. 7 Two Contramodule Constructions of Acyclic Complexes of Projectives
  8. 8 Summary of the Examples Obtained

## Introduction

Bounded above acyclic complexes of projective objects are contractible. So are
bounded below acyclic complexes of injective objects. On the other hand, there
is an easy, thematic example of a doubly unbounded, acyclic, noncontractible
complex of finitely generated projective-injective modules over the algebra of
dual numbers $R=k[\epsilon]/(\epsilon^{2})$ (over any field $k$):

(1) $\dotsb\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muR\overset{\epsilon*}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}R\overset{\epsilon*}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu}R\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

We refer to [9, Prologue], [10, Sections 7.4–7.5] and the references therein
for a discussion of the role of the complex (1) in the context of derived
Koszul duality and derived categories of the second kind.

Do there exist bounded below, noncontractible, acyclic complexes of projective
modules; and if so, under what rings? Dual-analogously, are there any bounded
above, noncontractible, acyclic complexes of injective modules? These
questions were posed, in the context of potential applications to the
Finitistic Dimension Conjecture, in the recent preprint of Shaul [16].
According to [16, Theorem 5.1], nonexistence of such complexes of
projective/injective modules over a two-sided Noetherian ring $S$ with a
dualizing complex would imply finiteness of the finitistic dimensions of $S$.

The aim of the present paper is to show that, over certain rather big rings
$S$, such complexes do exist. The examples of rings $S$ which we obtain are
certainly noncommutative and non-Noetherian. The more explicit ones among them
are rings of column-finite or row/column-zero-convergent infinite matrices
with entries in the rings of commutative polynomials or formal power series in
infinitely many variables.

On the other hand, in the world of coalgebras over fields, we demonstrate
examples of bounded above, acyclic, noncontractible complexes of injective
comodules and bounded below, acyclic, noncontractible complexes of projective
contramodules over certain _cocommutative_ coalgebras dual to algebras of
formal power series in infinitely many variables. These examples go back to
[6, Section 0.2.7], where they were very briefly discussed in the context of
semi-infinite homological algebra and derived comodule-contramodule
correspondence.

Almost all the examples presented in this paper are based on one idea, namely,
that of the dual Koszul complex of the ring of polynomials in infinitely many
variables. A straightforward realization of this idea is possible in the
worlds of comodules and contramodules, but we need an additional trick with a
passage to infinite matrices in order to produce examples of complexes of
projective/injective modules. The only exception is the (much simpler)
_universal_ example, communicated to the author by A. Canonaco. We reproduce
it at the end of the paper in Example 8.4.

The approach to the Finitistic Dimension Conjecture developed in [15, 16] goes
back to Rickard’s paper [14], where it was shown that if the injective modules
over a finite-dimensional algebra generate its unbounded derived category as a
triangulated category with coproducts, then the finitistic dimension is
finite. A counterexample in [14, Theorem 3.5] shows that for the ring of
commutative polynomials in infinitely many variables, the generation property
fails. Our examples in this paper follow in the footsteps of [6, Section
0.2.7] and [14, Theorem 3.5]. We also provide some details of the claims in
[6, Section 0.2.7] which were skipped in the book [6].

### Acknowledgement

This paper was inspired by Liran Shaul’s talk at the Algebra seminar in
Prague, organized by Jan Trlifaj. I want to thank both the speaker and the
organizer of the seminar. I also wish to thank Alberto Canonaco for
communicating his example to me and giving a kind permission to reproduce it
here (see Example 8.4). The author is supported by the GAČR project 23-05148S
and the Czech Academy of Sciences (RVO 67985840).

##  1\. Projective, Flat, and Injective Bounded Acyclicity Problems

The general convention in this paper is that complexes are presumed to be
cohomologically graded, so the differential raises the degree. A complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(C^{n},\>d_{n}\colon
C^{n}\to C^{n+1})$ is called _bounded above_ if $C^{n}=0$ for $n\gg 0$, and
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is _bounded below_ if
$C^{n}=0$ for $n\ll 0$. In this notation, it is a standard fact that every
bounded above acyclic complex of projective modules/objects (in an abelian or
exact category) is contractible, and every bounded below acyclic complex of
injective modules/objects is contractible. When we occasionally consider
homologically graded complexes, we use the notation with lower indices,
$P_{\text{\smaller\smaller$\scriptstyle\bullet$}}=(P_{n},\>d_{n}\colon
P_{n}\to P_{n-1})$.

Let $S$ be an associative ring. The two “wrong-sided bounded
projective/injective acyclicity problems” posed in [16, Theorem 5.1(4–5)] are:

  * •

Is every bounded above acyclic complex of injective $S$-modules contractible?

  * •

Is every bounded below acyclic complex of projective $S$-modules contractible?

In addition to the above two, we would like to ask a similar question about
flat $S$-modules. Here one has to be careful: even a two-sided bounded acyclic
complex of flat modules need not be contractible. However, such a complex is
always _pure acyclic_ , or in other words, has flat modules of cocycles. Thus
we ask:

  * •

Is every bounded below acyclic complex of flat $S$-modules pure acyclic?

Given a ring $S$ and a left $S$-module $M$, the _character module_
$M^{+}=\operatorname{Hom}_{\mathbb{Z}}(M,\mathbb{Q}/\mathbb{Z})$ is a right
$S$-module. The following lemma is well-known.

######  Lemma 1.1.

A left $S$-module $F$ is flat if and only if the right $S$-module $F^{+}$ is
injective. ∎

The next proposition explains the connection between the injective, flat, and
projective wrong-sided bounded acyclicity questions, and shows that presenting
a counterexample to the “projective” question is enough.

######  Proposition 1.2.

Given a ring $S$, consider the following three properties:

  1. (1)

Every bounded above acyclic complex of injective right $S$-modules is
contractible.

  2. (2)

Every bounded below acyclic complex of flat left $S$-modules is pure acyclic.

  3. (3)

Every bounded below acyclic complex of projective left $S$-modules is
contractible.

Then the implications (1) $\Longrightarrow$ (2) $\Longrightarrow$ (3) hold.

###### Proof.

(1) $\Longrightarrow$ (2) Let
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(0\to F^{0}\to F^{1}\to
F^{2}\to\dotsb)$ be a bounded below acyclic complex of flat left $S$-modules.
Then, by the direct implication of Lemma 1.1,
$F^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}=(\dotsb\to F^{2,+}\to
F^{1,+}\to F^{0,+}\to 0)$ is a bounded above acyclic complex of injective
right $S$-modules. A complex of injective modules is contractible if and only
if its modules of cocycles are injective. If this is the case for the complex
$F^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}$, then the inverse
implication of Lemma 1.1 tells that the modules of cocycles of the complex
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are flat; so
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a pure acyclic complex
of flat modules.

(2) $\Longrightarrow$ (3) By Neeman’s theorem [5, Theorem 8.6 (iii)
$\Rightarrow$ (i)], any pure acyclic complex of projective modules is
contractible. (Cf. [16, proof of Theorem A.7].) ∎

##  2\. The Injective Construction of Acyclic Complex of Projectives

Let $k$ be a field, $(x_{\alpha})_{\alpha\in A}$ be an infinite set of
variables, and $R=k[x_{\alpha}:\alpha\in A]$ be the commutative ring of
polynomials in the variables $x_{\alpha}$ over $k$. Endow the one-dimensional
vector space $k$ over $k$ with the $R$-module structure by the obvious rule:
all the elements $x_{\alpha}\in R$ act by zero in $k$.

######  Theorem 2.1 (Rickard).

For any injective $R$-module $J$ and all integers $n\geq 0$, one has
$\operatorname{Ext}_{R}^{n}(J,k)=0$.

###### Proof.

For a countably infinite set of variables $x_{\alpha}$, this is formulated and
proved in [14, Theorem 3.5]. The general case of a possibly uncountable index
set $A$ is similar. One represents $A$ as the union of its finite subsets
$B\subset A$, so the ring $R$ the direct limit of the related polynomial rings
$R_{B}$ in finitely many variables, considers the direct limit of finite
Koszul complex indexed by the finite subsets $B\subset A$, etc. (Cf. the proof
of Theorem 3.1 below for some further details.) ∎

Let $\mathsf{A}$ be an additive category and $M\in\mathsf{A}$ be an object.
Then we denote by $\mathsf{add}(M)$ the full subcategory in $\mathsf{A}$
formed by the direct summands of finite direct sums of copies of $M$. The
following lemma is a straightforward category-theoretic generalization of a
well-known module-theoretic observation going back to Dress [3].

######  Lemma 2.2.

Let $\mathsf{A}$ be an idempotent-complete additive category and
$M\in\mathsf{A}$ be an object.

(a) Let $S=\operatorname{Hom}_{\mathsf{A}}(M,M)^{\mathrm{op}}$ be the opposite
ring to the endomorphism ring of the object $M\in\mathsf{A}$; so the ring $S$
acts on the object $M$ on the right. Then the covariant functor
$\operatorname{Hom}_{\mathsf{A}}(M,{-})\colon\mathsf{A}\longrightarrow
S{\operatorname{\mathsf{--Mod}}}$ restricts to an equivalence of additive
categories

$\operatorname{Hom}_{\mathsf{A}}(M,{-})\colon\mathsf{add}(M)\simeq
S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$

between the full subcategory $\mathsf{add}(M)\subset\mathsf{A}$ and the full
subcategory of finitely generated projective left $S$-modules
$S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$ in the category of left
$S$-modules $S{\operatorname{\mathsf{--Mod}}}$.

