# Differential graded Koszul duality:
an introductory survey

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

This is an overview on derived nonhomogeneous Koszul duality over a field,
mostly based on the author’s memoir [64]. The paper is intended to serve as a
pedagogical introduction and a summary of the covariant duality between DG-
algebras and curved DG-coalgebras, as well as the triality between DG-modules,
CDG-comodules, and CDG-contramodules. Some personal reminiscences are included
as a part of historical discussion.

###### Contents

  1. 1 Algebras and Modules
    1. 1.1 Augmented algebras
    2. 1.2 Bar-constructions of modules
    3. 1.3 Posing the problem
  2. 2 Coalgebras and Comodules
    1. 2.1 Coalgebras and comodules
    2. 2.2 DG-coalgebras and DG-comodules
    3. 2.3 Coalgebra structure on the bar-construction
    4. 2.4 Derived Koszul duality formulated for complexes of modules
    5. 2.5 Nonhomogeneous quadratic dual DG-coalgebra
    6. 2.6 Cobar-construction; DG-algebra and DG-module structures
    7. 2.7 Derived Koszul duality formulated for complexes of comodules
  3. 3 Structure Theory of Coalgebras; Conilpotency
    1. 3.1 Brief remarks about coalgebras
    2. 3.2 Local finite-dimensionality
    3. 3.3 Conilpotent coalgebras
    4. 3.4 Cosemisimple coalgebras and the coradical filtration
  4. 4 DG-Algebras and DG-Coalgebras
    1. 4.1 Augmented DG-algebras
    2. 4.2 Coaugmented DG-coalgebras
    3. 4.3 Conilpotent DG-coalgebras
    4. 4.4 Duality between DG-algebras and DG-coalgebras
    5. 4.5 Quillen equivalence between DG-algebras and DG-coalgebras
  5. 5 Twisting Cochains
    1. 5.1 The Hom DG-algebra and twisting cochains
    2. 5.2 Bar-cobar adjunction and acyclic twisting cochains
    3. 5.3 Twisted differential on the tensor product
    4. 5.4 Derived Koszul duality on the comodule side in the augmented case
  6. 6 CDG-Rings and CDG-Coalgebras
    1. 6.1 Posing the problem of nonaugmented Koszul duality
    2. 6.2 CDG-rings and CDG-modules
    3. 6.3 CDG-coalgebras and CDG-comodules
    4. 6.4 Nonaugmented bar-construction for algebras and modules
    5. 6.5 Nonaugmented bar-construction for DG-algebras
    6. 6.6 Curved, noncoaugmented cobar-construction
    7. 6.7 Duality between DG-algebras and CDG-coalgebras
    8. 6.8 Twisting cochains in the curved context
    9. 6.9 Derived Koszul duality on the comodule side
  7. 7 Derived Categories of the Second Kind
    1. 7.1 Importance of derived categories of the second kind
    2. 7.2 History of derived categories of the second kind, I
    3. 7.3 Philosophy of derived categories of the second kind, I
    4. 7.4 Philosophy of derived categories of the second kind, II
    5. 7.5 Philosophy of derived categories of the second kind, III
    6. 7.6 Coderived and contraderived categories of CDG-modules
    7. 7.7 Coderived category of CDG-comodules
    8. 7.8 History, II, and conclusion
    9. 7.9 Becker’s derived categories of the second kind
  8. 8 Contramodules over Coalgebras
    1. 8.1 The basics of contramodules
    2. 8.2 Duality-analogy of comodules and contramodules
    3. 8.3 Comodules and contramodules over power series in one variable
    4. 8.4 Nonseparated contramodules
    5. 8.5 Contramodule Nakayama lemma and irreducible contramodules
    6. 8.6 Contratensor product
    7. 8.7 Underived co-contra correspondence
    8. 8.8 History of contramodules
  9. 9 CDG-Contramodules
    1. 9.1 Graded contramodules
    2. 9.2 Definition of CDG-contramodules
    3. 9.3 Contraderived category of CDG-contramodules
    4. 9.4 Derived co-contra correspondence
    5. 9.5 Twisted differential on the graded Hom space
    6. 9.6 Derived Koszul duality on the contramodule side
    7. 9.7 Koszul triality
    8. 9.8 Some comments on the proof

## Introduction

Koszul duality is a fundamental phenomenon in mathematics, including such
fields as algebraic topology, algebraic and differential geometry, and
representation theory. This phenomenon is so general that it does not seem to
admit a “maximal natural” generality. Whatever formulation one comes up with,
it can be likely further extended by building something on top or underneath.

Koszul duality is a synthetic subject. Depending on context, its formulation
involves such concepts as differential graded structures, curved diffferential
graded structures, comodules and contramodules, conilpotency, coderived and
contraderived categories, operads and properads, and whatnot.

Koszul duality has to be distinguished from the _comodule-contramodule
correspondence_. The co-contra correspondence is a less familiar though no
less fundamental, but simpler phenomenon, often accompanying the Koszul
duality. The following rule of thumb for distinguishing the two phenomena may
be helpful. Viewed as a covariant category equivalence, the co-contra
correspondence takes injective objects to projective objects or vice versa.
Koszul duality takes projective objects to irreducible objects and/or
irreducible objects to injective objects etc. [75, Prologue].

The aim of this paper is to introduce the reader to the subject of derived
Koszul duality in the context of differential graded algebras and modules, as
well as coalgebras, comodules, and contramodules; and survey some of its main
results. We also discuss curved differential structures, but _no_
$\mathrm{A}_{\infty}$-structures are considered in this paper (the reader can
find discussions of these in the Koszul duality context in the dissertation
[47] and the memoirs [64, 65]). _No_ relative Koszul duality settings (as in
[63, Sections 0.4 and 11] or [75]) are discussed in this survey, either; so
everything in this paper happens over a field. We also do not consider operads
and their generalizations, referring instead to the book [48] and the papers
[31, 89].

This introductory survey is intended for an audience versed in homological
algebra generally, but largely unfamiliar with the subject of derived Koszul
duality. It is almost entirely based on the author’s memoir [64]. The more
elementary topic of _underived_ quadratic duality [83, 61, 7, 58] is only
briefly touched in this paper. _Homogeneous_ versions of derived Koszul
duality [7], [64, Appendix A] (involing graded modules over positively or
negatively graded algebras, with the differentials preserving the grading) are
not elaborated upon in this paper, either. Model structures and compact
generators are also only briefly touched.

We start with posing the problem in a basic particular case, formulating the
main results in an approximate form in the simplest settings, and then proceed
to introduce further ingredients and make more precise and general assertions.
Some sketches of proofs are included in this survey, but most results receive
a brief outline of an idea of the proof and a reference to a more detailed
treatment.

All the Koszul duality functors, triangulated equivalences etc. in this paper
are _covariant_ functors or category equivalences. A discussion of
_contravariant_ version of _homogeneous_ Koszul duality can be found in [64,
Section A.2].

One disclaimer is in order. As usual in differential graded homological
algebra, sign rules (plus or minus) present a tedious problem. In this survey,
we skip many descriptions of sign rules, referring the reader to the original
publications such as [64] for the details. So our formulations may be
imprecise in that not all the signs are properly spelled out.

### Acknowledgements

I am grateful to Andrey Lazarev for suggesting the idea of writing this survey
to me. I wish to thank Jan Št’ovíček for asking me questions about derived
Koszul duality and subsequently offering me to present the material at his
seminar on $\infty$-categories. This stimulated me to come up with a
pedagogical introductory exposition. An acknowledgement is also due to Darya
Krinitsina for asking me a question about the proofs in [64, Section 6] which
is answered by Lemmas 9.11–9.12 in the last section of this survey. Last but
not least, I want to thank the anonymous referee for careful reading of the
manuscript and several helpful suggestions. The author is supported by the
GAČR project 20-13778S and research plan RVO: 67985840.

##  1\. Algebras and Modules

Throughout this paper, we work over a fixed ground field $k$. Unless otherwise
mentioned, all _algebras_ in this paper are associative and unital, all
_modules_ are unital, and all _homomorphisms of algebras_ are presumed to take
the unit to the unit.

In this section we start posing the problem of derived nonhomogeneous Koszul
duality in the simplest particular case of complexes of modules over an
augmented $k$-algebra $A$.

###  1.1. Augmented algebras

Let $A$ be an (associative, unital) algebra over a field $k$. An
_augmentation_ $\alpha$ on $A$ is a (unital) $k$-algebra homomorphism
$\alpha\colon A\longrightarrow k$. So $\alpha(1)=1$, and the existence of an
augmentation implies that $1\neq 0$ in $A$; hence $\alpha$ is a surjective
map. We denote by $A^{+}=\ker(\alpha)\subset A$ the augmentation ideal; so
$A^{+}$ is a two-sided ideal in $A$, and $A=k\oplus A^{+}$ as a $k$-vector
space. The $k$-algebra homomorphism $\alpha\colon A\longrightarrow k$ endows
the one-dimensional $k$-vector space $k$ with left and right $A$-module
structures.

The following definition goes back to the papers [20, Chapter II] and [1]. The
_bar-construction_
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
of an augmented $k$-algebra $A=(A,\alpha)$ is defined as the complex

$k\overset{0}{\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu}A^{+}\overset{\partial}{\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu}A^{+}\otimes_{k}A^{+}\overset{\partial}{\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu}A^{+}\otimes_{k}A^{+}\otimes_{k}A^{+}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

with the diffential given by the formulas $\partial(a\otimes b)=ab$,
$\partial(a\otimes b\otimes c)=ab\otimes c-a\otimes bc$, …,

$\partial(a_{1}\otimes\dotsb\otimes a_{n})=a_{1}a_{2}\otimes
a_{3}\otimes\dotsb\otimes a_{n}-\dotsb\\\ +(-1)^{i+1}a_{1}\otimes\dotsb\otimes
a_{i-1}\otimes a_{i}a_{i+1}\otimes a_{i+2}\otimes\dotsb\otimes a_{n}+\dotsb\\\
+(-1)^{n}a_{1}\otimes\dotsb\otimes a_{n-2}\otimes a_{n-1}a_{n},$

etc., for all $a$, $b$, $c$, $a_{i}\in A^{+}$, $n\geq 1$, and $1\leq i\leq
n-1$. The leftmost differential $\partial\colon A^{+}\longrightarrow k$ is the
zero map.

The complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
computes the $k$-vector spaces $\operatorname{Tor}^{A}_{*}(k,k)$ (where $k$ is
endowed with left and right $A$-module structures via $\alpha$). In other
words, there are natural isomorphisms of $k$-vector spaces

$H^{-n}\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)\simeq\operatorname{Tor}^{A}_{n}(k,k)\qquad\text{for
all $n\geq 0$}.$

Let $(A,\alpha)$ and $(B,\beta)$ be two augmented $k$-algebras, and let
$f\colon A\longrightarrow B$ be a $k$-algebra homomorphism compatible with the
augumentations (i. e., satisfying the equation $\alpha=\beta f$). Then there
is the induced map of augmentation ideals $f^{+}\colon A^{+}\longrightarrow
B^{+}$ and the induced map of the bar-constructions
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(f)\colon\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)\longrightarrow\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\beta}(B)$.
The latter map is a morphism of complexes of $k$-vector spaces.

######  Example 1.1.

The morphism of complexes
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(f)$ may
well be a quasi-isomorphism even when a morphism of augmented $k$-algebras
$f\colon A\longrightarrow B$ is _not_ an isomorphism.

For example, the complex $0\longleftarrow B^{+}\longleftarrow
B^{+}\otimes_{k}B^{+}\longleftarrow
B^{+}\otimes_{k}B^{+}\otimes_{k}B^{+}\longleftarrow\dotsb$ is acyclic whenever
the augmentation ideal $B^{+}$, viewed as an associative $k$-algebra, has a
unit of its own. The sequence of $k$-linear maps $h\colon B^{+}{}^{\otimes
n}\longrightarrow B^{+}{}^{\otimes n+1}$ given by the rules
$h(b_{1}\otimes\dotsb\otimes b_{n})=e\otimes b_{1}\otimes\dotsb\otimes b_{n}$
for all $b_{i}\in B^{+}$, $1\leq i\leq n$, $n\geq 1$, where $e$ is the unit in
$B^{+}$, provides a contracting homotopy.

Choose any nonzero associative, unital $k$-algebra, denote it by $B^{+}$, and
adjoin a new unit to it formally, producing the $k$-algebra $B=k\oplus B^{+}$.
Then the natural homomorphisms of augmented $k$-algebras $k\longrightarrow
B\longrightarrow k$ induce quasi-isomorphisms of the bar-constructions.

More generally, let $A$ and $B$ be two associative, unital $k$-algebras.
Consider the direct sum $A\oplus B$, and endow it with a $k$-algebra structure
as the product of $A$ and $B$ in the category of $k$-algebras. Choose an
augmentation $\alpha\colon A\longrightarrow k$, and let the augmentation of
$A\oplus B$ be constructed as the composition $A\oplus B\longrightarrow
A\overset{\alpha}{\longrightarrow}k$. Then the natural homomorphism of
augmented $k$-algebras $A\oplus B\longrightarrow A$ (the direct summand
projection) induces a quasi-isomorphism of the bar-constructions.

###  1.2. Bar-constructions of modules

Let $A$ be an associative $k$-algebra with an augmentation $\alpha\colon
A\longrightarrow k$. Let $M$ be a left $A$-module.

