# The dg Leavitt algebra, singular Yoneda category and singularity category

Xiao-Wu Chen and Zhengfang Wang,
with an appendix by Bernhard Keller and Yu Wang

###### Abstract.

For any finite dimensional algebra $\Lambda$ given by a quiver with relations,
we prove that its dg singularity category is quasi-equivalent to the perfect
dg derived category of a dg Leavitt path algebra. The result might be viewed
as a deformed version of the known description of the dg singularity category
of a radical-square-zero algebra in terms of a Leavitt path algebra with
trivial differential.

The above result is achieved in two steps. We first introduce the singular
Yoneda dg category of $\Lambda$, which is quasi-equivalent to the dg
singularity category of $\Lambda$. The construction of this new dg category
follows from a general operation for dg categories, namely an explicit dg
localization inverting a natural transformation from the identity functor to a
dg endofunctor. This localization turns out to be quasi-equivalent to a dg
quotient category. Secondly, we prove that the endomorphism algebra of the
quotient of $\Lambda$ modulo its Jacobson radical in the singular Yoneda dg
category is isomorphic to the dg Leavitt path algebra. The appendix is devoted
to an alternative proof of the result using Koszul-Moore duality and derived
localizations.

###### Key words and phrases:

dg Leavitt path algebra, singular Yoneda category, singularity category, dg
localization, dg quotient

###### 2010 Mathematics Subject Classification:

16E45, 18G80, 18E35, 16G20

###### Contents

  1. 1 Introduction
    1. 1.1 The background and main results
    2. 1.2 Conventions and structure
  2. 2 The Cohn and Leavitt algebras
  3. 3 The dg Cohn and Leavitt algebras
  4. 4 Quivers and path algebras
  5. 5 Pretriangulated dg categories
    1. 5.1 DG categories and functors
    2. 5.2 Exact and pretriangulated dg categories
    3. 5.3 The perfect dg derived categories
  6. 6 An explicit dg localization
  7. 7 The Yoneda dg category
    1. 7.1 The bar and Yoneda dg categories
    2. 7.2 The dg derived categories
    3. 7.3 The dg tensor algebra as an endomorphism algebra
  8. 8 Noncommutative differential forms
  9. 9 The singular Yoneda dg category
    1. 9.1 The dg singularity categories
    2. 9.2 The dg Leavitt algebra as an endomorphism algebra
  10. 10 Applications to finite dimensional algebras
    1. 10.1 Finite dimensional algebras
    2. 10.2 The quiver case
    3. 10.3 An example
  11. A Dg Leavitt path algebras for singularity categories, by Bernhard Keller and Yu Wang
    1. A.1 Modules and comodules
    2. A.2 Koszul–Moore duality
    3. A.3 Description of the singularity category
    4. A.4 Description of the singularity category as a derived localization
    5. A.5 Conjectural approach via Neeman–Ranicki’s theorem
      1. A.5.1 For rings concentrated in degree $0$
      2. A.5.2 The differential graded case

##  1\. Introduction

###  1.1. The background and main results

Let $\mathbb{K}$ be a field and $\Lambda$ a finite dimensional algebra over
$\mathbb{K}$. The _singularity category_ $\mathbf{D}_{\rm sg}(\Lambda)$ of
$\Lambda$ is defined as the Verdier quotient of the bounded derived category
$\mathbf{D}^{b}(\Lambda\mbox{-mod})$ of finitely generated left
$\Lambda$-modules by the full subcategory of perfect complexes. This notion
was first introduced in [16] and then rediscovered in [60] motivated by the
homological mirror symmetry conjecture. The singularity category measures the
homological singularity of the algebra: it vanishes if and only if the algebra
$\Lambda$ is of finite global dimension.

The homotopy category $\mathbf{K}_{\rm ac}(\Lambda\mbox{-Inj})$ [48] of
acyclic complexes of arbitrary injective $\Lambda$-modules is a _compactly
generated completion_ of $\mathbf{D}_{\rm sg}(\Lambda)$. This means that
$\mathbf{K}_{\rm ac}(\Lambda\mbox{-Inj})$ is compactly generated and that its
full subcategory of compact objects is triangle equivalent to $\mathbf{D}_{\rm
sg}(\Lambda)$. However, in general, we do not know whether $\mathbf{D}_{\rm
sg}(\Lambda)$ determines $\mathbf{K}_{\rm ac}(\Lambda\mbox{-Inj})$ uniquely as
a triangulated category.