(b) Let $S=\operatorname{Hom}_{\mathsf{A}}(M,M)$ be the endomorphism ring of
the object $M\in\mathsf{A}$; so the ring $S$ acts on the object $M$ on the
left. Then the contravariant functor
$\operatorname{Hom}_{\mathsf{A}}({-},M)\colon\mathsf{A}^{\mathsf{op}}\longrightarrow
S{\operatorname{\mathsf{--Mod}}}$ restricts to an anti-equivalence of additive
categories

$\operatorname{Hom}_{\mathsf{A}}({-},M)\colon\mathsf{add}(M)^{\mathsf{op}}\simeq
S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$

between the full subcategory $\mathsf{add}(M)\subset\mathsf{A}$ and the full
subcategory of finitely generated projective left $S$-modules
$S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}\subset
S{\operatorname{\mathsf{--Mod}}}$. ∎

The following corollary sums up the “injective coresolution construction of a
bounded below acyclic complex of projective modules”.

######  Corollary 2.3.

Let $R=k[x_{\alpha}:\alpha\in A]$ be the ring of polynomials in infinitely
many variables over a field $k$, and let

(2) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muk\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muJ^{0}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muJ^{1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muJ^{2}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

be an injective coresolution of the one-dimensional $R$-module $k$. Let $J$ be
an injective $R$-module such that the $R$-module $J^{n}$ is a direct summand
of $J$ for all $n\geq 0$. Let

(3) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(J,J^{0})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(J,J^{1})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(J,J^{2})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

be the complex obtained by applying the functor
$\operatorname{Hom}_{R}(J,{-})$ to the truncated coresolution (2). Then (3) is
a bounded below, noncontractible, acyclic complex of finitely generated
projective left modules over the ring
$S=\operatorname{Hom}_{R}(J,J)^{\mathrm{op}}$.

###### Proof.

The complex (3) is acyclic by Theorem 2.1. The left $S$-module
$\operatorname{Hom}_{R}(J,J^{n})$ is a direct summand of the left $S$-module
$\operatorname{Hom}_{R}(J,J)=S$ for every $n\geq 0$, since the $R$-module
$J^{n}$ is a direct summand of $J$. So (3) is even a complex of cyclic
projective left $S$-modules (i. e., projective $S$-modules with one
generator).

It remains to explain why the complex of $S$-modules (3) is not contractible.
For this purpose, one observes that, given a full subcategory $\mathsf{B}$ in
an additive category $\mathsf{A}$, a complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in $\mathsf{B}$ is
contractible in $\mathsf{A}$ if and only if it is contractible in
$\mathsf{B}$. Indeed, any contracting homotopy for
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ as a complex in
$\mathsf{A}$ would be a collection of morphisms in $\mathsf{A}$ between
objects from $\mathsf{B}$, which means a collection of morphisms in
$\mathsf{B}$.

By Lemma 2.2(a) (for $\mathsf{A}=R{\operatorname{\mathsf{--Mod}}}$ and $M=J$),
the functor $\operatorname{Hom}_{R}(J,{-})$ is an equivalence of categories
$\mathsf{add}(J)\simeq S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$. The
truncated coresolution (2),

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muJ^{0}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muJ^{1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muJ^{2}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

is a noncontractible (since nonacyclic) complex in
$R{\operatorname{\mathsf{--Mod}}}$ with the terms belonging to
$\mathsf{add}(J)$, so it is a noncontractible complex in $\mathsf{add}(J)$.
Applying the equivalence of additive categories $\mathsf{add}(J)\simeq
S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$, we obtain a noncontractible
complex (3) in $S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$, which is
consequently also noncontractible in $S{\operatorname{\mathsf{--Mod}}}$. It is
important for this argument that the functor
$\operatorname{Hom}_{R}(J,{-})\colon\mathsf{add}(J)\longrightarrow
S{\operatorname{\mathsf{--Mod}}}$ is fully faithful. ∎

##  3\. Dual Rickard’s Acyclicity Theorem

The aim of this section is to prove the following dual version of Rickard’s
theorem [14, Theorem 3.5].

######  Theorem 3.1.

Let $R=k[x_{\alpha}:\alpha\in A]$ be the ring of polynomials in infinitely
many variables over a field $k$. As above, we endow the one-dimensional
$k$-vector space $k$ with the obvious $R$-module structure. Then, for any flat
$R$-module $P$ and all integers $n\geq 0$, one has
$\operatorname{Ext}^{n}_{R}(k,P)=0$.

###### Proof.

For every $\alpha\in A$, consider the two-term Koszul complex of free
$R$-modules with one generator

(4) $\dotsb\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muR\overset{x_{\alpha}*}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu}R\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

situated in the cohomological degrees $-1$ and $0$. For every finite subset of
indices $B\subset A$, denote by
$K^{B}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R)$ the tensor product,
taken over the ring $R$, of the complexes (4) with $\alpha\in B$. As a finite
subset $B\subset A$ varies, the complexes
$K^{B}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R)$ form an inductive
system, indexed by the poset of all finite subsets $B\subset A$ ordered by
inclusion.

Put $K_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R)=\varinjlim_{B\subset
A}K^{B}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R)$. Then
$K_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R)$ is a bounded above
complex of free $R$-modules. One has $K_{n}(R)=0$ for $n<0$, $K_{0}(R)=R$, and
$K_{n}(R)$ is a free $R$-module with a set of generators of the cardinality
equal to the cardinality of the set $A$ for all $n>0$. (More invariantly,
$K_{n}(R)$ is the free $R$-module spanned by the set of all subsets in $A$ of
the finite cardinality $n$).

For any finite subset $B\subset A$, the complex
$K^{B}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(A)$ is a finite
resolution of the $R$-module $R/\sum_{\alpha\in B}x_{\alpha}R$ by finitely
generated free $R$-modules. Passing to the direct limit, one can easily see
that $K_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R)$ is a free
$R$-module resolution of the one-dimensional $R$-module $k=R/\sum_{\alpha\in
A}x_{\alpha}R$.

The following three lemmas are straightforward or standard.

######  Lemma 3.2.

Let $R$ be an associative ring, $\Xi$ be a directed poset, and
$(F_{\xi})_{\xi\in\Xi}$ be an inductive system of projective $R$-modules whose
direct limit $F=\varinjlim_{\xi\in\Xi}F_{\xi}$ is also a projective
$R$-module. Let $P$ be an arbitrary $R$-module. Then the higher derived
inverse limit functors vanish on the projective system
$\operatorname{Hom}_{R}(F_{\xi},P)_{\xi\in\Xi}$,

$\varprojlim\nolimits_{\xi\in\Xi}\operatorname{Hom}_{R}(F_{\xi},P)=\operatorname{Hom}_{R}(F,P)\quad\text{and}\quad\varprojlim\nolimits^{i}_{\xi\in\Xi}\operatorname{Hom}_{R}(F_{\xi},P)=0\quad\text{for
all\/ $i\geq 1$}.$

###### Proof.

Dropping the condition that the $R$-module $F=\varinjlim_{\xi\in\Xi}F_{\xi}$
is projective (but keeping the conditions that the $R$-modules $F_{\xi}$ are
projective), one would have
$\varprojlim\nolimits^{i}_{\xi\in\Xi}\operatorname{Hom}_{R}(F_{\xi},P)=\operatorname{Ext}^{i}_{R}(F,P)$
for every $i\geq 0$. ∎

######  Lemma 3.3.

Let $A$ be an infinite set and $(V_{B})_{B\subset A}$ be a projective system
of abelian groups, indexed by the poset of all finite subsets $B\subset A$
ordered by inclusion. Assume that there exists an integer $n\geq 0$ such that
$V_{B}=0$ whenever the cardinality of $B$ exceeds $n$. Then the whole derived
inverse limit functor vanishes on the projective system $(V_{B})_{B\subset
A}$,

$\varprojlim\nolimits^{i}_{B\subset A}V_{B}=0\qquad\text{for all\/ $i\geq
0$}.$

###### Proof.

This is a special case of the assertion that the derived functors of inverse
limit are preserved by the passage to a cofinal subsystem. This can be deduced
from fact that the derived inverse limits vanish on so-called weakly flabby
(faiblement flasque) projective systems [4, Théorème 1.8]. A stronger result
that the derived inverse limit (in an abelian category with exact product
functors) only depends on the pro-object represented by the given projective
system can be found in [13, Corollary 7.3.7]. ∎

######  Lemma 3.4.

Let $\Xi$ be a directed poset and
$(C^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\xi})_{\xi\in\Xi}$ be a
projective system of complexes of abelian groups with $C^{n}_{\xi}=0$ for
$n<0$. Then there are two spectral sequences ${}^{\prime}\\!E_{r}^{pq}$ and
${}^{\prime\prime}\\!E_{r}^{pq}$, starting from the pages

$\displaystyle{}^{\prime}\\!E_{2}^{pq}$
$\displaystyle=\varprojlim\nolimits^{p}H^{q}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\xi}),\qquad$
$\displaystyle p,\,q\geq 0,$ $\displaystyle{}^{\prime\prime}\\!E_{1}^{pq}$
$\displaystyle=\varprojlim\nolimits^{q}C^{p}_{\xi},\qquad$ $\displaystyle
p,\,q\geq 0,$

with the differentials
${}^{\prime}\\!d_{r}^{pq}\colon{}^{\prime}\\!E_{r}^{p,q}\longrightarrow{}^{\prime}\\!E_{r}^{p+r,q-r+1}$
and
${}^{\prime\prime}\\!d_{r}^{pq}\colon{}^{\prime\prime}\\!E_{r}^{p,q}\longrightarrow{}^{\prime\prime}\\!E_{r}^{p+r,q-r+1}$,
converging to the associated graded groups to two different filtrations
${}^{\prime}\\!F^{p}E^{n}$ and ${}^{\prime\prime}\\!F^{p}E^{n}$ on the same
graded abelian group $E^{n}$, $n=p+q$.