The _bar-construction_
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)$
of $A$ with the coefficients in $M$ is the complex

$M\overset{\partial}{\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu}A^{+}\otimes_{k}M\overset{\partial}{\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu}A^{+}\otimes_{k}A^{+}\otimes_{k}M\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

with the diffferential given by the formulas $\partial(a\otimes m)=am$,
$\partial(a\otimes b\otimes m)=ab\otimes\nobreak m-a\otimes bm$, …,

$\partial(a_{1}\otimes\dotsb\otimes a_{n}\otimes m)=a_{1}a_{2}\otimes
a_{3}\otimes\dotsb\otimes a_{n}\otimes
m-\dotsb+(-1)^{n+1}a_{1}\otimes\dotsb\otimes a_{n-1}\otimes a_{n}m,$

etc., for all $a$, $b$, $a_{i}\in A^{+}$, $m\in M$, and $n\geq 1$.

The complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)$
computes the $k$-vector spaces $\operatorname{Tor}^{A}_{*}(k,M)$ (where $k$ is
endowed with a right $A$-module structure via $\alpha$). In other words, there
are natural isomorphisms of $k$-vector spaces

$H^{-n}\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)\simeq\operatorname{Tor}^{A}_{n}(k,M)\qquad\text{for
all $n\geq 0$}.$

######  Examples 1.2.

The complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)$
may well be acyclic even when $M\neq 0$.

(1) For example, the complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)$
is acyclic whenever the augmentation ideal $A^{+}$, viewed as an associative
$k$-algebra, has a unit of its own. The sequence of $k$-linear maps $h\colon
A^{+}{}^{\otimes n}\otimes_{k}M\longrightarrow A^{+}{}^{\otimes
n+1}\otimes_{k}M$ given by the rules $h(a_{1}\otimes\dotsb\otimes a_{n}\otimes
m)=e\otimes a_{1}\otimes\dotsb\otimes a_{n}\otimes m$ for all $a_{i}\in
A^{+}$, $m\in$M, $1\leq i\leq n$, $n\geq 0$, where $e$ is the unit in $A^{+}$,
provides a contracting homotopy.

More generally, let $A$ and $B$ be two associative, unital $k$-algebras, and
let $\alpha\colon A\longrightarrow k$ be an augmentation of $A$. Consider the
direct sum $A\oplus B$, and endow it with an augmented $k$-algebra structure
$(\alpha,0)\colon A\oplus B\longrightarrow k$ as in Example 1.1. Pick a left
$B$-module $M$, and endow it with an $(A\oplus B)$-module structure via the
natural $k$-algebra homomorphism (the direct summand projection) $A\oplus
B\longrightarrow B$. Then the complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{(\alpha,0)}(A\oplus
B,\>M)$ is acyclic.

(2) To give a couple of other examples, consider the algebra of polynomials in
one variable $A=k[x]$ over the field $k$, endowed with the augmentation
$\alpha\colon A\longrightarrow k$ given by the rule $\alpha(x)=0$. For any
element $a\in k$, denote by $k_{a}=A/(x-a)$ the one-dimensional $A$-module in
which the generator $x\in A$ acts by the operator of multiplication with $a$.
In particular, in the $A$-module $k=k_{0}$ the algebra $A$ acts via the
augmentation $\alpha$. Put $M=k_{a}$, where $a\neq 0$. Then
$\operatorname{Tor}^{A}_{n}(k,M)=0$ for all $n\geq 0$, hence the complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)$
is acyclic.

Alternatively, consider the $A$-module $M=k[x,x^{-1}]$ of Laurent polynomials
in $x$ (or the $A$-module $M=k(x)$ of rational functions in $x$). Once again,
in these cases $\operatorname{Tor}^{A}_{n}(k,M)=0$ for all $n\geq 0$, and the
complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)$
is acyclic.

###  1.3. Posing the problem

Let $A=(A,\alpha)$ be an augmented $k$-algebra, and let
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex of left
$A$-modules. Then the bar-construction
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a bicomplex of $k$-vector spaces. Let us totalize this bicomplex by taking
infinite direct sums along the diagonals.

The problem of derived (nonhomogeneous) Koszul duality can be formulated as
follows. We would like to endow the complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
with some natural structure, and define an equivalence relation on complexes
with such structures, in a suitable way so that the assignment

$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$

would be a triangulated equivalence between the unbounded derived category
$\mathsf{D}(A{\operatorname{\mathsf{--mod}}})$ of complexes of $A$-modules and
the triangulated category of complexes with the said structure up to the said
equivalence relation.

What structure should it be, and what should be the equivalence relation? The
first question is easier to answer: the bar-construction
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
has a natural _DG-coalgebra_ structure, and the complexes
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
are _DG-comodules_ over
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$,
as will be explained below in Sections 2.3–2.4.

The second question is harder, because the conventional notion of quasi-
isomorphism is _not_ up to the task, as Examples 1.2 illustrate. The complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)$
can be quasi-isomorphic to zero for a quite nonzero one-term complex of
$A$-modules $M=M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The relevant
definition of the _coderived category_ of DG-comodules will be spelled out in
Section 7.7.

##  2\. Coalgebras and Comodules

Unless otherwise mentioned, all _coalgebras_ in this paper are coassociative,
counital coalgebras over the field $k$, all _comodules_ are counital, and all
_homomorphisms of coalgebras_ are compatible with the counits.

In this section we present the simplest initial formulations of derived
nonhomogeneous Koszul duality for complexes of modules over an augmented
algebra $A$ and for complexes of comodules over a conilpotent coalgebra $C$.

###  2.1. Coalgebras and comodules

The standard reference sources on coalgebras and comodules over a field are
the books [88, 49]. The present author’s overview [71] can be used as an
additional source.

A (_coassociative, counital_) _coalgebra_ $C$ over a field $k$ is a $k$-vector
space endowed with $k$-linear maps of _comultiplication_ and _counit_

$\mu\colon C\longrightarrow C\otimes_{k}C\qquad\text{and}\qquad\epsilon\colon
C\longrightarrow k$

satisfying the following _coassociativity_ and _counitality_ axioms. Firstly,
the two compositions

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

must be equal to each other,
$(\mu\otimes\mathrm{id}_{C})\circ\mu=(\mathrm{id}_{C}\otimes\mu)\circ\mu$.
Secondly, both the compositions

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

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

A _left comodule_ $M$ over a coalgebra $C$ is a $k$-vector space endowed with
a $k$-linear _left coaction_ map

$\nu\colon M\longrightarrow C\otimes_{k}M$

satisfying the following coassociativity and counitality axioms. Firstly, the
two compositions

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

must be equal to each other,
$(\mu\otimes\mathrm{id}_{M})\circ\nu=(\mathrm{id}_{C}\otimes\nu)\circ\nu$.
Secondly, the composition

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

must be equal to the identity map,
$(\epsilon\otimes\mathrm{id}_{M})\circ\nu=\mathrm{id}_{M}$.

A _right comodule_ $N$ over $C$ is a $k$-vector space endowed with a _right
coaction_ map

$\nu\colon N\longrightarrow N\otimes_{k}C$

satisfying the similar coassociativity and counitality axioms.

Let $V$ be a $k$-vector space. Then the comultiplication map $\mu$ on a
coalgebra $C$ induces a left coaction map $C\otimes_{k}V\longrightarrow
C\otimes_{k}C\otimes_{k}V$ on the $k$-vector space $C\otimes_{k}V$ and a right
coaction map $V\otimes_{k}C\longrightarrow V\otimes_{k}C\otimes_{k}C$ on the
$k$-vector space $V\otimes_{k}C$. The left $C$-comodule $C\otimes_{k}V$ and
the right $C$-comodule $V\otimes_{k}C$ are called the _cofree $C$-comodules_
cogenerated by the vector space $V$.

For any left $C$-comodule $L$, the $k$-vector space of all left $C$-comodule
maps $L\longrightarrow C\otimes_{k}V$ is naturally isomorphic to the
$k$-vector space of all $k$-linear maps $L\longrightarrow V$,

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

Hence the “cofree comodule” terminology.

###  2.2. DG-coalgebras and DG-comodules

The following definitions, going back at least to the paper [23], can be also
found in [64, Sections 2.1 and 2.3].

A _graded coalgebra_ over $k$ is a graded $k$-vector space
$C=\bigoplus_{i\in\mathbb{Z}}C^{i}$ endowed with a coalgebra structure such
that both the comultiplication map $\mu\colon C\longrightarrow C\otimes_{k}C$
and the counit map $\epsilon\colon C\longrightarrow k$ are morphisms of graded
vector spaces (i. e., homogeneous linear maps of degree $0$). Here the
standard induced grading is presumed on the tensor product $C\otimes_{k}C$,
while the one-dimensional $k$-vector space $k$ is endowed with the grading
where it is placed in the degree $i=0$.

A _graded left comodule_ over a graded coalgebra $C$ is a graded $k$-vector
space $M=\bigoplus_{i\in\mathbb{Z}}M^{i}$ endowed with a left $C$-comodule
structure such that the coaction map $\nu\colon M\longrightarrow
C\otimes_{k}M$ is a morphism of graded vector spaces. _Graded right
$C$-comodules_ are defined similarly.

In particular, for any graded $k$-vector space $V$, the graded $k$-vector
space $C\otimes_{k}V$ has a natural structure of a cofree graded left
$C$-comodule, while the tensor product $V\otimes_{k}C$ is a cofree graded
right $C$-comodule.

A _DG-coalgebra_ $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(C,d)$ over
$k$ is a complex of $k$-vector spaces endowed with a coalgebra structure such
that both the comultiplication map $\mu\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{k}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and the counit map $\epsilon\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow k$ are
morphisms of complexes of vector spaces (i. e., homogeneous linear maps of
degree $0$ commuting with the differentials). Here the standard induced
differential $d(c^{\prime}\otimes c^{\prime\prime})=d(c^{\prime})\otimes
c^{\prime\prime}+(-1)^{i}c^{\prime}\otimes d(c^{\prime\prime})$ for all
$c^{\prime}\in C^{i}$ and $c^{\prime\prime}\in C^{j}$ is presumed on the
tensor product of complexes
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{k}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
while the differential on the $k$-vector space $k$ is zero.

A _left DG-comodule_
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(M,d_{M})$ over a DG-
coalgebra $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex of
$k$-vector spaces endowed with a left $C$-comodule structure such that the
coaction map $\nu\colon
M^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{k}M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a morphism of complexes of $k$-vector spaces. _Right DG-comodules_ over
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are defined similarly.

###  2.3. Coalgebra structure on the bar-construction

Let $V$ be a $k$-vector space. Then the direct sum of the tensor powers of
$V$,

$k\oplus
V\oplus(V\otimes_{k}V)\oplus(V\otimes_{k}V\otimes_{k}V)\oplus\dotsb\,\simeq\,\bigoplus\nolimits_{n=0}^{\infty}V^{\otimes
n}$

can be naturally endowed with a structure of graded associative algebra over
$k$. The multiplication in this algebra, denoted by
$T(V)=\bigoplus_{n=0}^{\infty}V^{\otimes n}$, is given by the rule

$(v_{1}\otimes\dotsb\otimes v_{p})(w_{1}\otimes\dotsb\otimes
w_{q})=v_{1}\otimes\dotsb\otimes v_{p}\otimes w_{1}\otimes\dotsb\otimes w_{q}$

for all $v_{i}$, $w_{j}\in V$, $1\leq i\leq p$, $1\leq j\leq q$, $p$, $q\geq
0$. The unit element in $T(V)$ is $1\in k=V^{\otimes 0}$. The algebra $T(V)$
is the free associative, unital algebra spanned by the vector space $V$.

The same graded $k$-vector space $\bigoplus_{n=0}^{\infty}V^{\otimes n}$ also
has a natural structure of graded coassociative coalgebra over $k$. The
comultiplication in this coalgebra, denoted by
$\rotatebox[origin={c}]{180.0}{$T$}(V)=\bigoplus_{n=0}^{\infty}V^{\otimes n}$,
is given by the rule

$\mu(v_{1}\otimes\dotsb\otimes v_{n})=\sum\nolimits_{p+q=n}^{p,q\geq
0}(v_{1}\otimes\dotsb\otimes v_{p})\otimes(v_{p+1}\otimes\dotsb\otimes
v_{p+q})$

for all $v_{i}\in V$, $1\leq i\leq n$, $n\geq 0$. The counit on
$\rotatebox[origin={c}]{180.0}{$T$}(V)$ is the direct summand projection map
$\rotatebox[origin={c}]{180.0}{$T$}(V)\longrightarrow V^{\otimes 0}=k$. We
refer to Remark 3.2 for a discussion of a cofreeness property of the tensor
coalgebra $\rotatebox[origin={c}]{180.0}{$T$}(V)$.

Let $(A,\alpha)$ be an augmented algebra over $k$. We observe that the
underlying graded vector space $\operatorname{Bar}_{\alpha}(A)$ of the bar-
construction
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$,
as defined in Section 1.1, coincides (up to a grading sign change) with the
graded $k$-vector space $\bigoplus_{n=0}^{\infty}A^{+}{}^{\otimes n}$.
Consequently, the graded vector space $\operatorname{Bar}_{\alpha}(A)$ has
natural structures of an associative algebra and a coassociative coalgebra.

It turns out that the bar differential $\partial$ on
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
does _not_ respect the multiplication on
$\operatorname{Bar}_{\alpha}(A)=T(A^{+})$, i. e., $\partial$ does _not_
satisfy any kind of Leibniz rule with respect to the multiplication on
$T(A^{+})$. However, the differential $\partial$ is compatible with the
comultiplication on
$\operatorname{Bar}_{\alpha}(A)=\rotatebox[origin={c}]{180.0}{$T$}(A^{+})$. In
other words, the complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
endowed with the graded coalgebra structure of
$\rotatebox[origin={c}]{180.0}{$T$}(A^{+})$ is a _DG-coalgebra_ in the sense
of the definition in Section 2.2.

###  2.4. Derived Koszul duality formulated for complexes of modules

Let $(A,\alpha)$ be an augmented $k$-algebra, and let
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex of left
$A$-modules. We will denote simply by $M$ the underlying graded $A$-module of
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.