###### Proof.

These are called “the two hypercohomology spectral sequences” (for the derived
functor of inverse limit); cf. [2, Section XVII.3]. The groups $E^{n}$ are the
cohomology groups of the complex obtained by applying the derived functor of
inverse limit to the whole complex of projective systems
$(C^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\xi})$. ∎

Now we can finish the proof of the theorem. By the definition, we have
$\operatorname{Ext}^{n}_{R}(k,P)=H^{n}\operatorname{Hom}_{R}(K_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R),P)$.
The complex
$\operatorname{Hom}_{R}(K_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R),P)$
is the inverse limit

$\operatorname{Hom}_{R}(K_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R),P)=\varprojlim\nolimits_{B\subset
A}\operatorname{Hom}_{R}(K_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{B}(R),P).$

For every $n\geq 0$, Lemma 3.2 (with the poset $\Xi$ of all finite subsets
$B\subset A$, finitely generated free $R$-modules $F_{B}$, and an infinitely
generated free $R$-module $F$) tells that $\varprojlim^{i}_{B\subset
A}\operatorname{Hom}_{R}(K_{n}^{B}(R),P)=0$ for all $i\geq 1$.

On the other hand, the complex
$\operatorname{Hom}_{R}(K_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{B}(R),P)$
has its only nonzero cohomology module situated in the cohomological degree
$n$ equal to the cardinality of $B$ (as $P$ is a flat module over the ring
$R_{B}=k[x_{\alpha}:\alpha\in B]$). By Lemma 3.3, we have
$\varprojlim^{i}_{B\subset
A}H^{n}\operatorname{Hom}_{R}(K_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{B}(R),P)=0$
for all $i\geq 0$ and $n\geq 0$.

In the context of Lemma 3.4, put
$C_{B}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Hom}_{R}(K_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{B}(R),P)$.
Then we know that ${}^{\prime}\\!E_{2}^{pq}=\varprojlim^{p}_{B\subset
A}H^{q}(C_{B}^{\text{\smaller\smaller$\scriptstyle\bullet$}})=0$ for all $p$,
$q\geq 0$, and ${}^{\prime\prime}\\!E_{1}^{pq}=\varprojlim^{q}_{B\subset
A}C_{B}^{p}=0$ for all $q\geq 1$. Thus $E^{n}=0$ and
$H^{n}(\varprojlim\nolimits_{B\subset
A}\operatorname{Hom}_{R}(K_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{B}(R),P))={}^{\prime\prime}\\!E_{2}^{n,0}=0$
for all $n\geq 0$. ∎

##  4\. The Projective Construction of Acyclic Complex of Projectives

Now we are ready to present the “projective resolution construction of a
bounded below acyclic complex of projective modules”.

######  Corollary 4.1.

Let $R=k[x_{\alpha}:\alpha\in A]$ be the ring of polynomials in infinitely
many variables over a field $k$, and let

(5) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muk\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muP_{0}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muP_{1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muP_{2}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

be a projective resolution of the one-dimensional $R$-module $k$. Let $P$ be a
projective $R$-module such that the $R$-module $P_{n}$ is a direct summand of
$P$ for all $n\geq 0$. Let

(6) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(P_{0},P)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(P_{1},P)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(P_{2},P)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

be the complex obtained by applying the contravariant functor
$\operatorname{Hom}_{R}({-},P)$ to the truncated resolution (5). Then (6) is a
bounded below, noncontractible, acyclic complex of finitely generated
projective left modules over the ring $S=\operatorname{Hom}_{R}(P,P)$.

###### Proof.

The complex (6) is acyclic by Theorem 3.1. The left $S$-module
$\operatorname{Hom}_{R}(P_{n},P)$ is a direct summand of the left $S$-module
$\operatorname{Hom}_{R}(P,P)=S$ for every $n\geq 0$, since the $R$-module
$P_{n}$ is a direct summand of $P$. So (6) is even a complex of cyclic
projective left $S$-modules.

The proof of the assertion that the complex of $S$-modules (6) is not
contractible is similar to the argument in the proof of Corollary 2.3. By
Lemma 2.2(b) (for $\mathsf{A}=R{\operatorname{\mathsf{--Mod}}}$ and $M=P$),
the functor $\operatorname{Hom}_{R}({-},P)$ is an anti-equivalence of
categories $\mathsf{add}(P)^{\mathsf{op}}\simeq
S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$. The truncated resolution
(5),

$0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muP_{0}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muP_{1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muP_{2}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

is a noncontractible (since nonacyclic) complex in
$R{\operatorname{\mathsf{--Mod}}}$ with the terms belonging to
$\mathsf{add}(P)$, so it is a noncontractible complex in $\mathsf{add}(P)$.
Applying the anti-equivalence of additive categories
$\mathsf{add}(P)^{\mathsf{op}}\simeq
S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$, we obtain a noncontractible
complex (6) in $S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$, which is
consequently also noncontractible in $S{\operatorname{\mathsf{--Mod}}}$. It is
important for this argument that the contravariant functor
$\operatorname{Hom}_{R}({-},P)\colon\mathsf{add}(P)^{\mathsf{op}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS{\operatorname{\mathsf{--Mod}}}$ is fully faithful. ∎

##  5\. Brief Preliminaries on Coalgebras

In this section and the next two, we consider comodules and contramodules over
coassociative, counital coalgebras $\mathcal{C}$ over a field $k$. We refer to
the book [17] and the survey papers [7, Section 1], [10, Sections 3 and 8] for
background material on coalgebras, comodules, and contramodules.

For any coalgebra $\mathcal{C}$, there are locally finite Grothendieck abelian
categories of left and right $\mathcal{C}$-comodules
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ and
${\operatorname{\mathsf{Comod--}}}\mathcal{C}$, and a locally presentable
abelian category of left $\mathcal{C}$-contramodules
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}$. There are enough injective
objects in $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$, and they are
precisely the direct summands of the _cofree_ left $\mathcal{C}$-comodules
$\mathcal{C}\otimes_{k}V$ (where $V$ ranges over the $k$-vector spaces). Dual-
analogously, there are enough projective objects in
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}$, and they are precisely the
direct summands of the _free_ left $\mathcal{C}$-contramodules
$\operatorname{Hom}_{k}(\mathcal{C},V)$ (where $V\in
k{\operatorname{\mathsf{--Vect}}}$).

The additive categories of injective left $\mathcal{C}$-comodules and
projective left $\mathcal{C}$-contramodules are naturally equivalent,

(7)
$\Psi_{\mathcal{C}}\colon\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{\mathsf{inj}}\,\simeq\,\mathcal{C}{\operatorname{\mathsf{--Contra}}}_{\mathsf{proj}}\,:\\!\Phi_{\mathcal{C}}.$

The equivalence is provided by the restrictions of the adjoint functors

$\Psi_{\mathcal{C}}\colon\mathcal{C}{\operatorname{\mathsf{--comod}}}\,\leftrightarrows\mathcal{C}{\operatorname{\mathsf{--Contra}}}\,:\\!\Phi_{\mathcal{C}},$

the functor $\Phi_{\mathcal{C}}$ being the left adjoint and
$\Psi_{\mathcal{C}}$ the right adjoint. The functors $\Psi_{\mathcal{C}}$ and
$\Phi_{\mathcal{C}}$ are constructed as

$\Psi_{\mathcal{C}}(\mathcal{M})=\operatorname{Hom}_{\mathcal{C}}(\mathcal{C},\mathcal{M})\quad\text{and}\quad\Phi_{\mathcal{C}}(\mathfrak{P})=\mathcal{C}\odot_{\mathcal{C}}\mathfrak{P}$

for all $\mathcal{M}\in\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ and
$\mathfrak{P}\in\mathcal{C}{\operatorname{\mathsf{--Contra}}}$. Here
$\odot_{\mathcal{C}}\colon{\operatorname{\mathsf{Comod--}}}\mathcal{C}\times\mathcal{C}{\operatorname{\mathsf{--Contra}}}\longrightarrow
k{\operatorname{\mathsf{--Vect}}}$ is the functor of _contratensor product_
over a coalgebra $\mathcal{C}$, while $\operatorname{Hom}_{\mathcal{C}}$
denotes the Hom functor in the comodule category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$. The equivalence of additive
categories (7) is called the (_underived_) _comodule-contramodule
correspondence_. We refer to [7, Sections 1.2 and 3.1], [10, Sections
8.6–8.7], or [6, Sections 0.2.6 and 5.1] for a more detailed discussion.