Then the underlying graded vector space $\operatorname{Bar}_{\alpha}(A,M)$ of
the bar-construction
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
as defined in Sections 1.2–1.3, can be viewed as a cofree graded left comodule
$\operatorname{Bar}_{\alpha}(A,M)=\rotatebox[origin={c}]{180.0}{$T$}(A^{+})\otimes_{k}M$
over the tensor coalgebra $\rotatebox[origin={c}]{180.0}{$T$}(A^{+})$ (as
explained in Sections 2.1–2.2). It turns out that the total differential on
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is compatible with the bar differential $\partial$ on the DG-coalgebra
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
and the left coaction of $\operatorname{Bar}_{\alpha}(A)$ in
$\operatorname{Bar}_{\alpha}(A,M)$. In other words, the complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
endowed with the cofree graded comodule structure of
$\rotatebox[origin={c}]{180.0}{$T$}(A^{+})\otimes_{k}M$ is a _left DG-
comodule_ over the DG-coalgebra
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$.

The following theorem is our first formulation of the derived Koszul duality.
At this point in our exposition, it is more of an advertisement than a precise
claim, in that it contains details which have not been defined yet but will be
defined below.

######  Theorem 2.1.

Let $(A,\alpha)$ be an augmented associative algebra over a field $k$. Then
the assignment
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
induces a triangulated equivalence between the derived category of left
$A$-modules $\mathsf{D}(A{\operatorname{\mathsf{--mod}}})$ and the _coderived_
category of left DG-comodules over the DG-coalgebra
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$,

$\mathsf{D}(A{\operatorname{\mathsf{--mod}}})\simeq\mathsf{D}^{\mathsf{co}}(\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A){\operatorname{\mathsf{--comod}}}).$

###### Proof.

This is a particular case of [64, Theorem 6.3(a)]; see [47, Théorème 2.2.2.2]
and [42, Section 4] for an earlier approach. The inverse functor,
$\mathsf{D}^{\mathsf{co}}(\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A){\operatorname{\mathsf{--comod}}})\allowbreak\longrightarrow\mathsf{D}(A{\operatorname{\mathsf{--mod}}})$,
assigns to a left DG-comodule
$N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over the DG-coalgebra
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
the graded left $A$-module $A\otimes_{k}N$ endowed with a natural differential
whose construction will be explained in Sections 5.3–5.4. The definition of
the coderived category will be given in Section 7.7. ∎

The triangulated equivalence of Theorem 2.1 takes the irreducible left
$A$-module $k$ (with the $A$-module structure defined in terms of the
augmentation $\alpha$) to the cofree left DG-comodule
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over the DG-coalgebra
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
(cf. the discussion of cofree comodules in Section 2.1). The same equivalence
takes the left DG-comodule $k$ over
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ (with the $C$-comodule
structure on $k$ defined in terms of the unique coaugmentation of $C$; see
Sections 2.6 and 4.4 below) to the free left $A$-module $A$.

For generalizations of Theorem 2.1, see Theorems 2.4 and 5.4 below.

###  2.5. Nonhomogeneous quadratic dual DG-coalgebra

The DG-coalgebra
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
produced by the bar-construction is rather big. In particular, it is
essentially _never_ finite-dimensional (unless $A=k$). The concept of
_nonhomogeneous quadratic duality_ , going back to [45], [83, Section 3], [25,
Section 2.3], and [61], sometimes allows to produce a smaller DG-coalgebra
that can be used in lieu of
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
in Koszul duality theorems such as Theorem 2.1.

Let $A=\bigoplus_{n=0}^{\infty}A_{n}$ be a nonnegatively graded $k$-algebra
with $A_{0}=k$. Then the direct summand inclusion $A_{1}\longrightarrow A$ can
be uniquely extended to a morphism of graded algebras $\pi\colon
T(A_{1})\longrightarrow A$. The algebra $A$ is said to be _quadratic_ if the
homomorphism $\pi$ is surjective and its kernel is generated by elements of
degree $2$ (simply speaking, this means that the algebra $A$ is generated by
elements of degree $1$ with relations in degree $2$).

The graded algebra $A$ is called _Koszul_ if
$\operatorname{Tor}^{A}_{ij}(k,k)=0$ for all $i\neq j$. Here the first grading
$i$ on the $\operatorname{Tor}$ spaces is the usual homological grading, while
the second grading $j$, called the _internal_ grading, is induced by the
grading of $A$. This condition for $i=1$ means precisely that the algebra $A$
is generated by $A_{1}$; assuming this is the case, the same condition for
$i=2$ means precisely that $A$ is defined by quadratic relations. So any
Koszul graded algebra is quadratic [58, Corollary 5.3 in Chapter 1].

Let $A=\bigoplus_{n=0}^{\infty}A_{n}$ be a quadratic algebra. Put $V=A_{1}$,
and denote by $I\subset V\otimes_{k}V$ the kernel of the (surjective)
multiplication map $A_{1}\otimes_{k}A_{1}\longrightarrow A_{2}$. So $V$ is the
space of generators of the quadratic algebra $A$, and $I$ is the space of
defining quadratic relations of $A$. The grading components of $A$ can be
expressed in terms of $V$ and $I$ by the formula

$A_{n}=V^{\otimes n}\Big{/}\sum\nolimits_{i=1}^{n-1}V^{\otimes
i-1}\otimes_{k}I\otimes_{k}V^{\otimes n-i-1},\qquad n\geq 1.$

Consider the tensor coalgebra $\rotatebox[origin={c}]{180.0}{$T$}(V)$, and
denote by $C=A^{?}$ the graded subcoalgebra in
$\rotatebox[origin={c}]{180.0}{$T$}(V)$ with the components $C^{0}=k$,
$C^{1}=V$, $C^{2}=I$, and generally

$C^{n}=\bigcap\nolimits_{i=1}^{n-1}V^{\otimes
i-1}\otimes_{k}I\otimes_{k}V^{\otimes n-i-1}\,\subset\,V^{\otimes n},\qquad
n\geq 1.$

The coalgebra $C$ is called the _quadratic dual coalgebra_ to a quadratic
algebra $A$ [82, Section 2.1].

######  Remark 2.2.

In most expositions, including the present author’s [61] and [58], the
quadratic duality is viewed as a contravariant functor constructed using the
passage to the dual vector space. The contravariant duality assigns to a
quadratic algebra $A$ the quadratic algebra $A^{!}$, which can be obtained as
the graded dual $k$-vector space to the coalgebra $C=A^{?}$, that is,
$A^{!}_{n}=(C^{n})^{*}$ for all $n\geq 0$. The construction of the quadratic
algebra $A^{!}$ works well for quadratic algebras $A$ with finite-dimensional
components (see, e. g., [58, Section 3 in Chapter 1]), and it can be made to
work without the assumption of locally finite dimension by considering
linearly compact topological vector spaces. Our general preference is to use
coalgebras instead, and consider the covariant quadratic duality between
algebras and coalgebras as constructed above. In particular, the intended
applications in [82] were quadratic algebras with infinite-dimensional
components; that is one reason why the covariant algebra-coalgebra quadratic
duality was introduced in [82]. In the context of derived nonhomogeneous
Koszul duality over a field, which is the subject of this survey, the language
of covariant duality is more illuminating, in our view.

Let $0=F_{-1}A\subset k=F_{0}A\subset F_{1}A\subset F_{2}A\subset\dotsb$ be a
$k$-algebra endowed with an increasing filtration $F$ such that
$A=\bigcup_{n\geq 0}F_{n}A$, $1\in F_{0}A$, and $F_{n}A\cdot F_{m}A\subset
F_{n+m}A$ for all $n$, $m\geq 0$. Then the associated graded vector space
$\mathrm{gr}^{F}A=\bigoplus_{n=0}^{\infty}F_{n}A/F_{n-1}A$ carries a naturally
induced associative algebra structure. A _nonhomogeneous quadratic algebra_ is
a filtered algebra $(A,F)$ such that the graded algebra $\mathrm{gr}^{F}A$ is
quadratic. A _nonhomogeneous Koszul algebra_ is a filtered algebra $(A,F)$
such that the graded algebra $\mathrm{gr}^{F}A$ is Koszul [83, Section 2],
[58, Chapter 5].

######  Example 2.3.

(1) Let $V$ be a $k$-vector space. Consider the graded algebra
$A=\bigoplus_{n=0}^{\infty}A_{n}$ with the components $A_{0}=k$, $A_{1}=V$,
and $A_{n}=0$ for all $n\geq 2$. The multiplication on the graded algebra $A$
is trivial: the multiplication map $A_{n}\otimes A_{m}\longrightarrow A_{n+m}$
vanishes for all $n$, $m>0$ (in the cases when $n=0$ or $m=0$ the
multiplication map is determined by the condition that $1\in k=A_{0}$ is the
unit element of $A$). The graded algebra $A$ is well-known to be Koszul (see,
e. g., [58, Corollary 2.4 or 4.3 in Chapter 2] for much more general results).
In this example, the quadratic dual coalgebra $C=A^{?}$ is the whole tensor
coalgebra, that is, $C=\rotatebox[origin={c}]{180.0}{$T$}(V)$.

(2) Any nonzero associative algebra $A$ can be endowed with the _trivial_
filtration, $F_{0}A=k\cdot 1$ and $F_{1}A=A$. Then the associated graded
algebra $\mathrm{gr}^{F}A$ has grading components $\mathrm{gr}^{F}_{0}A=k$,
$\mathrm{gr}^{F}_{1}A=A/(k\cdot\nobreak 1)$, and $\mathrm{gr}^{F}_{n}A=0$ for
$n\geq 2$. So the algebra $\mathrm{gr}^{F}A$ has the form of example (1) above
(with $V=A/(k\cdot\nobreak 1)$), and the multiplication on it is trivial in
the sense we have explained. Thus the passage to the associated graded algebra
of an algebra $A$ with respect to the trivial filtration $F$ destroys all the
information about the multiplication in $A$.

It is clear from this discussion that _any_ nonzero associative algebra $A$
endowed with the trivial filtration $F$ is a nonhomogeneous Koszul algebra.
Some associative algebras $A$ admit nontrivial filtrations making them
nonhomogeneous Koszul algebras with a more interesting associated graded
algebra $\mathrm{gr}^{F}A$ and a smaller quadratic dual coalgebra
$C=(\mathrm{gr}^{F}A)^{?}$ (cf. Example 2.5 below).

Let $(A,F)$ be a nonhomogeneous quadratic algebra and
$C=(\mathrm{gr}^{F}A)^{?}$ be the quadratic dual coalgebra to the quadratic
algebra $\mathrm{gr}^{F}A$. Assume additionally that the algebra $A$ is
endowed with an augmentation $\alpha\colon A\longrightarrow k$ with the
augmentation ideal $A^{+}\subset A$. Then there is a natural isomorphism
$\mathrm{gr}^{F}_{1}A=F_{1}A/F_{0}A\simeq A^{+}\cap F_{1}A\subset A$. So the
vector space $V=\mathrm{gr}^{F}_{1}A$ can be viewed as a subspace in $A^{+}$,
and the graded coalgebra $C$ can be viewed as a subcoalgebra in
$\rotatebox[origin={c}]{180.0}{$T$}(A^{+})$,

$C\subset\rotatebox[origin={c}]{180.0}{$T$}(V)\subset\rotatebox[origin={c}]{180.0}{$T$}(A^{+})=\operatorname{Bar}_{\alpha}(A).$

One observes that the subcoalgebra $C\subset\operatorname{Bar}_{\alpha}(A)$ is
in fact a DG-subcoalgebra
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\subset\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$,
that is $\partial(C)\subset C$. Essentially, the first reason for that is
because the multiplication map $A^{+}\otimes_{k}A^{+}\longrightarrow A^{+}$
restricted to the subspace

$I\subset\mathrm{gr}^{F}_{1}A\otimes_{k}\mathrm{gr}^{F}_{1}A\simeq(A^{+}\cap
F_{1}A)\otimes_{k}(A^{+}\cap F_{1}A)\subset A^{+}\otimes_{k}A^{+}$

lands within the subspace $A^{+}\cap F_{1}A\subset A^{+}$ (by the definition
of $I$ as the kernel of the multiplication map
$\mathrm{gr}^{F}_{1}A\otimes_{k}\mathrm{gr}^{F}_{1}A\longrightarrow\mathrm{gr}^{F}_{2}A=F_{2}A/F_{1}A$
in the graded algebra $\mathrm{gr}^{F}A$). Further details need to be checked;
we refer to [83, Section 3], [61, Proposition 2.2] or [58, Proposition 4.1 in
Chapter 5]. A generalization to nonaugmented algebras $A$ will be discussed in
Example 6.11 below.

Let $(A,F,\alpha)$ be an augmented nonhomogeneous quadratic algebra and
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex of left
$A$-modules. Then one can easily see that
$C\otimes_{k}M\subset\rotatebox[origin={c}]{180.0}{$T$}(A^{+})\otimes_{k}M=\operatorname{Bar}_{\alpha}(A,M)$
is a subcomplex in
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)$,
that is $\partial(C\otimes_{k}M)\subset C\otimes_{k}M$. In anticipation of the
discussion in Section 5, we denote the complex $C\otimes_{k}M$ with the
differential induced by the differential $\partial$ on
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)$
by
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
This notation is indented to emphasize the fact that the differential on
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is _not_ simply the tensor product differential on the tensor product of two
complexes
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{k}M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
but rather some twisted version of it taking the action of $A$ in $M$ into
account.

Let $(A,F,\alpha)$ be a nonhomogeneous Koszul algebra. Then the DG-coalgebra
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is quasi-isomorphic to the
ambient DG-coalgebra
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$,
and the complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is quasi-isomorphic to the ambient bar-complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M)$.
Accordingly, one has

$H^{-n}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Tor}_{n}^{A}(k,k)\quad\text{and}\quad
H^{-n}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}M)\simeq\operatorname{Tor}_{n}^{A}(k,M)$

for any left $A$-module $M$ and all $n\geq 0$.

######  Theorem 2.4.

Let $(A,F,\alpha)$ be an augmented nonhomogeneous Koszul algebra over a field
$k$. Then the assignment
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
induces a triangulated equivalence between the derived category of left
$A$-modules $\mathsf{D}(A{\operatorname{\mathsf{--mod}}})$ and the _coderived_
category of left DG-comodules over the DG-coalgebra
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,

$\mathsf{D}(A{\operatorname{\mathsf{--mod}}})\simeq\mathsf{D}^{\mathsf{co}}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}}{\operatorname{\mathsf{--comod}}}).$

###### Proof.

This is a particular case of [64, Theorems 6.5(a) and 6.6]. The inverse
functor,
$\mathsf{D}^{\mathsf{co}}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}}{\operatorname{\mathsf{--comod}}})\longrightarrow\mathsf{D}(A{\operatorname{\mathsf{--mod}}})$,
assigns to a left DG-comodule
$N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the complex of left
$A$-modules $A\otimes^{\tau}N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
which means the graded left $A$-module $A\otimes_{k}N$ endowed with a twisted
differential, as explained below in Sections 5.3–5.4. The definition of the
coderived category will be given in Section 7.7. ∎

Any associative algebra $A$ can be viewed as a nonhomogeneous Koszul algebra
with the trivial filtration, as explained in Example 2.3(2). For an augmented
algebra $A$ endowed with the trivial filtration, one has
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$.
So Theorem 2.1 can be obtained as a particular case of Theorem 2.4 for the
trivial filtration $F$.

For a generalization of Theorem 2.4, see Theorem 5.4 below.

######  Example 2.5.

Let $\mathfrak{g}$ be a Lie algebra over a field $k$ and
$\Lambda(\mathfrak{g})$ be the exterior algebra spanned by the $k$-vector
space $\mathfrak{g}$. Then the Chevalley–Eilenberg differential $\partial$ on
$\Lambda(\mathfrak{g})$ makes $\Lambda(\mathfrak{g})$ a complex computing the
homology spaces $H_{*}(\mathfrak{g},k)$ of the Lie algebra $\mathfrak{g}$ with
coefficients in the trivial $\mathfrak{g}$-module $k$ [17, Chapter III], [45].
The differential $\partial$ does not respect the exterior multiplication on
$\Lambda(\mathfrak{g})$, but it is an odd coderivation of the exterior
comultiplication; so
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(\Lambda(\mathfrak{g}),\partial)$
is a DG-coalgebra.

The enveloping algebra $U(\mathfrak{g})$ is endowed with the natural
(Poincaré–Birkhoff–Witt) filtration $F$, making $U(\mathfrak{g})$ (the
thematic example of) a nonhomogeneous Koszul algebra over a field. There is
also a natural augmentation $\alpha$ on $U(\mathfrak{g})$, corresponding to
the action of $\mathfrak{g}$ in trivial $\mathfrak{g}$-modules. The
construction above assigns to the augmented nonhomogeneous quadratic algebra
$(U(g),F,\alpha)$ the DG-coalgebra $(\Lambda(\mathfrak{g}),\partial)$.

Moreover, for any $\mathfrak{g}$-module $M$, the homological
Chevalley–Eilenberg complex $(\Lambda(\mathfrak{g})\otimes_{k}M,\>\partial)$
of the Lie algebra $\mathfrak{g}$ with coefficients in $M$ is a DG-comodule
over the $(\Lambda(\mathfrak{g}),\partial)$. The construction above assigns
the DG-comodule
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}M=(\Lambda(\mathfrak{g})\otimes_{k}M,\>\partial)$
to the $U(\mathfrak{g})$-module $M$.

Thus Theorem 2.4 claims that the construction of the homological
Chevalley–Eilenberg complex induces an equivalence between the derived
category of $\mathfrak{g}$-modules and the coderived category of DG-comodules
over the DG-coalgebra $(\Lambda(\mathfrak{g}),\partial)$. For a finite-
dimensional Lie algebra $\mathfrak{g}$, the coalgebra $\Lambda(\mathfrak{g})$
is finite-dimensional; so one can pass to the dual algebra
$\Lambda(\mathfrak{g}^{*})$ and say that the construction of the cohomological
Chevalley–Eilenberg complex $(\Lambda(\mathfrak{g}^{*})\otimes_{k}M,\>d)$
induces an equivalence between the derived category of $\mathfrak{g}$-modules
and the coderived category of DG-modules over the DG-algebra
$\Lambda^{\text{\smaller\smaller$\scriptstyle\bullet$}}(\mathfrak{g}^{*})=(\Lambda(\mathfrak{g}^{*}),d)$,

(1)
$\mathsf{D}(\mathfrak{g}{\operatorname{\mathsf{--mod}}})\simeq\mathsf{D}^{\mathsf{co}}(\Lambda^{\text{\smaller\smaller$\scriptstyle\bullet$}}(\mathfrak{g}^{*}){\operatorname{\mathsf{--mod}}}).$

This example can be found in [64, Example 6.6].

Let us emphasize that these assertions are certainly _not_ true for the
conventional derived category of DG-(co)modules in place of the coderived
category. In fact, the derived category
$\mathsf{D}(\mathfrak{g}{\operatorname{\mathsf{--mod}}})$ is _not_ equivalent
to the derived category
$\mathsf{D}(\Lambda^{\text{\smaller\smaller$\scriptstyle\bullet$}}(\mathfrak{g}^{*}){\operatorname{\mathsf{--mod}}})$
already when $\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra
over a field $k$ of characteristic $0$, or indeed, a one-dimensional abelian
Lie algebra (cf. Example 1.2(2)). We will continue this discussion in Example
7.3 below.

A relative version of Theorem 2.4, with the ground field $k$ replaced by an
arbitrary associative ring $R$, can be found in the book [75]. See [75,
Sections 3.8 and 6.6].

###  2.6. Cobar-construction; DG-algebra and DG-module structures

Let $C$ be a (coassociative, counital) coalgebra over a field $k$, as defined
in Section 2.1. A _coaugmentation_ $\gamma$ on $C$ is a (counital) coalgebra
homomorphism $\gamma\colon k\longrightarrow C$. So the composition
$k\overset{\gamma}{\longrightarrow}C\overset{\epsilon}{\longrightarrow}k$ is
the identity map. We denote by $C^{+}=\operatorname{coker}(\gamma)$ the
cokernel of the augmentation map. Generally speaking, $C^{+}$ is a
coassociative coalgebra without counit (dually to the augmentation ideal,
which is an associative algebra without unit). The coalgebra homomorphism
$\gamma\colon k\longrightarrow C$ endows the one-dimensional $k$-vector space
$k$ with left and right $C$-comodule structures.

The _cobar-construction_
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
of a coaugmented coalgebra $C=(C,\gamma)$ is defined as the complex [20,
Chapter II], [1]

$k\overset{0}{\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}C^{+}\overset{\partial}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}C^{+}\otimes_{k}C^{+}\overset{\partial}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}C^{+}\otimes_{k}C^{+}\otimes_{k}C^{+}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

with the differential given by the formulas $\partial(c)=\mu^{+}(c)$,
$\partial(c_{1}\otimes c_{2})=\mu^{+}(c_{1})\otimes
c_{2}-c_{1}\otimes\mu^{+}(c_{2})$, …,

$\partial(c_{1}\otimes\dotsb\otimes c_{n})=\mu^{+}(c_{1})\otimes
c_{2}\otimes\dotsb\otimes c_{n}-\dotsb\\\ +(-1)^{i-1}c_{1}\otimes\dotsb\otimes
c_{i-1}\otimes\mu^{+}(c_{i})\otimes c_{i+1}\otimes\dotsb\otimes
c_{n}+\dotsb+(-1)^{n-1}c_{1}\otimes\dotsb\otimes
c_{n-1}\otimes\mu^{+}(c_{n}),$

etc., for all $c$, $c_{i}\in C^{+}$, $1\leq i\leq n$, and $n\geq 1$. Here
$\mu^{+}\colon C^{+}\longrightarrow C^{+}\otimes_{k}C^{+}$ is the
comultiplication map on $C^{+}$ (induced by the comultiplication map
$\mu\colon C\longrightarrow C\otimes_{k}C$). The leftmost differential
$\partial\colon k\longrightarrow C^{+}$ is the zero map.

Let $(C,\gamma)$ be a coaugmented coalgebra and $M$ be a left $C$-comodule.
The _cobar-construction_
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,M)$
of $C$ with the coefficients in $M$ is the complex

$M\overset{\partial}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}C^{+}\otimes_{k}M\overset{\partial}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}C^{+}\otimes_{k}C^{+}\otimes_{k}M\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

with the differentials given by the formulas $\partial(m)=\nu^{+}(m)$,
$\partial(c\otimes m)=\mu^{+}(c)\otimes m-c\otimes\nu^{+}(m)$, …,

$\partial(c_{1}\otimes\dotsb\otimes c_{n}\otimes m)=\mu^{+}(c_{1})\otimes
c_{2}\otimes\dotsb\otimes c_{n}\otimes m-\dotsb\\\
+(-1)^{n-1}c_{1}\otimes\dotsb\otimes c_{n-1}\otimes\mu^{+}(c_{n})\otimes
m+(-1)^{n}c_{1}\otimes\dotsb\otimes c_{n}\otimes\nu^{+}(m),$

etc., for all $c_{i}\in C^{+}$, $1\leq i\leq n$, and $n\geq 0$. Here
$\nu^{+}\colon M\longrightarrow C^{+}\otimes_{k}M$ is the left
$C^{+}$-coaction map obtained by composing the left $C$-coaction map
$\nu\colon M\longrightarrow C\otimes_{k}M$ with the natural surjection
$C\otimes_{k}M\longrightarrow C^{+}\otimes_{k}M$.

Dually to the discussion in Sections 2.3–2.4, one observes that the underlying
graded vector space $\operatorname{Cob}_{\gamma}(C)$ of the cobar-construction
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
coincides with the graded $k$-vector space
$\bigoplus_{n=0}^{\infty}C^{+}{}^{\otimes n}$. Consequently, the graded vector
space $\operatorname{Cob}_{\gamma}(C)$ has natural structures of an
associative algebra and a coassociative coalgebra.

It turns out that the cobar differential $\partial$ on
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
does _not_ respect the comultiplication on
$\operatorname{Cob}_{\gamma}(C)=\rotatebox[origin={c}]{180.0}{$T$}(C^{+})$.
However, the differential $\partial$ is compatible with the multiplication on
$\operatorname{Cob}_{\gamma}(C)=T(C^{+})$, i. e., it satisfies the Leibniz
rule with signs

$\partial(ab)=\partial(a)b+(-1)^{|a|}a\partial(b)$

for any elements $a$ and $b\in T(C^{+})$ of degrees $|a|$ and $|b|$. In other
words, the complex
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
endowed with the graded algebra structure of $T(C^{+})$ is a DG-algebra over
$k$.

The graded algebra of cohomology of the DG-algebra
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
is naturally isomorphic to (the graded algebra with the opposite
multiplication to) the Ext algebra of the left $C$-comodule $k$,

$H^{*}\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)\simeq\operatorname{Ext}_{C}^{*}(k,k),$

where $k$ is endowed with the left $C$-comodule structure via $\gamma$ and the
Ext is computed in the abelian category of left $C$-comodules. Similarly, for
any left $C$-comodule $M$, the graded module of cohomology of the DG-module
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,M)$
over the DG-algebra
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
(see the discussion in the next Section 2.7) is naturally isomorphic to the
Ext module $\operatorname{Ext}_{C}^{*}(k,M)$ over the algebra
$\operatorname{Ext}_{C}^{*}(k,k)$,

$H^{*}\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,M)\simeq\operatorname{Ext}_{C}^{*}(k,M).$

###  2.7. Derived Koszul duality formulated for complexes of comodules

Let $N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex of left
$C$-comodules. Then the cobar-construction
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a bicomplex of $k$-vector spaces. We totalize this bicomplex by taking
infinite direct sums along the diagonals.

Denote simply by $N$ the underlying graded $C$-comodule of a complex of left
$C$-comodules $N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The
underlying graded vector space $\operatorname{Cob}_{\gamma}(C,N)$ of the
cobar-construction
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
can be viewed as a free graded left module $T(C^{+})\otimes_{k}N$ over the
tensor algebra $T(C^{+})$. It turns out that the total differential on
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is compatible with the cobar differential $\partial$ on the DG-algebra
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(C)$ and the
left action of $\operatorname{Cob}_{\gamma}(C)$ in
$\operatorname{Cob}_{\gamma}(C,N)$; i. e., a Leibniz rule with signs similar
to the one in Section 2.6 is satisfied for this action. In other words, the
complex
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
endowed with the free graded module structure of $T(C^{+})\otimes_{k}N$ is a
left DG-module over the DG-algebra
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$.

We formulate two versions of the derived Koszul duality for complexes of
comodules, the nonconilpotent and the conilpotent one. The definitions of the
coderived and the absolute derived categories will be given in Sections 7.6
and 7.7.