In fact, we are only interested in one special kind of coalgebras, namely, the
_symmetric coalgebra_ $\mathcal{S}\mathit{ym}(U)$ of a $k$-vector space $U$.
To define the symmetric coalgebra, consider the _tensor coalgebra_
$\mathcal{T}\\!\mathit{en}(U)=\bigoplus_{n=0}^{\infty}U^{\otimes n}$, as
defined, e. g., in [10, Section 2.3] (where the notation is slightly
different). The tensor coalgebra is the cofree conilpotent coalgebra cospanned
by $U$ [10, Remark 3.2]; it is also naturally graded. The symmetric coalgebra
is simplest defined as the graded subcoalgebra in
$\mathcal{T}\\!\mathit{en}(U)$ whose grading components
$\mathcal{S}\mathit{ym}_{n}(U)\subset\mathcal{T}\\!\mathit{en}_{n}(U)=U^{\otimes
n}$ are the subspaces of symmetric tensors
$\mathcal{S}\mathit{ym}_{n}(U)\subset U^{\otimes n}$ in the tensor powers of
the vector space $U$. So the whole symmetric coalgebra is
$\mathcal{S}\mathit{ym}(U)=\bigoplus_{n=0}^{\infty}\mathcal{S}\mathit{ym}_{n}(U)=k\oplus
U\oplus\mathcal{S}\mathit{ym}_{2}(U)\oplus\dotsb$.

Following the discussion in [7, Section 1.3–1.4] or [10, Section 8.3],
coalgebras $\mathcal{C}$ can be described (and in fact, defined) in terms of
their vector space dual algebras
$\mathcal{C}^{*}=\operatorname{Hom}_{k}(\mathcal{C},k)$, which carry natural
linearly compact ( $=$ pseudocompact) topologies. In particular, if $U$ is a
finite-dimensional $k$-vector space with a basis $x_{1}^{*}$, …, $x_{m}^{*}$,
then the dual algebra $\mathcal{S}\mathit{ym}(U)^{*}$ to the symmetric
coalgebra $\mathcal{S}\mathit{ym}(U)$ is the topological algebra of formal
Taylor power series $\mathcal{S}\mathit{ym}(U)^{*}=k[[x_{1},\dotsc,x_{m}]]$.

Generally speaking, for an infinite-dimensional $k$-vector space $W$, one has
$\mathcal{S}\mathit{ym}(W)=\varinjlim_{U\subset W}\mathcal{S}\mathit{ym}(U)$
and $\mathcal{S}\mathit{ym}(W)^{*}=\varprojlim_{U\subset
W}\mathcal{S}\mathit{ym}(U)^{*}$, where $U$ ranges over the finite-dimensional
vector subspaces of $W$. So, if $\\{x_{\alpha}^{*}:\alpha\in A\\}$ is a
$k$-vector space basis of $W$, indexed by some set $A$, then
$\mathcal{S}\mathit{ym}(W)^{*}=\varprojlim_{B\subset A}k[[x_{\alpha}:\alpha\in
B]]$, where $B$ ranges over the finite subsets of $A$. Here, given two finite
subsets $B^{\prime}\subset B^{\prime\prime}\subset A$, the transition map
$k[[x_{\alpha}:\alpha\in B^{\prime\prime}]]\longrightarrow
k[[x_{\alpha}:\alpha\in B^{\prime}]]$ in the projective system takes
$x_{\alpha}$ to $x_{\alpha}$ for all $\alpha\in B^{\prime}$ and $x_{\beta}$ to
$0$ for all $\beta\in B^{\prime\prime}\setminus B^{\prime}$. Such rings
$\mathcal{S}\mathit{ym}(W)^{*}=\varprojlim_{B\subset A}k[[x_{\alpha}:\alpha\in
B]]$ are the “commutative rings of formal power series in infinitely many
variables” that we are interested in.

##  6\. Comodule and Contramodule Acyclicity Theorems

As above, we denote by $W$ an infinite-dimensional $k$-vector space with a
basis $\\{x_{\alpha}^{*}:\alpha\in A\\}$ indexed by a set $A$. Given a finite
set $B$, we let $\widehat{R}_{B}=k[[x_{\alpha}:\alpha\in B]]$ be the
(topological) ring of commutative formal Taylor power series in finitely many
variables indexed by $B$. Furthermore, we put
$\widehat{R}=\varprojlim_{B\subset A}\widehat{R}_{B}$ (with the transition
maps described in the previous section). So, denoting by $U_{B}\subset W$ the
finite-dimensional vector subspace spanned by $\\{x_{\alpha}^{*}:\alpha\in
B\\}$, we have $\widehat{R}_{B}=\mathcal{S}\mathit{ym}(U_{B})^{*}$ and
$\widehat{R}=\varprojlim_{B\subset A}k[[x_{\alpha}:\alpha\in
B]]=\mathcal{S}\mathit{ym}(W)^{*}$. Let us also introduce the notation
$\mathcal{C}_{B}=\mathcal{S}\mathit{ym}(U_{B})$ and
$\mathcal{C}=\mathcal{S}\mathit{ym}(W)$ for the symmetric coalgebras.

As in the proof of Theorem 3.1, we start with considering the two-term Koszul
complex of free $\widehat{R}_{B}$-modules with one generator

(8) $\dotsb\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\widehat{R}_{B}\overset{x_{\alpha}*}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}\widehat{R}_{B}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

situated in the cohomological degrees $-1$ and $0$ (where $\alpha\in B$).
Denote by
$K^{B}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\widehat{R}_{B})$ the
tensor product, taken over the ring $\widehat{R}_{B}$, of the complexes (8).
As the elements $\\{x_{\alpha}:\alpha\in B\\}$ form a regular sequence in the
formal power series ring $\widehat{R}_{B}$, the complex
$K^{B}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\widehat{R}_{B})$ is a
finite resolution of the one-dimensional $\widehat{R}_{B}$-module
$k=\widehat{R}_{B}/\sum_{\alpha\in B}x_{\alpha}\widehat{R}_{B}$ by finitely
generated free $\widehat{R}_{B}$-modules.

The (augmented) Koszul complex
$K^{B}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\widehat{R}_{B})\longrightarrow
k$ is a complex of linearly compact topological $k$-vector spaces; so it can
be obtained by applying the vector space dualization functor
$\operatorname{Hom}_{k}({-},k)$ to a certain complex of discrete vector
spaces. The latter complex has the form

(9) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muk\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{S}\mathit{ym}(U_{B})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{S}\mathit{ym}(U_{B})\otimes_{k}U_{B}\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{S}\mathit{ym}(U_{B})\otimes_{k}\Lambda^{2}(U_{B})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{S}\mathit{ym}(U_{B})\otimes_{k}\Lambda^{m}(U_{B})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0,$

where $m=\dim U_{B}$ and $\Lambda^{n}(V)$, $n\geq 0$, denotes the exterior
powers of a vector space $V$. The complex (9) is an injective/cofree
$\mathcal{C}_{B}$-comodule coresolution of the trivial one-dimensional
comodule $k$ over the conilpotent coalgebra
$\mathcal{C}_{B}=\mathcal{S}\mathit{ym}(U_{B})$.

Passing to the direct limit of the finite complexes (9) over all the finite
subsets $B\subset A$, we obtain a bounded below complex

(10) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{S}\mathit{ym}(W)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{S}\mathit{ym}(W)\otimes_{k}W\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{S}\mathit{ym}(W)\otimes_{k}\Lambda^{2}(W)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{S}\mathit{ym}(W)\otimes_{k}\Lambda^{n}(W)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

The complex (10) is an injective/cofree $\mathcal{C}$-comodule coresolution of
the trivial one-dimensional comodule $k$ over the conilpotent coalgebra
$\mathcal{C}=\mathcal{S}\mathit{ym}(W)$.

One can easily check that the coresolutions (9) and (10) are well-defined and
functorial for any $k$-vector spaces $U$ (in place of $U_{B}$) and $W$, and do
not depend on the choice of any bases in the vector spaces. In fact, the
differential
$\mathcal{S}\mathit{ym}(W)\otimes_{k}\Lambda^{n}(W)\longrightarrow\mathcal{S}\mathit{ym}(W)\otimes_{k}\Lambda^{n+1}(W)$
can be constructed as the composition
$\mathcal{S}\mathit{ym}(W)\otimes_{k}\Lambda^{n}(W)\longrightarrow\mathcal{S}\mathit{ym}(W)\otimes_{k}W\otimes_{k}\Lambda^{n}(W)\longrightarrow\mathcal{S}\mathit{ym}(W)\otimes_{k}\Lambda^{n+1}(W)$
of the map induced by the comultiplication map
$\mathcal{S}\mathit{ym}(W)\longrightarrow\mathcal{S}\mathit{ym}(W)\otimes_{k}W$
and the map induced by the multiplication map
$W\otimes_{k}\Lambda^{n}(W)\longrightarrow\Lambda^{n+1}(W)$.