######  Theorem 2.6.

Let $(C,\gamma)$ be a coaugmented coassociative coalgebra over a field $k$.
Then the assignment
$N^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
induces a triangulated equivalence between the _coderived_ category of left
$C$-comodules $\mathsf{D}^{\mathsf{co}}(C{\operatorname{\mathsf{--comod}}})$
and the _absolute derived_ category of left DG-modules over the DG-algebra
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$,

$\mathsf{D}^{\mathsf{co}}(C{\operatorname{\mathsf{--comod}}})\simeq\mathsf{D}^{\mathsf{abs}}(\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C){\operatorname{\mathsf{--mod}}}).$

###### Proof.

This is a particular case of [64, Theorem 6.7(a)]. The absolute derived
category
$\mathsf{D}^{\mathsf{abs}}(\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C){\operatorname{\mathsf{--mod}}})$
coincides with the coderived category
$\mathsf{D}^{\mathsf{co}}(\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C){\operatorname{\mathsf{--mod}}})$
by [64, Theorem 3.6(a)]; see Theorem 7.8(a) below. ∎

######  Theorem 2.7.

Let $C$ be a _conilpotent_ coassociative coalgebra over a field $k$, endowed
with its unique coaugmentation $\gamma$. Then the same assignment
$N^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
induces a triangulated equivalence between the _coderived_ category of left
$C$-comodules $\mathsf{D}^{\mathsf{co}}(C{\operatorname{\mathsf{--comod}}})$
and the conventional _derived_ category of left DG-modules over the DG-algebra
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$,

$\mathsf{D}^{\mathsf{co}}(C{\operatorname{\mathsf{--comod}}})\simeq\mathsf{D}(\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C){\operatorname{\mathsf{--mod}}}).$

###### Proof.

This is a particular case of [64, Theorem 6.4(a)]; see [47, Théorème 2.2.2.2]
and [42, Section 4] for an earlier approach. It will be explained in the next
Section 3 what a “conilpotent coalgebra” is. ∎

In both the theorems, the inverse functor assigns to a left DG-module
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over the DG-algebra
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
the complex of left $C$-comodules
$C\otimes^{\tau}M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, which is a
notation for the graded left $C$-comodule $C\otimes_{k}M$ endowed with a
twisted differential, as constructed below in Sections 5.3–5.4.

In both Theorems 2.6 and 2.7, the triangulated equivalence takes the
irreducible left $C$-comodule $k$ (with the $C$-comodule structure defined in
terms of the coaugmentation $\gamma$) to the free left DG-module
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over the DG-algebra
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$.
The same equivalences take the left DG-module $k$ over
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ (with the $A$-module
structure on $k$ defined in terms of the natural augmentation of
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$; see Section 4.4) to the
cofree (injective) left $C$-comodule $C$.

For generalizations of Theorems 2.6–2.7, see Section 5.4 below.

######  Remark 2.8.

One can obtain the definition of a coalgebra from the definition of an algebra
over a field by inverting the arrows, but inverting the arrows in the
formulation of Theorem 2.1 does not exactly produce the formulation of Theorem
2.6 or 2.7. The roles of algebras and coalgebras in derived nonhomogeneous
Koszul duality are _not_ symmetric up to inverting the arrows, as there are
subtle differences between the constructions on the algebra and coalgebra
side.

Of course, one can say that the associative algebras are the monoid objects in
the monoidal (tensor) category of vector spaces, while the coassociative
coalgebras are the monoid objects in the monoidal category opposite to vector
spaces. Then the point is that the monoidal category of infinite-dimensional
vector spaces is _not_ equivalent to its opposite monoidal category, and
moreover, the properties of these categories are different in a way important
for nonhomogeneous Koszul duality.

To begin with, the tensor product of vector spaces preserves infinite direct
sums, but not infinite products. Furthermore, the functor of tensor product of
vector spaces (with one of the two objects fixed) has a right adjoint, but not
a left adjoint functor. It is also relevant that the (filtered) direct limits
of vector spaces are exact functors, but the inverse limits aren’t. These
differences cause the lack of symmetry between the Koszul duality theorems
formulated above and in Sections 2.4–2.5.

In particular, it is illuminating to observe that the constructions of the DG-
comodule
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
and the DG-module
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
are _not_ transformed into one another by inverting the arrows, for the subtle
reason that the direct sum (rather than the direct product) totalization of
the respective bicomplex is used in _both_ the constructions. The fact that
the tensor products preserve the direct sums, but not the direct products of
vector spaces explains this choice of the totalization procedures, but it
breaks the symmetry between the algebra and coalgebra sides.

##  3\. Structure Theory of Coalgebras; Conilpotency

All coassociative coalgebras over a field are unions of their finite-
dimensional subcoalgebras. For this reason, the structure theory of coalgebras
stands approximately on the same level of complexity as the structure theory
of finite-dimensional algebras. So it is much simpler than the structure
theory of infinite-dimensional algebras and rings. Some details of this
argument are elaborated upon in this section.

###  3.1. Brief remarks about coalgebras

The theory of coalgebras is notoriously counterintuitive. Most algebraists
seem to find it difficult to invert the arrows properly in their mind in order
to pass from the familiar ring theory to the coalgebra theory. One needs to
get used to that.

Here we only mention a couple of obvious points. In ring theory, subrings are
aplenty and quotient rings are rare. To pass to the quotient ring, one needs
to have an ideal in the ring. All ideals are subalgebras (without unit), but
most nonunital subalgebras are not ideals. Furthermore, all two-sided ideals
are both left ideals and right ideals, but most left or right ideals are not
(two-sided) ideals.

In coalgebras, the situation is reversed. A quotient coalgebra is the dual
concept to a subalgebra, and a subcoalgebra is the dual concept to a quotient
algebra. Quotient coalgebras are aplenty, but subcoalgebras are rare.

To pass to the quotient coalgebra, it suffices to have a coideal in the
coalgebra. If one is satisfied with obtaning a noncounital quotient coalgebra,
then the definition of a coideal reduces to this: a vector subspace $J$ in a
coalgebra $C$ is said to be a _coideal_ (in the noncounital sense) if
$\mu(J)\subset J\otimes_{k}C+C\otimes_{k}J$ (where $\mu\colon C\longrightarrow
C\otimes_{k}C$ is the comultiplication map). The definition of a coideal in
the counital sense includes an additional condition that the counit
$\epsilon\colon C\longrightarrow k$ must vanish on a coideal $J$.

For comparison, a _subcoalgebra_ $D\subset C$ is a vector subspace such that
$\mu(D)\subset D\otimes_{k}D$. One immediately observes that any subcoalgebra
is a coideal (in the noncounital sense), but most coideals are not
subcoalgebras. Furthermore, any left coideal (i. e., a left subcomodule in the
left $C$-comodule $C$) is a two-sided coideal, and any right coideal is a two-
sided coideal (in the noncounital sense), but most (two-sided) coideals are
neither left nor right coideals.

It is also worth mentioning that the comodule structures induced via
homomorphisms of coalgebras are pushed forward rather than pulled back. So if
$f\colon C\longrightarrow D$ is a homomorphism of coalgebras, then any (left
or right) $C$-comodule acquires the induced structure of a left $D$-comodule.
The resulting exact, faithful functor
$C{\operatorname{\mathsf{--comod}}}\longrightarrow
D{\operatorname{\mathsf{--comod}}}$ on the abelian categories of comodules is
called the _corestriction of scalars_ with respect to $f$.

We refer to [88, Section 1.4] for further details.

###  3.2. Local finite-dimensionality

The following classical result can be found in [88, Corollary 2.1.4 and
Theorem 2.2.1].

######  Lemma 3.1.

(a) Any coassociative coalgebra $C$ is the (directed) union of its finite-
dimensional subcoalgebras. The sum of any two finite-dimensional subcoalgebras
in $C$ is a finite-dimensional subcoalgebra in $C$.

(b) Any (left) comodule over a coassociative coalgebra $C$ is the union of its
finite-dimensional $C$-subcomodules.

(c) Any finite-dimensional $C$-comodule $M$ is a comodule over some finite-
dimensional subcoalgebra $E\subset C$ (so the image of the coaction map
$\nu\colon M\longrightarrow C\otimes_{k}M$ is contained in $E\otimes_{k}M$).

###### Proof.

Let us sketch a proof of the first assertion of part (a); the other assertions
are similar or simpler. Given a coassociative, counital coalgebra $C$,
consider an arbitrary finite-dimensional vector subspace $V\subset C$, and
denote by $E\subset C$ the full preimage of $C\otimes_{k}V\otimes_{k}C$ under
the iterated comultiplication map
$\mu^{(2)}=(\mu\otimes\mathrm{id}_{C})\circ\mu=(\mathrm{id}_{C}\otimes\mu)\circ\mu\colon
C\longrightarrow C\otimes_{k}C\otimes_{k}C$. Then $E=E(V)$ is a subcoalgebra
in $C$, one has $E\subset V$ (so $E$ is finite-dimensional), and
$C=\bigcup_{V\subset C}E(V)$. ∎

Specifying a coalgebra structure on a finite-dimensional $k$-vector space $E$
is equivalent to specifying an algebra structure on the dual vector space
$E^{*}$. So the dual vector space to a finite-dimensional coalgebra is a
finite-dimensional algebra, and the dual vector space to a finite-dimensional
algebra is a finite-dimensional coalgebra. For infinite-dimensional vector
spaces the situation is more complicated: the dual vector space to a coalgebra
has a natural algebra structure, but _not_ vice versa.

###  3.3. Conilpotent coalgebras

A nonunital associative algebra $A$ is said to be _nilpotent_ if there exists
an integer $n\geq 1$ such that the product of any $n$ elements in $A$
vanishes, that is $A^{n}=0$. An augmented algebra $A$ is said to be
_nilpotent_ if its augmentation ideal $A^{+}$ is a nilpotent algebra without
unit, that is, the product of any $n$ elements in $A^{+}$ vanishes.

Notice that the augmentation of a nilpotent augmented algebra $A$ is _unique_.
Since the product of any $n$ elements in $A^{+}$ vanishes, there are no
nonzero $k$-algebra homomorphisms $A^{+}\longrightarrow k$; hence only one
unital $k$-algebra homomorphism $A\longrightarrow k$.

Coalgebras are better suited for nilpotency than algebras, in the sense that
one can speak of a coalgebra being conilpotent without assuming existence of a
related finite integer $n$. One can say that a (unital or nonunital) finite-
dimensional coalgebra $E$ is _conilpotent_ if its dual finite-dimensional
algebra is nilpotent. Then an arbitrary coalgebra $C$ is said to be
_conilpotent_ if all its finite-dimensional subcoalgebras are conilpotent, i.
e., $C$ is a union of finite-dimensional conilpotent coalgebras. So a finite-
dimensional conilpotent coalgebra always has a finite conilpotency degree, but
an infinite-dimensional conilpotent coalgebra need _not_ have it.

Here is an explicit definition of a conilpotent coalgebra without counit that
is easy to work with on the technical level. A noncounital coalgebra $D$ is
said to be _conilpotent_ if for every element $x\in D$ there exists an integer
$n\geq 1$ such that the image of $x$ under the iterated comultiplication map
$\mu^{(n)}\colon D\longrightarrow D^{\otimes n+1}$ vanishes. A coaugmented
coalgebra $(C,\gamma)$ is called _conilpotent_ if the noncounital coalgebra
$C^{+}=C/\gamma(k)$ is conilpotent in the sense of the previous definition. A
conilpotent coaugmented coalgebra has a unique coaugmentation; moreover, any
counital coalgebra homomorphism between two conilpotent coalgebras preserves
the coaugmentations.

A conilpotent coalgebra $C$ comes endowed with a natural increasing filtration
$F$. Namely, $F_{-1}C=0$, $F_{0}C=\gamma(k)\simeq k$, and for every $n\geq 0$
the subspace $F_{n}C\subset C$ consists of all elements $c\in C$ whose image
in $(C^{+})^{\otimes n+1}$ under the composition of maps $C\longrightarrow
C^{\otimes n+1}\longrightarrow(C^{+})^{\otimes n+1}$ vanishes. Then one has
$\mu(F_{n}C)\subset\sum_{p+q=n}F_{p}C\otimes_{k}F_{q}C\subset C\otimes_{k}C$,
so the filtration $F$ on $C$ is compatible with the comultiplication. The
equation $C=\bigcup_{n\geq 0}F_{n}C$ expresses the condition that $C$ is
conilpotent.

A noncounital conilpotent coalgebra $D$ comes endowed with a similarly defined
natural increasing filtration $F$. Namely, one has $F_{0}D=0$, and for every
$n\geq 0$ the subspace $F_{n}D\subset D$ is the kernel of the iterated
comultiplication map $\mu^{(n)}\colon D\longrightarrow D^{\otimes n+1}$. Then
one has $\mu(F_{n}D)\subset\sum_{p+q=n}F_{p}D\otimes_{k}F_{q}D\subset
D\otimes_{k}D$ and $D=\bigcup_{n\geq 1}F_{n}D$.

######  Remark 3.2.

Let $V$ be a $k$-vector space. In the discussion of the tensor algebra $T(V)$
and tensor coalgebra $\rotatebox[origin={c}]{180.0}{$T$}(V)$ in Section 2.3,
it was mentioned that $T(V)$ is the free associative algebra spanned by $V$,
but a discussion of a possible cofreeness property of the tensor coalgebra
$\rotatebox[origin={c}]{180.0}{$T$}(V)$ was postponed. Now it is the time for
such a discussion.

The first important observation is that the coaugmented coalgebra
$\rotatebox[origin={c}]{180.0}{$T$}(V)$ is always conilpotent. In fact, one
has $F_{n}\rotatebox[origin={c}]{180.0}{$T$}(V)=\bigoplus_{i=0}^{n}V^{\otimes
i}$ for every $n\geq 0$. Moreover, $\rotatebox[origin={c}]{180.0}{$T$}(V)$ is
the cofree conilpotent coalgebra cospanned by $V$, in the following sense. Let
$C$ be a conilpotent coalgebra, and let $f\colon C\longrightarrow V$ be a
$k$-linear map annihilating the coaugmentation, i. e., such that the
composition $f\gamma\colon k\longrightarrow V$ vanishes. Then there exists a
unique coalgebra homomorphism $t\colon
C\longrightarrow\rotatebox[origin={c}]{180.0}{$T$}(V)$ whose composition with
the direct summand projection
$\rotatebox[origin={c}]{180.0}{$T$}(V)\longrightarrow V^{\otimes 1}=V$ is
equal to $f$.

Let us emphasize that the assertion above is _not_ true for nonconilpotent
coalgebras $C$. For such coalgebras, a homomorphism like $t$ does not exist in
general.

###  3.4. Cosemisimple coalgebras and the coradical filtration

The classical structure theory of finite-dimensional associative algebras over
a field tells that any such algebra $A$ has a nilpotent Jacobson radical $J$,
while the quotient algebra $A/J$ decomposes uniquely as a finite product of
simple algebras. The Jacobson radical $J\subset A$ is simultaneously the
unique maximal nilpotent ideal and the unique minimal ideal in $A$ for which
the quotient algebra $A/J$ is semisimple.

Moreover, if $f\colon A\longrightarrow B$ is a surjective homomorphism of
finite-dimensional $k$-algebras, then the Jacobson radicals $J(A)\subset A$
and $J(B)\subset B$ are well-behaved with respect to $f$ in the sense that
$f(J(A))=J(B)$. The induced surjective homomorphism of semisimple algebras
$A/J(A)\longrightarrow B/J(B)$ is a projection onto a direct factor; in fact,
it represents the algebra $B/J(B)$ as a direct factor of the algebra $A/J(A)$
in a unique way.

Passing to the dual coalgebra, one obtains what can be called “the structure
theory of finite-dimensional coassociative coalgebras” $E$. Any such coalgebra
has a unique maximal cosemisimple subcoalgebra $E^{\mathrm{ss}}$, which
decomposes uniquely as a direct sum of cosimple coalgebras. The (noncounital)
quotient coalgebra $E/E^{\mathrm{ss}}$ is conilpotent.

Moreover, if $E\subset D$ is a subcoalgebra in a finite-dimensional coalgebra,
then $E^{\mathrm{ss}}=E\cap D^{\mathrm{ss}}$. The cosemisimple coalgebra
$D^{\mathrm{ss}}$ can be decomposed as a direct sum of its subcoalgebra
$E^{\mathrm{ss}}$ and a complementary subcoalgebra in a unique way.

Let us formulate the general definition of a _cosemisimple coalgebra_ , in the
form of a list of equivalent conditions in the next lemma. Here a nonzero
coassociative, counital coalgebra over $k$ is called _cosimple_ if it has no
nonzero proper subcoalgebras.

######  Lemma 3.3.

For any coassociative, counital coalgebra $C$ over a field $k$, the following
conditions are equivalent:

  1. (1)

the abelian category of left $C$-comodules is semisimple;

  2. (2)

the abelian category of right $C$-comodules is semisimple;

  3. (3)

$C$ is the sum of its cosimple subcoalgebras;

  4. (4)

$C$ uniquely decomposes as a direct sum of its (finite-dimensional) cosimple
subcoalgebras, with all the cosimple subcoalgebras of $C$ appearing as direct
summands in this direct sum decomposition. ∎

Now let $C$ be an arbitrary (infinite-dimensional) coassociative coalgebra.
Representing $C$ as the directed union of its finite-dimensional subcoalgebras
$E$ and passing to the direct limit, one obtains the following structural
result.

######  Proposition 3.4.

Any coassociative, counital coalgebra $C$ over a field $k$ contains a unique
maximal cosemisimple subcoalgebra $C^{\mathrm{ss}}$, which is can be obtained
as the sum of all co(semi)simple subcoalgebras of $C$. Simultaneously,
$C^{\mathrm{ss}}$ is the unique minimal subcoalgebra among all the
subcoalgebras $D\subset C$ for which that the (noncounital) quotient coalgebra
$C/D$ is conilpotent. ∎

Following the discussion in Section 3.3, the conilpotent noncounital coalgebra
$D=C/C^{\mathrm{ss}}$ is endowed with a natural increasing filtration $F$.
Denoting by $F_{n}C$ the full preimage of $F_{n}(C/C^{\mathrm{ss}})$ under the
natural surjective homomorphism $C\longrightarrow C/C^{\mathrm{ss}}$, one
obtains a natural increasing filtration $F$ on $C$, called the _coradical
filtration_. What we call the “maximal cosemisimple subcoalgebra”
$C^{\mathrm{ss}}$ of a coalgebra $C$ is otherwise known as the _coradical_ of
$C$ [88, Chapters VIII–IX], [49, Chapter 5].

######  Remark 3.5.

The definitions of the coradical filtration in [88, 49] are formulated in
terms of the so-called _wedge_ operation on vector subspaces of a coalgebra.
Let $C$ be a coassociative coalgebra over $k$ and $X$, $Y\subset C$ be two
vector subspaces. According to [88, Section 9.0], [49, proof of Theorem
5.2.2], the wedge $X\wedge Y\subset C$ is defined as the kernel of the
composition $C\longrightarrow C\otimes_{k}C\longrightarrow C/X\otimes_{k}C/Y$
of the comultiplication map $C\longrightarrow C\otimes_{k}C$ with the natural
surjection $C\otimes_{k}C\longrightarrow C/X\otimes_{k}C/Y$. According to [88,
Proposition 9.0.0(c)], the wedge operation is associative. By the definition,
put $\bigwedge^{1}X=X$ and $\bigwedge^{n}X=(\bigwedge^{n-1}X)\wedge X$ for
$n\geq 2$.

Notice that the definition of the wedge makes perfect sense for noncounital
coalgebras $D$ as well. Arguing by induction on $n$, one easily shows that
$\bigwedge^{n}X\subset D$ is the kernel of the composition $D\longrightarrow
D^{\otimes n}\longrightarrow(D/X)^{\otimes n}$ of the iterated
comultiplication map $D\longrightarrow D^{\otimes n}$ with the natural
surjection $D^{\otimes n}\longrightarrow(D/X)^{\otimes n}$ (for every $n\geq
1$). Then, for a noncounital conilpotent coalgebra $D$, the definition of the
natural increasing filtration $F$ on $D$ given in Section 3.3 can be rephrased
as $F_{n}D=\bigwedge^{n+1}0\subset D$.

Let $p\colon C\longrightarrow D$ be a homomorphism of noncounital coalgebras.
Then one can easily see that the wedge operation commutes with taking the full
preimages of vector subspaces under $p$: for any subspaces $X$, $Y\subset D$
one has $p^{-1}(X\wedge Y)=p^{-1}X\wedge p^{-1}Y$, hence
$p^{-1}(\bigwedge^{n}X)=\bigwedge^{n}(p^{-1}X)$.

Now let $C$ be a coassociative, counital coalgebra over $k$. The coradical
filtration is defined in [88, Section 9.1] by the rule
$C_{n}=\bigwedge^{n+1}C^{\mathrm{ss}}\subset C$, where $C^{\mathrm{ss}}\subset
C$ is what we call the maximal cosemisimple subcoalgebra (called the coradical
in [88, 49]). In the book [49, §5.2], the coradical filtration is defined by
induction using the rules $C_{0}=C^{\mathrm{ss}}$ and, essentially,
$C_{n}=C^{\mathrm{ss}}\wedge C_{n-1}\subset C$; this is clearly equivalent to
the definition in [88]. Taking $D=C/C^{\mathrm{ss}}$, one concludes that
$C_{n}=\bigwedge^{n+1}C^{\mathrm{ss}}=\bigwedge^{n+1}p^{-1}(0)=p^{-1}(\bigwedge^{n+1}0)=p^{-1}(F_{n}D)=F_{n}C$
(where $p\colon C\longrightarrow D$ is the natural surjection). Thus the
definition of the coradical filtration by the rule
$C_{n}=\bigwedge^{n+1}C^{\mathrm{ss}}$ in [88, 49] agrees with our definition
of it by the rule $F_{n}C=p^{-1}(F_{n}D)$ given above in this section.

A counital coalgebra $C$ is conilpotent (in the sense of the definition in
Section 3.3) if and only if there is an isomorphism of coalgebras
$C^{\mathrm{ss}}\simeq k$, or in other words, if and only if the coalgebra
$C^{\mathrm{ss}}$ is one-dimensional.

##  4\. DG-Algebras and DG-Coalgebras

The aim of this section is to formulate a simple version of Koszul duality
between algebras and coalgebras (rather than modules and comodules). The
notion of a _filtered quasi-isomorphism_ of DG-coalgebras [30] plays a key
role.

###  4.1. Augmented DG-algebras

Let $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(A,d)$ be a DG-algebra
over a field $k$. So $A=\bigoplus_{i\in\mathbb{Z}}A^{i}$ is a graded
$k$-algebra and $d\colon A\longrightarrow A$ is a homogeneous $k$-linear map
of degree $1$ satisfying the Leibniz rule with signs with respect to the
multiplication (in other words, $d$ is an odd derivation of $A$) such that
$d^{2}=0$.

An _augmentation_ $\alpha$ on
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a (unital) DG-algebra
homomorphism $\alpha\colon
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow k$. So
$\alpha(a)=0$ for all $a\in A^{i}$, $i\neq 0$, $\alpha(ab)=\alpha(a)\alpha(b)$
for all $a$, $b\in A$, $\alpha(1)=1$, and $\alpha(d(a))=0$ for all $a\in
A^{-1}$. We denote by
$A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}=\ker(\alpha)\subset
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the augmentation ideal. So
$A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}$ is a homogeneous two-
sided DG-ideal in $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, and
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}=k\oplus
A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}$ as a complex of
$k$-vector spaces.

The _bar-construction_
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of an augmented DG-algebra
$(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},\alpha)$ is the bicomplex

$k\overset{0}{\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu}A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muA^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\otimes_{k}A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muA^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\otimes_{k}A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\otimes_{k}A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

with the bar differential $\partial$ given by the formulas similar to those in
Section 1.1 (except for the $\pm$ signs, which are more complicated in the
case of a DG-algebra $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$) and
the differential $d$ induced by the differential $d$ on
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The differential $d$ on
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
acts on every tensor power
$A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}{}^{\otimes n}$ of the
augmentation ideal $A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}$ by
the tensor product of $n$ copies of the differential $d$ on
$A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}$ (in the sense of the
tensor product of complexes).

The total complex of the bar-construction
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is produced by taking infinite direct sums along the diagonals. The underlying
graded vector space of
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is endowed with the graded coalgebra structure of the tensor coalgebra
$\operatorname{Bar}_{\alpha}(A)=\rotatebox[origin={c}]{180.0}{$T$}(A^{+}[1])$.
Here the shift of cohomological grading $[1]$ in the latter formula is a way
to express the passage to the total complex of the bicomplex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
with its total grading equal to the grading induced by the grading of $A$
minus the grading by the number of the tensor factors. This makes
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
a DG-coalgebra (as defined in Section 2.2). We refer to [64, Section 6.1] for
further details, incuding the (not so trivial) sign rules for the
differentials $\partial$ and $d$.

###  4.2. Coaugmented DG-coalgebras

Let $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(C,d)$ be a DG-coalgebra
over $k$ (see Section 2.2). A _coaugmentation_ $\gamma$ on
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a (counital) DG-
coalgebra homomorphism $\gamma\colon k\longrightarrow C$. So $\gamma(k)\subset
C^{0}$, $\gamma$ is a homomorphism of coalgebras, the composition
$k\overset{\gamma}{\longrightarrow}C\overset{\epsilon}{\longrightarrow}k$ is
the identity map, and $d(\gamma(1))=0$. We put
$C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}=\operatorname{coker}(\gamma)$;
so $C^{+}$ is, generally speaking, a DG-coalgebra without counit, and
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=k\oplus
C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}$ as a complex of
$k$-vector spaces.

The _cobar-construction_
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of a coaugmented DG-coalgebra
$(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},\gamma)$ is the bicomplex

$k\overset{0}{\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\otimes_{k}C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\otimes_{k}C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\otimes_{k}C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

with the cobar differential $\partial$ given by the formulas similar to those
in Section 2.6 (except for the $\pm$ signs) and the differential $d$ induced
by the differential $d$ on $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
The differential $d$ on
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
acts on every tensor power
$C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}{}^{\otimes n}$ of the
complex $C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}$ by the tensor
product of $n$ copies of the differential $d$ on
$C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}$.

The total complex of the cobar-construction
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is produced by taking infinite direct sums along the diagonals. The underlying
graded vector space of
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is endowed with the graded algebra structure of the tensor algebra
$\operatorname{Cob}_{\gamma}(C)=T(C^{+}[-1])$. Here the shift of cohomological
grading $[-1]$ is a way to express the passage to the total complex of the
bicomplex
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
with its total grading equal to the grading induced by the grading of $C$ plus
the grading by the number of tensor factors. This makes
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
a DG-algebra. We refer to [64, Section 6.1] for further details, including the
sign rules for the differentials $\partial$ and $d$.

###  4.3. Conilpotent DG-coalgebras

A noncounital graded coalgebra $D$ is called _conilpotent_ if its underlying
noncounital ungraded coalgebra is conilpotent (see the definition in Section
3.3). A coaugmented graded coalgebra $(C,\gamma)$ is called _conilpotent_ if
its underlying coaugmented ungraded coalgebra is conilpotent; equivalently,
this means that the noncounital graded coalgebra $D=C/\gamma(k)$ is
conilpotent.

A coaugmented DG-coalgebra
$(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},\gamma)$ is called
_conilpotent_ if its underlying graded coalgebra $C$ is conilpotent. In other
words, a DG-coalgebra $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
conilpotent if and only if the graded coalgebra $C$ is conilpotent _and_ its
coaugmentation $\gamma\colon k\longrightarrow C$ is a DG-coalgebra map, that
is, $d(\gamma(1))=0$.