Applying the vector space dualization functor $\operatorname{Hom}_{k}({-},k)$
to the complex (10), we obtain a bounded above complex

(11) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muk\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},k)\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},W^{*})\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},\Lambda^{2}(W)^{*})\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},\Lambda^{n}(W)^{*})\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

The complex (11) is a projective/free $\mathcal{C}$-contramodule resolution of
the trivial one-dimensional $\mathcal{C}$-contramodule $k$.

Applying the functor $\Phi_{\mathcal{C}}=\mathcal{C}\odot_{\mathcal{C}}{-}$ to
the truncated $\mathcal{C}$-contramodule resolution (11), we obtain a bounded
above complex of injective/cofree $\mathcal{C}$-comodules

(12) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}\otimes_{k}W^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}\otimes_{k}\Lambda^{2}(W)^{*}\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}\otimes_{k}\Lambda^{n}(W)^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

Applying the functor
$\Psi_{\mathcal{C}}=\operatorname{Hom}_{\mathcal{C}}(\mathcal{C},{-})$ to the
truncated $\mathcal{C}$-comodule coresolution (10), we obtain a bounded below
complex of projective/free $\mathcal{C}$-contramodules

(13) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},k)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},W)\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},\Lambda^{2}(W))\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},\Lambda^{n}(W))\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

######  Theorem 6.1.

For any infinite-dimensional $k$-vector space $W$, the complex of cofree
comodules (12) is acyclic (i. e., its cohomology spaces vanish in all the
degrees).

###### Proof.

This was stated in [6, Section 0.2.7] (as a part of introductory/preliminary
material for the book). The proof is not difficult.

The complex (12) is the direct limit of its subcomplexes

(14) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}_{B}\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}_{B}\otimes_{k}W^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}_{B}\otimes_{k}\Lambda^{2}(W)^{*}\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}_{B}\otimes_{k}\Lambda^{n}(W)^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

taken over the directed poset of all finite subsets $B\subset A$. The complex
(14), which is a complex of comodules over the subcoalgebra
$\mathcal{C}_{B}=\mathcal{S}\mathit{ym}(U_{B})$ of the coalgebra
$\mathcal{C}=\mathcal{S}\mathit{ym}(W)$, can be obtained by applying the
cotensor product functor
$\mathcal{C}_{B}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{-}$ to the
complex (12) (see [7, Sections 2.5–2.6] or [6, Section 0.2.1 or 1.2.1]).

The complex (14) is _not_ acyclic, but its cohomology spaces gradually vanish
as the size of the finite subset $B\subset A$ grows. Indeed, applying the
vector space dualization functor $\operatorname{Hom}_{k}({-},k)$ to the finite
complex (9), we obtain a finite Koszul complex that was denoted above by
$K^{B}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\widehat{R}_{B})$. It
has the form

(15) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muk\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},k)\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},U_{B}^{*})\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},\Lambda^{2}(U_{B})^{*})\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},\Lambda^{m}(U_{B})^{*})\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu0$

and can be viewed as a projective/free $\mathcal{C}_{B}$-contramodule
resolution of the trivial one-dimensional $\mathcal{C}_{B}$-contramodule $k$.

Applying the functor
$\Phi_{\mathcal{C}_{B}}=\mathcal{C}_{B}\odot_{\mathcal{C}_{B}}{-}$ to the
truncated $\mathcal{C}_{B}$-contramodule resolution (15), we obtain a finite
complex of injective/cofree $\mathcal{C}_{B}$-comodules

(16) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}_{B}\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}_{B}\otimes_{k}U_{B}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}_{B}\otimes_{k}\Lambda^{2}(U_{B})^{*}\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathcal{C}_{B}\otimes_{k}\Lambda^{m}(U_{B})^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu0.$

The only cohomology space of the complex (16) is the one-dimensional
$k$-vector space $\Lambda^{m}(U_{B})^{*}$ situated in the cohomological degree
$-m$, i. e., at the rightmost term.

Consider the direct sum decomposition $W=U_{B}\oplus V_{B}$, where
$V_{B}\subset W$ is the subspace with the basis $\\{x_{\alpha}^{*}:\alpha\in
A\setminus B\\}$. Consider the graded dual vector space to the exterior
algebra $\bigoplus_{n=0}^{\infty}\Lambda^{n}(V_{B})$, and view it as a complex

(17) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muk\overset{0}{\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu}V_{B}^{*}\overset{0}{\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu}\Lambda^{2}(V_{B})^{*}\overset{0}{\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu}\dotsb\overset{0}{\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu}\Lambda^{n}(V_{B})^{*}\overset{0}{\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu}\dotsb$

with zero differential. Then the complex (14) is the tensor product, taken
over the field $k$, of the complexes (16) and (17). Accordingly, the
cohomology spaces of the complex (14) are concentrated in the cohomological
degrees $\leq-m$, where $m$ is the cardinality of the set $B$.

As the size of the subset $B\subset A$ grows, the cohomology of the complex
(14) move away and disappear at the cohomological degree $-\infty$. So the
direct limit (12) of the complexes (14) is acyclic. ∎

######  Theorem 6.2.

For any infinite-dimensional $k$-vector space $W$, the complex of free
contramodules (13) is acyclic (i. e., its cohomology spaces vanish in all the
degrees).

###### Proof.

This was also stated in [6, Section 0.2.7]. The proof is only slightly more
complicated than the proof of the previous theorem, in that one needs to deal
with inverse limits. However, we have done all the preparatory work already.

The complex (13) is the inverse limit of its quotient complexes

(18) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},k)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},W)\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},\Lambda^{2}(W))\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},\Lambda^{n}(W))\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

taken over the directed poset of all finite subsets $B\subset A$. The complex
(18), which is a complex of contramodules over the subcoalgebra
$\mathcal{C}_{B}\subset\mathcal{C}$, can be obtained by applying the Cohom
functor $\operatorname{Cohom}_{\mathcal{C}}(\mathcal{C}_{B},{-})$ to the
complex (13) (see [7, Sections 2.5–2.6] or [6, Section 0.2.4 or 3.2.1]).

Similarly to the previous proof, the complex (18) is _not_ acyclic, but its
cohomology spaces gradually vanish as the size of the finite subset $B\subset
A$ grows. For the sake of completeness of the exposition, let us start with
applying the functor
$\Psi_{\mathcal{C}_{B}}=\operatorname{Hom}_{\mathcal{C}_{B}}(\mathcal{C}_{B},{-})$
to the truncated $\mathcal{C}_{B}$-comodule coresolution (9). We obtain a
finite complex of projective/free $\mathcal{C}_{B}$-contramodules

(19) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},k)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},U_{B})\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},\Lambda^{2}(U_{B}))\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C}_{B},\Lambda^{m}(U_{B}))\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0.$

The only cohomology space of the complex (19) is the one-dimensional
$k$-vector space $\Lambda^{m}(U_{B})$ situated in the cohomological degree
$m$, i. e, at the rightmost term. In fact, the complex of contramodules (19)
can be obtained by applying the vector space dualization functor
$\operatorname{Hom}_{k}({-},k)$ to the complex of comodules (16).

Consider the exterior algebra $\bigoplus_{n=0}^{\infty}\Lambda^{n}(V_{B})$,
where as in the previous proof $W=U_{B}\oplus V_{B}$, and view it as a complex

(20) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk\overset{0}{\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}V_{B}\overset{0}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}\Lambda^{2}(V_{B})\overset{0}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}\dotsb\overset{0}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}\Lambda^{n}(V_{B})\overset{0}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu}\dotsb$

with zero differential. Then the complex (18) is the complex of $k$-vector
space morphisms, $\operatorname{Hom}_{k}({-},{-})$, from the complex (16) into
the complex (20). Accordingly, the cohomology spaces of the complex (18) are
concentrated in the cohomological degrees $\geq m$.

The rest of the argument proceeds along the lines of the proof of Theorem 3.1,
based on Lemmas 3.2–3.4. As mentioned above, the complex (13) is the inverse
limit of the complexes (18) taken over the directed poset $\Xi$ of all finite
subsets $B\subset A$ with respect to inclusion. At every cohomological degree
$n\geq 0$, Lemma 3.2 (for $R=k$, $F_{B}=\mathcal{C}_{B}$, and
$P=\Lambda^{n}(W)$) tells that $\varprojlim^{i}_{B\subset
A}\operatorname{Hom}_{k}(\mathcal{C}_{B},\Lambda^{n}(W))=0$ for all $i\geq 1$.

Denote by $C_{B}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the complex
(18). By Lemma 3.3, we have $\varprojlim^{i}_{B\subset
A}H^{n}(C_{B}^{\text{\smaller\smaller$\scriptstyle\bullet$}})=0$ for all
$i\geq 0$ and $n\geq 0$. Now in the context of Lemma 3.4 we have
${}^{\prime}\\!E_{2}^{pq}=0$ for all $p$, $q\geq 0$, and
${}^{\prime\prime}\\!E_{1}^{pq}=0$ for all $q\geq 1$. Therefore, $E^{n}=0$ and
$H^{n}(\varprojlim_{B\subset
A}C_{B}^{\text{\smaller\smaller$\scriptstyle\bullet$}})={}^{\prime\prime}\\!E_{2}^{n,0}=0$
for all $n\geq 0$. ∎

##  7\. Two Contramodule Constructions of Acyclic Complexes of Projectives

We have essentially already constructed the promised bounded above,
noncontractible, acyclic complex of injective comodules and bounded below,
noncontractible, acyclic complex of projective contramodules over the
cocommutative coalgebra $\mathcal{C}=\mathcal{S}\mathit{ym}(W)$. Let us state
this as a corollary.