By construction, the canonical increasing filtration $F$ on a conilpotent
(noncounital or coaugmented) graded coalgebra is a filtration by homogeneous
vector subspaces. The canonical increasing filtration $F$ on a conilpotent DG-
coalgebra $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a filtration
by subcomplexes $F_{n}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\subset
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.

###  4.4. Duality between DG-algebras and DG-coalgebras

Denote by $k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}$ the category of DG-
algebras over $k$ (with the usual DG-algebra homomorphisms). Furthermore, let
$k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}$ denote the
category of augmented DG-algebras (with DG-algebra homomorphisms preserving
the augmentations).

Similarly, denote by $k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}$ the
category of DG-coalgebras over $k$, and by
$k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{coaug}}$ the
category of coaugmented DG-coalgebras. Let
$k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}}\subset
k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{coaug}}$ denote the
full subcategory whose objects are the conilpotent DG-coalgebras.

One easily observes that the cobar-construction
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of any coaugmented DG-coalgebra
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is naturally an augmented
DG-algebra (with the direct summand projection
$\operatorname{Cob}_{\gamma}(C)\longrightarrow(C^{+})^{\otimes 0}=k$ providing
the augmentation). The bar-construction
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of any augmented DG-algebra $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is not only naturally coaugmented, but even a conilpotent DG-coalgebra
(essentially, because the tensor coalgebra is conilpotent; see Remark 3.2).

So the bar-construction is a functor

$\operatorname{Bar}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon
k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}},$

while the cobar-construction is a functor

$\operatorname{Cob}_{\gamma}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon
k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{coaug}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}.$

######  Lemma 4.1.

The restriction of the cobar-construction to conilpotent DG-coalgebras,

$\operatorname{Cob}_{\gamma}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon
k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}$

is naturally a left adjoint functor to the bar-construction
$\operatorname{Bar}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon
k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}\longrightarrow
k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}}$.

###### Proof.

Essentially, the assertion holds because $T(V)$ is the free (graded) algebra
spanned by a (graded) vector space $V$, while
$\rotatebox[origin={c}]{180.0}{$T$}(V)$ is the cofree conilpotent (graded)
coalgebra cospanned by $V$. A more detailed explanation, based on the notion
of a twisting cochain, will be offered in Section 5.2 below. ∎

One would like to define equivalence relations (i. e., the classes of
morphisms to be inverted) in the categories of augmented DG-algebras and
conilpotent DG-coalgebras so that, after inverting these classes of morphisms,
the adjoint functors
$\operatorname{Bar}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and
$\operatorname{Cob}_{\gamma}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
become equivalences of categories. One important obstacle is that it is _not_
good enough to just invert the usual quasi-isomorphisms in both the categories
$k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}$ and
$k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}}$, for the
reason demonstrated by Example 1.1.

The problem is resolved by using the concept of a _filtered quasi-isomorphism_
introduced by Hinich in [30, Section 4]. In the terminology of [30], an
_admissible filtration_ on a coaugmented DG-coalgebra
$(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},\gamma)$ is an exhaustive
comultiplicative increasing filtration by subcomplexes
$F_{n}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ such that
$F_{-1}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=0$ and
$F_{0}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\gamma(k)$. This means
that $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\bigcup_{n\geq
0}F_{n}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$\mu(F_{n}C^{\text{\smaller\smaller$\scriptstyle\bullet$}})\subset\sum_{p+q=n}^{p,q\geq
0}F_{p}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes
F_{q}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for every $n\geq 0$.

In particular, any coaugmented DG-coalgebra admitting an admissible filtration
is conilpotent. Conversely, the canonical increasing filtration on any
conilpotent DG-coalgebra is admissible.

Let $f\colon C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a morphism of conilpotent
DG-coalgebras. The morphism $f$ is said to be a _filtered quasi-isomorphism_
if there exist admissible filtrations $F$ on both the DG-coalgebras
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ such that $f(F_{n}C)\subset
F_{n}D$ and the induced map
$F_{n}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}/F_{n-1}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
F_{n}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}/F_{n-1}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a quasi-isomorphism of complexes of $k$-vector spaces for every $n\geq 0$.

Notice that there is _no_ reason to expect that the composition of filtered
quasi-isomorphisms should be a filtered quasi-isomorphism. Nevertheless,
leaving set-theoretical issues aside, to any class of morphisms $\mathsf{S}$
in a category $\mathsf{C}$ one can assign the category
$\mathsf{C}[\mathsf{S}^{-1}]$ obtained by formally inverting all morphisms
from $\mathsf{S}$; in particular, one can formally invert all filtered quasi-
isomorphisms of DG-coalgebras.

By contrast, a morphism of DG-algebras $f\colon A\longrightarrow B$ is said to
be a quasi-isomorphism if it is a quasi-isomorphism of the underlying
complexes of vector spaces. Clearly, for any pair of composable morphisms of
DG-algebras $f$ and $g$, if two of the morphisms $f$, $g$, and $gf$ are quasi-
isomorphisms, then so is the third one.

The following theorem goes back to [30, Theorem 3.2] and [47, Théorème
1.3.1.2]. In the stated form, it can be found in [64, Theorem 6.10(b)].

######  Theorem 4.2.

Let $\mathsf{Quis}$ be the class of all quasi-isomorphisms of augmented DG-
algebras and $\mathsf{FQuis}$ be the class of all filtered quasi-isomorphisms
of conilpotent DG-coalgebras over $k$. Then the adjoint functors
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}$
and
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}$
induce mutually inverse equivalences of categories

$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}\colon
k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}[\mathsf{Quis}^{-1}]\,\simeq\,k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}}[\mathsf{FQuis}^{-1}]\,:\\!\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}.$

∎

######  Remarks 4.3.

Clearly, any filtered quasi-isomorphism of DG-coalgebras is a quasi-
isomorphism of their underlying complexes of vector spaces (i. e., a quasi-
isomorphism in the usual sense of the word). By the converse is _not_ true.

The functor
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}\colon
k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}\longrightarrow
k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}}$ takes
quasi-isomorphisms to filtered quasi-isomorphisms. In particular, it follows
that it takes quasi-isomorphisms to quasi-isomorphisms.

The functor
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}\colon
k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}}\longrightarrow
k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}$ takes filtered
quasi-isomorphisms to quasi-isomorphisms. This important property of filtered
quasi-isomorphisms of DG-coalgebras is a part of Theorem 4.2.

However, the functor
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}$
does _not_ take quasi-isomorphisms to quasi-isomorphisms, generally speaking.
For example, let $f\colon(A,\alpha)\longrightarrow(B,\beta)$ be a morphism of
augmented algebras (or augmented DG-algebras) as in Example 1.1, that is, $f$
is _not_ a quasi-isomorphism, but
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(f)$ is. Put
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\beta}(B^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
and $g=\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(f)$;
so
$g\colon(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},\gamma)\longrightarrow(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},\delta)$
is a quasi-isomorphism (but not a filtered quasi-isomorphism!) of conilpotent
DG-coalgebras.

It is a part of Theorem 4.2 that the adjunction morphism
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\eta}(\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\varepsilon}(E^{\text{\smaller\smaller$\scriptstyle\bullet$}}))\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a quasi-isomorphism of
DG-algebras for any augmented DG-algebra
$(E^{\text{\smaller\smaller$\scriptstyle\bullet$}},\varepsilon)$. So the
adjunction morphisms
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}}))\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\delta}(\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\beta}(B^{\text{\smaller\smaller$\scriptstyle\bullet$}}))\longrightarrow
B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are quasi-isomorphisms. It
follows that
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(g)\colon\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\delta}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is _not_ a quasi-isomorphism of DG-algebras.

###  4.5. Quillen equivalence between DG-algebras and DG-coalgebras

The derived Koszul duality and comodule-contramodule correspondence results
can be usually expressed in the language of Quillen equivalences between model
categories [64, Section 8.4]. In particular, the Koszul duality between
augmented DG-algebras and conilpotent DG-coalgebras stated in Theorem 4.2 can
be expressed as a Quillen equivalence. Let us briefly sketch the related
assertions.

We suggest the book [33] as a standard reference source on model categories
and related concepts. The category
$k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}$ of DG-algebras over $k$ has a
standard model category structure in which the quasi-isomorphisms are the weak
equivalences and the surjective DG-algebra maps (i. e., the DG-algebra maps
that are surjective as maps of graded vector spaces) are the fibrations [29,
37]. It is clear from the explicit descriptions of the classes of cofibrations
and trivial cofibrations in $k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}$,
which can be found in the formulation of [64, Theorem 9.1(a)], that the model
category $k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}$ is cofibrantly
generated. The similar assertions hold for the category of augmented DG-
algebras $k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}$ [64,
Theorem 9.1(b)].

The category
$k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}}$ of
conilpotent DG-coalgebras over $k$ also has a natural model structure [30,
Theorem 3.1], [47, Théorème 1.3.1.2(a)], [64, Theorem 9.3(b)]. The weak
equivalences can be described as the morphisms which get inverted when one
inverts the filtered quasi-isomorphisms. The cofibrations are the injective
DG-coalgebra maps (i. e., the DG-coalgebra maps that are injective as maps of
graded vector spaces). Explicit descriptions of all the classes of morphisms
involved can be found in the formulation of [64, Theorem 9.3(b)].

The model category
$k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}}$ is
likewise cofibrantly generated. In fact, the injective morphisms of finite-
dimensional DG-coalgebras are the generating cofibrations (as one can see from
the fact that any DG-coalgebra is the directed union of its finite-dimensional
DG-subcoalgebras; cf. Section 3.2). The injective filtered quasi-isomorphisms
strictly compatible with the filtrations, acting between finite-dimensional
DG-coalgebras, are the generating trivial cofibrations (this follows from the
description of the trivial cofibrations as the retracts of the injective
filtered quasi-isomorphisms strictly compatible with the filtrations, as per
[64, Theorem 9.3(b)]).

The pair of adjoint functors in Lemma 4.1 and Theorem 4.2 is a Quillen
equivalence between
$k{\operatorname{\mathsf{--alg}}}_{\mathsf{dg}}^{\mathsf{aug}}$ and
$k{\operatorname{\mathsf{--coalg}}}_{\mathsf{dg}}^{\mathsf{conilp}}$ [30,
Theorem 3.2], [47, Théorème 1.3.1.2(b)], [64, end of Section 9.3].

##  5\. Twisting Cochains

The concept of a _twisting cochain_ explains the adjunction of Lemma 4.1 and
the constructions of inverse equivalences in Theorems 2.1, 2.4, 2.6, and 2.7.
More generally, the notion of an _acyclic twisting cochain_ , defined in this
section, allows to conveniently formulate the Koszul duality theorems in wide
contexts.

###  5.1. The Hom DG-algebra and twisting cochains

Let $C$ be a coassociative coalgebra and $A$ be an associative algebra over a
field $k$. Then the vector space $\operatorname{Hom}_{k}(C,A)$ of all
$k$-linear maps $C\longrightarrow A$ acquires an associative $k$-algebra
structure. Given two linear maps $f$, $g\in\operatorname{Hom}_{k}(C,A)$, their
product $fg\in\operatorname{Hom}_{k}(C,A)$ is constructed as the composition

$C\overset{\mu}{\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}C\otimes_{k}C\overset{f\otimes g}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}A\otimes_{k}A\overset{m}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu}A,$

where $\mu\colon C\longrightarrow C\otimes_{k}C$ is the comultiplication and
$m\colon A\otimes_{k}A\longrightarrow A$ is the multiplication map. Given a
counit $\epsilon\colon C\longrightarrow k$ on $C$ and a unit $e\colon
k\longrightarrow A$ in $A$, the unit element $1\in\operatorname{Hom}_{k}(C,A)$
is constructed as the composition
$C\overset{\epsilon}{\longrightarrow}k\overset{e}{\longrightarrow}A$.

Given two graded vector spaces $U$ and $V$ over $k$, one denotes by
$\operatorname{Hom}_{k}(U,V)$ the graded Hom space. So
$\operatorname{Hom}_{k}(U,V)=\bigoplus_{i\in\mathbb{Z}}\operatorname{Hom}_{k}^{i}(U,V)$,
where $\operatorname{Hom}_{k}^{i}(U,V)$ is the vector space of all homogeneous
$k$-linear maps $U\longrightarrow V$ of degree $i$. Then, for any graded
coalgebra $C$ and any graded algebra $A$, the graded Hom space
$\operatorname{Hom}_{k}(C,A)$ is a graded algebra over $k$.

Given two complexes of vector spaces
$U^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$V^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, the graded Hom space
$\operatorname{Hom}_{k}(U,V)$ is endowed with a differential in the usual way,
producing the complex
$\operatorname{Hom}_{k}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U^{\text{\smaller\smaller$\scriptstyle\bullet$}},V^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
For any DG-coalgebra $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and a
DG-algebra $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over $k$, the
complex
$\operatorname{Hom}_{k}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a DG-algebra.

Let $E^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(E,d)$ be a DG-algebra.
A homogeneous element $a\in E^{1}$ of degree $1$ in $E$ is said to be a
_Maurer–Cartan element_ (or a _Maurer–Cartan cochain_) if it satisfies the
_Maurer–Cartan equation_ $a^{2}+d(a)=0$ in $E^{2}$.

Let $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a DG-coalgebra and
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a DG-algebra over $k$.
By the definition, a _twisting cochain_ $\tau$ for
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a Maurer–Cartan cochain
in the DG-algebra
$\operatorname{Hom}_{k}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
So $\tau\colon C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a homogeneous $k$-linear
map of degree $1$ (i. e., $\tau_{i}\colon C^{i}\longrightarrow A^{i+1}$,
$i\in\mathbb{Z}$) satisfying the Maurer–Cartan equation.