######  Corollary 7.1.

Let $W$ be an infinite-dimensional vector space over a field $k$ and
$\mathcal{C}=\mathcal{S}\mathit{ym}(W)$ be the symmetric coalgebra. Then

(a) the complex (12) is a bounded above, noncontractible, acyclic complex of
injective comodules over $\mathcal{C}$;

(b) the complex (13) is a bounded below, noncontractible, acyclic complex of
projective contramodules over $\mathcal{C}$.

###### Proof.

Part (a): the complex (12) is acyclic by Theorem 6.1. It remains to explain
why the complex of $\mathcal{C}$-comodules (12) is not contractible.

The truncated resolution (11),

$0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},k)\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},W^{*})\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},\Lambda^{2}(W)^{*})\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{C},\Lambda^{n}(W)^{*})\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

is a noncontractible (since nonacyclic) complex in the abelian category
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}$ with the terms belonging to
the full subcategory of projective objects
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}_{\mathsf{proj}}$, so it is a
noncontractible complex in
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}_{\mathsf{proj}}$. Applying the
equivalence of additive categories
$\Phi_{\mathcal{C}}\colon\mathcal{C}{\operatorname{\mathsf{--Contra}}}_{\mathsf{proj}}\simeq\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{\mathsf{inj}}$
(7), we obtain a noncontractible complex (12) in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{\mathsf{inj}}$, which is
consequently also noncontractible in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$.

Part (b): the complex (13) is acyclic by Theorem 6.2. It remains to explain
why the complex of $\mathcal{C}$-contramodules (13) is not contractible.

The truncated coresolution (10),

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{C}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{C}\otimes_{k}W\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{C}\otimes_{k}\Lambda^{2}(W)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{C}\otimes_{k}\Lambda^{n}(W)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

is a noncontractible (since nonacyclic) complex in the abelian category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ with the terms belonging to the
full subcategory of injective objects
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{\mathsf{inj}}$, so it is a
noncontractible complex in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{\mathsf{inj}}$. Applying the
equivalence of additive categories
$\Psi_{\mathcal{C}}\colon\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{\mathsf{inj}}\simeq\mathcal{C}{\operatorname{\mathsf{--Contra}}}_{\mathsf{proj}}$
(7), we obtain a noncontractible complex (13) in
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}_{\mathsf{proj}}$, which is
consequently also noncontractible in
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}$. ∎

Now let us present the two contramodule constructions of bounded below,
noncontractible, acyclic complexes of projective modules. Recall the notation
$\operatorname{Hom}_{\mathcal{C}}({-},{-})$ for the Hom spaces in the category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$. The notation
$\operatorname{Hom}^{\mathcal{C}}({-},{-})$ stands for the Hom spaces in the
category $\mathcal{C}{\operatorname{\mathsf{--Contra}}}$.

######  Corollary 7.2.

Let $W$ be an infinite-dimensional vector space over a field $k$ and
$\mathcal{C}=\mathcal{S}\mathit{ym}(W)$ be the symmetric coalgebra. Let

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muk\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{J}^{0}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{J}^{1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{J}^{2}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

be a notation for the injective coresolution (10) of the trivial one-
dimensional $\mathcal{C}$-comodule $k$. Denote by $\mathcal{J}$ the cofree
$\mathcal{C}$-comodule $\mathcal{C}\otimes_{k}W$ cospanned by $W$. Let

(21) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{\mathcal{C}}(\mathcal{J},\mathcal{J}^{0})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{\mathcal{C}}(\mathcal{J},\mathcal{J}^{1})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{\mathcal{C}}(\mathcal{J},\mathcal{J}^{2})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

be the complex obtained by applying the functor
$\operatorname{Hom}_{\mathcal{C}}(\mathcal{J},{-})$ to the truncated
coresolution (10). Then (21) is a bounded below, noncontractible, acyclic
complex of finitely generated projective left modules over the ring
$S=\operatorname{Hom}_{\mathcal{C}}(\mathcal{J},\mathcal{J})^{\mathrm{op}}$.

###### Proof.

For any $\mathcal{C}$-comodule $\mathcal{M}$, we have
$\operatorname{Hom}_{\mathcal{C}}(\mathcal{C}\otimes_{k}W,\>\mathcal{M})=\operatorname{Hom}_{k}(W,\operatorname{Hom}_{k}(\mathcal{C},\mathcal{M}))\allowbreak=\operatorname{Hom}_{k}(W,\Psi_{\mathcal{C}}(\mathcal{M}))$.
Thus the complex (21) can be obtained by applying the vector space Hom functor
$\operatorname{Hom}_{k}(W,{-})$ to the complex (13), and it follows from
Theorem 6.2 that the complex (21) is acyclic.

Furthermore, by construction, the $\mathcal{C}$-comodule
$\mathcal{J}^{0}=\mathcal{C}$ is a direct summand of $\mathcal{J}$, while the
$\mathcal{C}$-comodules $\mathcal{J}^{n}=\mathcal{C}\otimes_{k}\Lambda^{n}(W)$
are isomorphic to $\mathcal{J}$ for $n\geq 1$. Hence the left $S$-module
$\operatorname{Hom}_{\mathcal{C}}(\mathcal{J},\mathcal{J}^{0})$ is a direct
summand of $\operatorname{Hom}_{\mathcal{C}}(\mathcal{J},\mathcal{J})=S$, and
the left $S$-modules
$\operatorname{Hom}_{\mathcal{C}}(\mathcal{J},\mathcal{J}^{n})$ are isomorphic
to $S$ for $n\geq 1$. So (21) is even a complex of cyclic projective left
$S$-modules.

The assertion that the complex of $S$-modules (21) is not contractible is
provable similarly to the argument in the proof of Corollary 2.3. By Lemma
2.2(a) (for $\mathsf{A}=\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ and
$M=\mathcal{J}$), the functor
$\operatorname{Hom}_{\mathcal{C}}(\mathcal{J},{-})$ is an equivalence of
categories $\mathsf{add}(\mathcal{J})\simeq
S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$. The truncated coresolution
(10),

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{J}^{0}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{J}^{1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{J}^{2}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

is a noncontractible (since nonacyclic) complex in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ with the terms belonging to
$\mathsf{add}(\mathcal{J})$, so it is a noncontractible complex in
$\mathsf{add}(\mathcal{J})$. Applying the equivalence of additive categories
$\mathsf{add}(\mathcal{J})\simeq
S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$, we obtain a noncontractible
complex (21) in $S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$, which is
consequently also noncontractible in $S{\operatorname{\mathsf{--Mod}}}$. It is
important for this argument that the functor
$\operatorname{Hom}_{\mathcal{C}}(\mathcal{J},{-})\colon\mathsf{add}(\mathcal{J})\longrightarrow
S{\operatorname{\mathsf{--Mod}}}$ is fully faithful.

Alternatively, put
$\mathfrak{P}=\operatorname{Hom}_{k}(\mathcal{C},W)\in\mathcal{C}{\operatorname{\mathsf{--Contra}}}_{\mathsf{proj}}$.
Then the co-contra correspondence (7) restricts to an equivalence of additive
categories $\mathsf{add}(\mathcal{J})\simeq\mathsf{add}(\mathfrak{P})$ taking
$\mathcal{J}$ to $\mathfrak{P}$. Hence the ring $S$ can be alternatively
described as
$S=\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P},\mathfrak{P})^{\mathrm{op}}$.
The complex of left $S$-modules (21) can be constructed by applying the
functor $\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P},{-})$ to the complex of
$\mathcal{C}$-contramodules (13), whose terms belong to
$\mathsf{add}(\mathfrak{P})$.