Various conditions of compatibility with (co)augmentations are usually imposed
on twisting cochains. In particular, let
$(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},\gamma)$ be a coaugmented
DG-coalgebra and $(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},\alpha)$
be an augmented DG-algebra. Twisting cochains $\tau\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ such that
$\alpha\circ\tau=0=\tau\circ\gamma$ will be important for us in this Section
5.

###  5.2. Bar-cobar adjunction and acyclic twisting cochains

We start with some examples before passing to the general case.

######  Examples 5.1.

(1) Let $(A,\alpha)$ be an augmented algebra over $k$, and let
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
be its bar-construction, as in Section 1.1. Let $\gamma\colon k\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be the natural
coaugmentation of
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
(as in Section 4.4). Then the composition $\tau$ of the direct summand
projection
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)\longrightarrow
A^{+}{}^{\otimes 1}=A^{+}$ and the inclusion $A^{+}\longrightarrow A$ is a
twisting cochain for the DG-coalgebra
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the algebra $A$ (viewed
as a DG-algebra in the obvious way). The equations of compatibility with the
augmentation and the coaugmentation $\alpha\circ\tau=0=\tau\circ\gamma$ are
satisfied in this example.

(2) More generally, let
$(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},\alpha)$ be an augmented
DG-algebra, and let
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
be its bar-construction, as in Section 4.1. Denote by $\gamma\colon
k\longrightarrow C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the natural
coaugmentation of
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Then the composition $\tau\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the direct summand
projection
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow
A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}{}^{\otimes
1}=A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}$ and the inclusion
$A^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a twisting cochain for
the DG-coalgebra $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the
DG-algebra $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The equations
$\alpha\circ\tau=0=\tau\circ\gamma$ are satisfied for this twisting cochain.

######  Examples 5.2.

(1) Let $(C,\gamma)$ be a coaugmented coalgebra over $k$, and let
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
be its cobar-construction, as in Section 2.6. Let $\alpha\colon
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow k$ be the
natural augmentation of
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
(as in Section 4.4). Then the composition $\tau$ of the natural surjection
$C\longrightarrow C^{+}$ and the direct summand inclusion
$C^{+}\longrightarrow\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
is a twisting cochain for the coalgebra $C$ (viewed as a DG-coalgebra in the
obvious way) and the DG-algebra
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The equations of
compatibility with the augmentation and the coaugmentation
$\alpha\circ\tau=0=\tau\circ\gamma$ are satisfied in this example.

(2) More generally, let
$(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},\gamma)$ be a coaugmented
DG-coalgebra, and let
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
be its cobar-construction, as in Section 4.2. Denote by $\alpha\colon
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow k$ the natural
augmentation of
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Then the composition $\tau\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the natural surjection
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}$ and the direct summand
inclusion
$C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},+}\longrightarrow\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a twisting cochain for the DG-coalgebra
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the DG-algebra
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The equations
$\alpha\circ\tau=0=\tau\circ\gamma$ are satisfied for this twisting cochain.

######  Example 5.3.

Let $(A,F,\alpha)$ be an augmented nonhomogeneous quadratic algebra, as
defined in Section 2.5, and let
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\subset\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
be the related (nonhomogeneous quadratic dual) DG-subcoalgebra. Then the
composition
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)\longrightarrow
A$ of the inclusion map
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
with twisting cochain
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)\longrightarrow
A$ from Example 5.1(1) is a twisting cochain $\tau\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow A$.
Equivalently, the twisting cochain $\tau$ can be constructed as the
composition of the direct summand projection $C\longrightarrow
C^{1}=V=F_{1}A/F_{0}A$ and the inclusion $F_{1}A/F_{0}A\simeq A^{+}\cap
F_{1}A\longrightarrow A$.

The natural coaugmentation
$k\longrightarrow\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
of the cobar DG-algebra
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
factorizes as $k\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$,
making $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ a coaugmented (in
fact, conilpotent) DG-coalgebra with the coaugmentation $\gamma\colon
k\longrightarrow C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The
twisting cochain $\tau\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow A$ satisfies
the equations $\alpha\circ\tau=0=\tau\circ\gamma$.

Now we can return to the proof of Lemma 4.1.

###### Proof of Lemma 4.1: further details.

Let $(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},\alpha)$ be an
augmented DG-algebra and
$(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},\gamma)$ be a conilpotent
DG-coalgebra. Then the underlying graded coalgebra of the bar-construction
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is
$\operatorname{Bar}_{\alpha}(A)=\rotatebox[origin={c}]{180.0}{$T$}(A^{+}[1])$
and the underlying graded algebra of the cobar-construction
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is $\operatorname{Cob}_{\gamma}(C)=T(C^{+}[-1])$.

Since $T(C^{+}[-1])$ is the free graded algebra spanned by $C^{+}[-1]$, graded
algebra homomorphisms $\operatorname{Cob}_{\gamma}(C)\longrightarrow A$
correspond bijectively to homogeneous $k$-linear maps
$C^{+}[-1]\longrightarrow A$ of degree $0$. Among these, the graded algebra
homomorphisms compatible with the augmentations on
$\operatorname{Cob}_{\gamma}(C)$ and $A$ correspond precisely to the linear
maps $C^{+}[-1]\longrightarrow A^{+}$. This means elements of the vector space
$\operatorname{Hom}_{k}^{1}(C^{+},A^{+})$.

Since $\rotatebox[origin={c}]{180.0}{$T$}(A^{+}[1])$ is the cofree conilpotent
graded coalgebra cospanned by $A^{+}[1]$ and $C$ is a conilpotent graded
coalgebra, graded coalgebra homomorphisms
$C\longrightarrow\operatorname{Bar}_{\alpha}(A)$ correspond bijectively to
homogeneous $k$-linear maps $C^{+}\longrightarrow A^{+}[1]$ of degree $0$.
This means elements of the same vector space
$\operatorname{Hom}_{k}^{1}(C^{+},A^{+})$.

Finally, one has to check that a graded algebra homomorphism
$\operatorname{Cob}_{\gamma}(C)\longrightarrow A$ commutes with the
differentials on
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
and $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ if and only if the
related element of
$\operatorname{Hom}_{k}^{1}(C^{+},A^{+})\subset\operatorname{Hom}_{k}^{1}(C,A)$
satisfies the Maurer–Cartan equation. Similarly, a graded coalgebra
homomorphism $C\longrightarrow\operatorname{Bar}_{\alpha}(A)$ commutes with
the differentials on $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
if and only if the related element of
$\operatorname{Hom}_{k}^{1}(C^{+},A^{+})\subset\operatorname{Hom}_{k}^{1}(C,A)$
satisfies the same Maurer–Cartan equation.

To sum up, both the augmented DG-algebra homomorphisms
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the (conilpotent) DG-
coalgebra homomorphisms
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
correspond bijectively to twisting cochains $\tau\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ satisfying the equations of
compatibility with the augmentation and the coaugmentation
$\alpha\circ\tau=0=\tau\circ\gamma$. ∎

Let $(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},\alpha)$ be an
augmented DG-algebra,
$(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},\gamma)$ be a conilpotent
DG-coalgebra, and $\tau\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a twisting cochain
satisfying the equations $\alpha\circ\tau=0=\tau\circ\gamma$. The twisting
cochain $\tau$ is said to be _acyclic_ if one of (or equivalently, both) the
related DG-algebra homomorphism
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the DG-coalgebra
homomorphism
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
become isomorphisms after the quasi-isomorphisms of DG-algebras, or
respectively the filtered quasi-isomorphisms of conilpotent DG-coalgebras, are
inverted, as in Theorem 4.2. Simply put, $\tau$ is called acyclic if the
related homomorphism of DG-algebras
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a quasi-isomorphism.

For example, the twisting cochain from Examples 5.2 is acyclic, by the
definition, whenever the coaugmented coalgebra $(C,\gamma)$ or the coaugmented
DG-coalgebra $(C^{\text{\smaller\smaller$\scriptstyle\bullet$}},\gamma)$ is
conilpotent. It is a part of Theorem 4.2 that the twisting cochain from
Examples 5.1 is acyclic for any augmented algebra $(A,\alpha)$ or an augmented
DG-algebra $(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},\alpha)$.

Concerning the twisting cochain $\tau$ from Example 5.3, it is acyclic for any
(augmented) nonhomogeneous _Koszul_ algebra $(A,F)$. This claim is a corollary
of the proof of Poincaré–Birkhoff–Witt theorem for nonhomogeneous
quadratic/Koszul algebras in [61, Sections 3.2–3.3] or [58, Proposition
7.2(ii) in Chapter 5].

###  5.3. Twisted differential on the tensor product

Let $(A,\alpha)$ be an augmented algebra and
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex of left
$A$-modules. Let us make some basic observations about the bar-constructions
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
and
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
as defined in Sections 1.1 and 1.3.

The underlying graded vector space $\operatorname{Bar}_{\alpha}(A,M)$ of the
complex
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is isomorphic to the tensor product
$\operatorname{Bar}_{\alpha}(A)\otimes_{k}M$ of the underlying graded vector
spaces of the complexes
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
and $M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. But the differential
on
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is _not_ the tensor product of the differentials on
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$
and $M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Rather, in the formula
for the differential of
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A,M^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
there are the summands coming from the differential of
$\operatorname{Bar}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}(A)$,
there is the summand coming from the differential of
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, but there is also one
additional summand, defined in terms of the action of $A$ in $M$.

Similarly, let $(C,\gamma)$ be a coaugmented coalgebra and
$N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex of left
$C$-comodules. Consider the cobar-constructions
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
and
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
as defined in Sections 2.6 and 2.7.

The underlying graded vector space $\operatorname{Cob}_{\gamma}(C,N)$ of the
complex
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is isomorphic to the tensor product
$\operatorname{Cob}_{\gamma}(C)\otimes_{k}N$ of the underlying graded vector
spaces of the complexes
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
and $N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. But the differential
on
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is _not_ the tensor product of the differentials on
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$
and $N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Rather, in the formula
for the differential of
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C,N^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
there are the summands coming from the differential of
$\operatorname{Cob}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}(C)$,
there is the summand coming from the differential on
$N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, but there is also one
additional summand, defined in terms of the coaction of $C$ in $N$.

The construction of the twisted differential on the tensor product of a DG-
module and a DG-comodule (twisted by a twisting cochain $\tau$) is abstracted
from these observations. Let
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a DG-algebra and
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a DG-coalgebra over $k$.
Let $\tau\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a twisting cochain.

Let $M^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(M,d_{M})$ be a right
DG-module over $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, and let
$N^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(N,d_{N})$ be a left DG-
comodule over $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Consider the
tensor product $M\otimes_{k}N$ of the underlying graded vector spaces of
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, and endow it with the
differential given by the formula

$d(x\otimes y)=d_{M}(x)\otimes y+(-1)^{|x|}x\otimes d_{N}(y)\pm
d^{\tau}(x\otimes y)$

for all homogeneous elements $x\in M$ and $y\in N$ of degrees $|x|$ and $|y|$.
Here $d^{\tau}\colon M\otimes_{k}N\longrightarrow M\otimes_{k}N$ is the
composition

$M\otimes_{k}N\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muM\otimes_{k}C\otimes_{k}N\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muM\otimes_{k}A\otimes_{k}N\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM\otimes_{k}N$

of the map induced by the comultiplication map $N\longrightarrow
C\otimes_{k}N$, the map induced by the twisting cochain $\tau\colon
C\longrightarrow A$, and the map induced by the multiplication map
$M\otimes_{k}A\longrightarrow M$. The reader can consult with [64, Section
6.2] for the sign rule.

Then one can check that $d^{2}(x\otimes y)=0$. So the tensor product of graded
vector spaces $M\otimes_{k}N$ endowed with the differential $d$ is a complex.
We denote this complex by
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.

Similarly, let $M^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(M,d_{M})$ be
a left DG-module over $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, and
let $N^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(N,d_{N})$ be a right
DG-comodule over $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Consider
the tensor product $N\otimes_{k}M$ of the respective underlying graded vector
spaces, and endow it with the differential given by the formula

$d(y\otimes x)=d_{N}(y)\otimes x+(-1)^{|y|}y\otimes d_{M}(x)\pm
d^{\tau}(y\otimes x),$

where $d^{\tau}\colon N\otimes_{k}M\longrightarrow N\otimes_{k}M$ is the
composition

$N\otimes_{k}M\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muN\otimes_{k}C\otimes_{k}M\overset{\tau}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}N\otimes_{k}A\otimes_{k}M\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muN\otimes_{k}M.$

Once again, choosing the sign properly (cf. [64, Section 6.2]), one can can
check that $d^{2}(y\otimes x)=0$. So the tensor product $N\otimes_{k}M$
endowed with the differential $d$ is a complex. We denote this complex by
$N^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.

###  5.4. Derived Koszul duality on the comodule side in the augmented case

Let $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a DG-algebra and
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a DG-coalgebra over $k$.
Suppose that we are given a twisting cochain $\tau\colon
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.

Given a left DG-comodule $N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, we consider the
complex of vector spaces
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
constructed in Section 5.3. Then the structure of right DG-module over
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ has been eaten up in the
construction of the twisted differential on the tensor product, but the left
DG-module structure of $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is inherited by the twisted
tensor product, making
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}N^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
a left DG-module over $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.

Similarly, given a left DG-module
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, we consider the complex of
vector spaces
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes^{\tau}M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
constructed in Section 5.3. Then the right DG-comodule structure over
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ has been consumed in the
construction of the twisted differential on the tensor product, but the left
DG-comodule structure of $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is inherited by the
twisted tensor product, making