Then the noncontractibility argument can be based on Corollary 7.1(b) and the
fact that the functor $\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P},{-})$ is
an equivalence of categories $\mathsf{add}(\mathfrak{P})\simeq
S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$ (by Lemma 2.2(a) for
$\mathsf{A}=\mathcal{C}{\operatorname{\mathsf{--Contra}}}$ and
$M=\mathfrak{P}$). Once again, it is important for this argument that the
functor
$\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P},{-})\colon\mathsf{add}(\mathfrak{P})\longrightarrow
S{\operatorname{\mathsf{--Mod}}}$ is fully faithful. In fact, the whole
functor
$\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P},{-})\colon\mathcal{C}{\operatorname{\mathsf{--Contra}}}\longrightarrow
S{\operatorname{\mathsf{--Mod}}}$ is fully faithful (on the whole abelian
category $\mathcal{C}{\operatorname{\mathsf{--Contra}}}$) by [11, Theorem
6.10]. The latter conclusion is based on the observations that $\mathfrak{P}$
is the coproduct of $\dim W$ copies of the projective generator
$\mathcal{C}^{*}=\operatorname{Hom}_{k}(\mathcal{C},k)$ of the abelian
category $\mathcal{C}{\operatorname{\mathsf{--Contra}}}$, and
$\mathcal{C}^{*}$ is abstractly $\kappa$-small in
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}$ for
$\mathcal{C}=\mathcal{S}\mathit{ym}(W)$ if $\kappa$ is the successor
cardinality of $\dim W$. ∎

######  Corollary 7.3.

Let $W$ be an infinite-dimensional vector space over a field $k$ and
$\mathcal{C}=\mathcal{S}\mathit{ym}(W)$ be the symmetric coalgebra. Let

$0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muk\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathfrak{P}_{0}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathfrak{P}_{1}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathfrak{P}_{2}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

be a notation for the projective resolution (11) of the trivial one-
dimensional $\mathcal{C}$-contramodule $k$. Denote by $\mathfrak{P}$ the free
$\mathcal{C}$-contramodule $\operatorname{Hom}_{k}(\mathcal{C},W^{*})$ spanned
by the vector space $W^{*}=\operatorname{Hom}_{k}(W,k)$. Let

(22) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P}_{0},\mathfrak{P})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P}_{1},\mathfrak{P})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P}_{2},\mathfrak{P})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

be the complex obtained by applying the contravariant functor
$\operatorname{Hom}^{\mathcal{C}}({-},\mathfrak{P})$ to the truncated
resolution (11). Then (22) is a bounded below, noncontractible, acyclic
complex of finitely generated projective left modules over the ring
$S=\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P},\mathfrak{P})$.

###### Proof.

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

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

[7, Section 3.1], [10, Section 8.6], or [6, Sections 0.2.6 and 5.1.1]. In
particular, we have natural isomorphisms

$\operatorname{Hom}^{\mathcal{C}}(\mathfrak{Q},\mathfrak{P})=\operatorname{Hom}_{k}(\mathcal{C}\odot_{\mathcal{C}}\mathfrak{Q},\>W^{*})=\operatorname{Hom}_{k}(\Phi_{\mathcal{C}}(\mathfrak{Q}),W^{*}).$

Thus the complex (22) can be obtained by applying the contravariant vector
space Hom functor $\operatorname{Hom}_{k}({-},W^{*})$ to the complex (12), and
it follows from Theorem 6.1 that the complex (22) is acyclic.

Furthermore, by construction, the $\mathcal{C}$-comodule
$\mathfrak{P}_{0}=\mathcal{C}^{*}$ is is a direct summand of $\mathfrak{P}$,
while the $\mathcal{C}$-contramodules
$\mathfrak{P}_{n}=\operatorname{Hom}_{k}(\mathcal{C},\Lambda^{n}(W)^{*})$ are
isomorphic to $\mathfrak{P}$ for $n\geq 1$. Hence the left $S$-module
$\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P}_{0},\mathfrak{P})$ is a direct
summand of $\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P},\mathfrak{P})=S$,
and the left $S$-modules
$\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P}_{n},\mathfrak{P})$ are
isomorphic to $S$ for $n\geq 1$. So (22) is even a complex of cyclic
projective left $S$-modules.

The assertion that the complex of $S$-modules (22) is not contractible is
provable similarly to the argument in the proof of Corollary 4.1. By Lemma
2.2(b) (for $\mathsf{A}=\mathcal{C}{\operatorname{\mathsf{--Contra}}}$ and
$M=\mathfrak{P}$), the functor
$\operatorname{Hom}^{\mathcal{C}}({-},\mathfrak{P})$ is an anti-equivalence of
categories $\mathsf{add}(\mathfrak{P})^{\mathsf{op}}\simeq
S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$. The truncated resolution
(11),

$0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathfrak{P}_{0}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathfrak{P}_{1}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\mathfrak{P}_{2}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

is a noncontractible (since nonacyclic) complex in
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}$ with the terms belonging to
$\mathsf{add}(\mathfrak{P})$, so it is a noncontractible complex in
$\mathsf{add}(\mathfrak{P}$). Applying the anti-equivalence of additive
categories $\mathsf{add}(\mathfrak{P})^{\mathsf{op}}\simeq
S{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$, we obtain a noncontractible
complex (22), which is consequently also noncontractible in
$S{\operatorname{\mathsf{--Mod}}}$. It is important for this argument that the
contravariant functor
$\operatorname{Hom}^{\mathcal{C}}({-},\mathfrak{P})\colon\mathsf{add}(\mathfrak{P})^{\mathsf{op}}\longrightarrow
S{\operatorname{\mathsf{--Mod}}}$ is fully faithful. ∎

##  8\. Summary of the Examples Obtained

Now we can summarize our constructions as follows.

######  Conclusion 8.1.

There exists an associative ring $S$ for which

(a) there is a bounded above acyclic complex of injective right $S$-modules
that is not contractible;

(b) there is a bounded below acyclic complex of flat left $S$-modules that is
not pure acyclic;

(c) there is a bounded below acyclic complex of (finitely generated)
projective left $S$-modules that is not contractible.

###### Proof.

Proposition 1.2 tells that any ring $S$ satisfying (c) also satisfies (a) and
(b). Various examples of associative rings $S$ satisfying (c) are provided by
Corollaries 2.3, 4.1, 7.2, and 7.3. ∎

What can one say about the rings $S$ appearing in Corollaries 2.3, 4.1, 7.2,
and 7.3 ? First of all, _none_ of them is commutative (while we have
cocommutative coalgebra examples in Corollary 7.1).

Let us denote the respective versions of the ring $S$ by $S_{\ref{injective-
module-coresolution-construction}}$, $S_{\ref{projective-module-resolution-
construction}}$, $S_{\ref{first-contramodule-construction-cor}}$, and
$S_{\ref{second-contramodule-construction-cor}}$. While the ring
$S_{\ref{injective-module-coresolution-construction}}$ (from Corollary 2.3)
appears to be complicated and hard to visualize, the rings
$S_{\ref{projective-module-resolution-construction}}$, $S_{\ref{first-
contramodule-construction-cor}}$, and $S_{\ref{second-contramodule-
construction-cor}}$ can be described rather explicitly.

In the context of Corollary 4.1, it makes sense to choose the infinite Koszul
complex
$K_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R)=\varinjlim_{B\subset
A}K^{B}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(R)$ to play the role of
the projective resolution $P_{\text{\smaller\smaller$\scriptstyle\bullet$}}$
(5) of the $R$-module $k$. In this case, one can take $P$ to be the free
$R$-module with $A$ generators, $P=\bigoplus_{\alpha\in A}R$. Then the
$R$-module $P_{0}=R$ is a direct summand of $P$, while the $R$-module $P_{n}$
is isomorphic to $P$ for $n\geq 1$, so the assumption of the corollary is
satisfied. The resulting ring $S_{\ref{projective-module-resolution-
construction}}=\operatorname{Hom}_{R}(P,P)$ is the ring of infinite, column-
finite $A\times A$ matrices with entries from the commutative polynomial ring
$R=k[x_{\alpha}:\alpha\in A]$ in infinitely many variables.

In the context of Corollaries 7.2 and 7.3, it makes sense to introduce the
notation $\mathcal{J}_{\ref{first-contramodule-construction-cor}}$ for the
cofree comodule $\mathcal{J}=\mathcal{C}\otimes_{k}W$ appearing in Corollary
7.2 and the notation $\mathfrak{P}_{\ref{first-contramodule-construction-
cor}}$ for the free contramodule
$\mathfrak{P}=\operatorname{Hom}_{k}(\mathcal{C},W)$ mentioned in the
discussion in its proof. Then the notation $\mathfrak{P}_{\ref{second-
contramodule-construction-cor}}$ can be used for the bigger free contramodule
$\mathfrak{P}=\operatorname{Hom}_{k}(\mathcal{C},W^{*})$ from Corollary 7.3,
and we can also denote by $\mathcal{J}_{\ref{second-contramodule-construction-
cor}}$ the corresponding cofree comodule
$\mathcal{J}=\mathcal{C}\otimes_{k}W^{*}$.

The ring $S_{\ref{first-contramodule-construction-
cor}}=\operatorname{Hom}_{\mathcal{C}}(\mathcal{J}_{\ref{first-contramodule-
construction-cor}},\mathcal{J}_{\ref{first-contramodule-construction-
cor}})^{\mathrm{op}}=\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P}_{\ref{first-
contramodule-construction-cor}},\mathfrak{P}_{\ref{first-contramodule-
construction-cor}})^{\mathrm{op}}$ is the ring of infinite, row-zero-
convergent $A\times A$ matrices with entries from the topological commutative
formal power series ring $\widehat{R}=\mathcal{C}^{*}=\varprojlim_{B\subset
A}k[[x_{\alpha}:\alpha\in B]]$ in infinitely many variables. Such rings of
row-zero-convergent matrices were discussed in the papers [11, Example 7.10]
and [12, Section 5].

Let $D$ denote the indexing set of a basis $\\{y_{\delta}\colon\delta\in D\\}$
in the $k$-vector space $W^{*}$. The cardinality $|D|$ of the set $D$ is equal
to $|k|^{|A|}$, where $|k|$ is the cardinality of the field $k$ and $|A|$ is
the cardinality of the set $A$. Then the ring $S_{\ref{second-contramodule-
construction-cor}}=\operatorname{Hom}_{\mathcal{C}}(\mathcal{J}_{\ref{second-
contramodule-construction-cor}},\mathcal{J}_{\ref{second-contramodule-
construction-
cor}})=\operatorname{Hom}^{\mathcal{C}}(\mathfrak{P}_{\ref{second-
contramodule-construction-cor}},\mathfrak{P}_{\ref{second-contramodule-
construction-cor}})$ is the ring of infinite, column-zero-convergent $D\times
D$ matrices with entries from the topological commutative formal power series
ring $\widehat{R}$ in infinitely many variables indexed by $A$.

Endowed with its natural topology, the ring $S_{\ref{first-contramodule-
construction-cor}}$ becomes a complete, separated right linear topological
ring (i. e., a topological ring with a base of neighborhoods of zero formed by
open right ideals). Such topological rings were discussed in the papers [8,
11, 12]. Moreover, the ring $S_{\ref{first-contramodule-construction-cor}}$ is
_topologically left perfect_ in the sense of [12, Section 14] (as one can see
from the discussion in [11, Section 7.3] or [12, Section 5] together with [8,
Example 12.3]). Similarly, the ring $S_{\ref{second-contramodule-construction-
cor}}$ is a complete, separated, left linear topological ring which is,
moreover, topologically right perfect. In other words, the ring
$S_{\ref{second-contramodule-construction-cor}}$ is the endomorphism ring of a
module with perfect decomposition [12, Section 10], while $S_{\ref{first-
contramodule-construction-cor}}$ is the opposite ring to the endomorphism ring
of such a module.

Viewed as abstract rings, both the rings $S_{\ref{first-contramodule-
construction-cor}}$ and $S_{\ref{second-contramodule-construction-cor}}$ are
_semiregular_ in the sense of [1, Section 4]; see the discussion in [12,
Remark 10.5]. (The semiregularity is a left-right symmetric property.) The
ring $S_{\ref{projective-module-resolution-construction}}$, on the other hand,
has vanishing Jacobson radical.

The ring $S_{\ref{second-contramodule-construction-cor}}$ differs from the
opposite ring to $S_{\ref{first-contramodule-construction-cor}}$ for the only
reason that the cardinality of the set $D$ is larger than that of the set $A$.
One can employ the bigger cofree comodule $\mathcal{J}_{\ref{second-
contramodule-construction-cor}}=\mathcal{C}\otimes_{k}W^{*}$ in lieu of the
cofree comodule $\mathcal{J}_{\ref{first-contramodule-construction-
cor}}=\mathcal{C}\otimes_{k}W$ in the construction of Corollary 7.2 (while
leaving the rest of the construction unchanged). This will produce a pair of
opposite rings $S$ and $S^{\mathrm{op}}$, both of them semiregular, both of
them satisfying all the claims of Conclusion 8.1.

######  Remark 8.2.

_None_ of the rings $S_{\ref{projective-module-resolution-construction}}$,
$S_{\ref{first-contramodule-construction-cor}}$, and $S_{\ref{second-
contramodule-construction-cor}}$ is Noetherian on either side. Indeed,
consider the ring $T=\operatorname{Hom}_{k}(k^{(E)},k^{(E)})$ of infinite,
column-finite $E\times E$ matrices with entries from the field $k$ (where $E$
is a set and $k^{(E)}$ is the $k$-vector space with a basis indexed by $E$).
Then $T$ is a quotient ring of $S_{\ref{projective-module-resolution-
construction}}$ (for $E=A$) and of $S_{\ref{second-contramodule-construction-
cor}}$ (for $E=D$), while the opposite ring to $T$ is a quotient ring of
$S_{\ref{first-contramodule-construction-cor}}$ (for $E=A$). The ring $T$ is
von Neumann regular, so it cannot be left or right Noetherian (as any one-
sided Noetherian von Neumann regular ring is semisimple Artinian).

######  Remark 8.3.

A simpler construction of rings and complexes of modules satisfying Conclusion
8.1 than the one discussed above exists (see Example 8.4 below). But the
following naïve attempt at constructing an example for Conclusion 8.1(b)
fails.

The commutative ring $R=k[x_{\alpha}:\alpha\in A]$ of polynomials in
infinitely many variables over a field $k$ is not Noetherian, but it is
coherent. Hence the class of flat $R$-modules is closed under infinite
products in $R{\operatorname{\mathsf{--Mod}}}$, and it follows that the
$R$-module $\operatorname{Hom}_{R}(Q,P)$ is flat for any projective $R$-module
$Q$ and flat $R$-module $P$. Thus the complex (6) from Corollary 4.1 is a
bounded below, acyclic complex of flat $R$-modules. One does not even need the
$R$-modules $P_{n}$ to be direct summands of $P$ for this claim to hold; it
suffices to take $P=R$.

However, this is _not_ an example for Conclusion 8.1(b), because the complex
of $R$-modules (6) is actually pure acyclic (for any flat $R$-module $P$).
Indeed, it suffices to show that, for any finitely presented $R$-module $M$,
applying the functor $M\otimes_{R}{-}$ preserves acyclicity of the complex
(6). Denote the complex (6) by
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.

Any finitely presented module over the ring of polynomials $R$ in infinitely
many variables has a finite projective resolution
$G_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ by finitely generated
projective $R$-modules. Since
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex of flat
$R$-modules and $G_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a finite
resolution, the complexes
$M\otimes_{R}F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$G_{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
are quasi-isomorphic. Finally, viewed as an object of the homotopy category of
complexes of $R$-modules $\mathsf{K}(R{\operatorname{\mathsf{--Mod}}})$, the
complex
$G_{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
belongs to the thick subcategory spanned by the complex
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ (since the complex
$G_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ belongs to the thick
subcategory spanned by the one-term complex of $R$-modules $R$). As the
complex $F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is acyclic, so is
the complex
$G_{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.

######  Example 8.4.

The following example has a different nature than all the previous examples in
this paper. It was communicated to the author by A. Canonaco and is reproduced
here with his kind permission.

Suppose that we have a bounded below complex of _free modules with one
generator_ over a ring $S$. Obviously, such a complex of (left) modules has
the form

(23) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS\overset{*z_{0}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}S\overset{*z_{1}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}S\overset{*z_{2}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu}S\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb,$

where $z_{0}$, $z_{1}$, $z_{2}$, … is some sequence of elements in $S$. For
the sequence of maps (23) to be a complex, the equation $z_{n}z_{n+1}=0$ has
to be satisfied in $S$ for all integers $n\geq 0$.

Now let $k$ be a field and $S_{\mathrm{uni}}$ be the $k$-algebra generated by
a sequence of elements $x_{0}$, $x_{1}$, $x_{2}$, … with the imposed relations
$x_{n}x_{n+1}=0$ for all $n\geq 0$, and no other relations. Then one can
easily see that any element $s\in S_{\mathrm{uni}}$ satisfying the equation
$sx_{0}=0$ vanishes, while any element $s\in S_{\mathrm{uni}}$ satisfying
$sx_{n+1}=0$ with $n\geq 0$ has the form $s=tx_{n}$ for some $t\in
S_{\mathrm{uni}}$. It suffices to represent $s$ as a $k$-linear combination of
monomials in the variables $x_{n}$, $n\geq 0$, etc. In other words, the
bounded below complex of free $S_{\mathrm{uni}}$-modules

(24) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS_{\mathrm{uni}}\overset{*x_{0}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}S_{\mathrm{uni}}\overset{*x_{1}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}S_{\mathrm{uni}}\overset{*x_{2}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}S_{\mathrm{uni}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

is acyclic. On the other hand, if $k$ is endowed with the right
$S_{\mathrm{uni}}$-module structure in which all the elements $x_{n}$ act by
zero in $k$, then applying the functor $k\otimes_{S_{\mathrm{uni}}}{-}$ to the
complex (24) produces a nonacyclic complex with zero differential. So the
complex (24) is _not_ contractible.

The bounded below complex of free $S_{\mathrm{uni}}$-modules with one
generator (24) is _universal_ in the following sense. Let $S$ be an
associative $k$-algebra and $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
be a complex of free $S$-modules with one generator such that $C^{i}=0$ for
$i<0$. Then there exists a $k$-algebra homomorphism $f\colon
S_{\mathrm{uni}}\longrightarrow S$ such that the complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is obtained by applying the
functor of extension of scalars $S\otimes_{S_{\mathrm{uni}}}{-}$ to the
complex (24). Indeed, the complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ has the form (23) for some
elements $z_{n}\in S$, $n\geq 0$ satisfying the equations $z_{n}z_{n+1}=0$,
and it remains to let $f\colon S_{\mathrm{uni}}\longrightarrow S$ be the
homomorphism taking $x_{n}$ to $z_{n}$ for every $n\geq 0$.

While the example in Example 8.4 is certainly simpler (to construct and prove
its properties) than the examples in Corollaries 2.3, 4.1, 7.2, and 7.3, _no_
example of a bounded below, noncontractible, acyclic complex of projective
modules (or of a bounded above, noncontractible, acyclic complex of injective
modules) can be _too_ simple. The results of [16, Appendix A] demonstrate
this.

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