As is well known, triangulated categories arising naturally in algebra usually
have a dg enhancement, that is, there is a pretriangulated dg category whose
zeroth cohomology yields the given triangulated category [56]. For instance,
the _dg singularity category_ $\mathbf{S}_{\rm dg}(\Lambda)$ [46, 11, 15] is a
canonical dg enhancement of $\mathbf{D}_{\rm sg}(\Lambda)$, which is defined
to be the dg quotient of the bounded dg derived category $\mathbf{D}_{\rm
dg}^{b}(\Lambda\mbox{-mod})$ by the full dg subcategory of perfect complexes.

In comparison with the singularity category, the dg singularity category
contains more information and has more invariants. For example, the above
completion $\mathbf{K}_{\rm ac}(\Lambda\mbox{-Inj})$ is uniquely determined by
$\mathbf{S}_{\rm dg}(\Lambda)$: there is a triangle equivalence; see [48, 23]

(1.1) $\displaystyle\mathbf{K}_{\rm
ac}(\Lambda\mbox{-Inj})\simeq\mathbf{D}(\mathbf{S}_{\rm dg}(\Lambda)^{\rm
op}),$

where $\mathbf{D}(\mathbf{S}_{\rm dg}(\Lambda)^{\rm op})$ is the derived
category of right dg $\mathbf{S}_{\rm dg}(\Lambda)$-modules. The main theorem
in [46] states that under very mild conditions the Hochschild cohomology of
$\mathbf{S}_{\rm dg}(\Lambda)$ is isomorphic to the singular Hochschild
cohomology [69] of $\Lambda$.

We are interested in describing the (dg) singularity categories of $\Lambda$.
Let us assume for a moment that $\Lambda=\mathbb{K}Q/I$, where $\mathbb{K}Q$
is the path algebra of a finite quiver $Q$ and $I$ is an admissible ideal of
$\mathbb{K}Q$.

Recall from [63] the description of $\mathbf{D}_{\rm sg}(\Lambda)$ when
$\Lambda$ is radical square zero, i.e. the ideal $I$ contains all paths of
length two. Then $\Lambda=\mathbb{K}Q_{0}\oplus\mathbb{K}Q_{1}$ has a basis
given by vertices and arrows in $Q$. Denote by $Q^{\circ}$ the finite quiver
without sinks, which is obtained from $Q$ by removing sinks repeatedly. The
corresponding Leavitt path algebra $L(Q^{\circ})$ in the sense of [1, 4, 5] is
naturally graded, and is viewed as a dg algebra with trivial differential. One
of the main results in [63] states a triangle equivalence

$\mathbf{D}_{\rm sg}(\Lambda)\simeq\mathbf{per}(L(Q^{\circ})).$

Here, $\mathbf{per}(L(Q^{\circ}))$ denotes the perfect derived category of
left dg $L(Q^{\circ})$-modules. Indeed, by the work [52, 22], such a triangle
equivalence lifts to a quasi-equivalence between the corresponding dg
enhancements

$\mathbf{S}_{\rm dg}(\Lambda)\simeq\mathbf{per}_{\rm dg}(L(Q^{\circ})).$

Combining (1.1) with the quasi-equivalence above, we recover the following
triangle equivalence in [24]:

$\mathbf{K}_{\rm ac}(\Lambda\mbox{-Inj})\simeq\mathbf{D}(L(Q^{\circ})),$

where $\mathbf{D}(L(Q^{\circ}))$ denotes the derived category of left dg
$L(Q^{\circ})$-modules.

We observe that usually the Leavitt path algebra $L(Q^{\circ})$ is infinite
dimensional in each degree, and that correspondingly the singularity category
of $\Lambda$ is usually Hom-infinite [21]. We mention that Leavitt path
algebras are related to noncommutative geometry [63], symbolic dynamic systems
[3], graph $C^{*}$-algebras [18, 2] and algebraic bivariant K-theory [26, 27,
28].

We will extend the above description to the general case. For
$\Lambda=\mathbb{K}Q/I$, we have a natural decomposition
$\Lambda=\mathbb{K}Q_{0}\oplus J$ with $J$ its Jacobson radical. We introduce
the _radical quiver_ $\widetilde{Q}$ of $\Lambda$: it has the same vertex set
as $Q$, that is, $\widetilde{Q}_{0}=Q_{0}$; for any vertices $i$ and $j$, the
arrows from $i$ to $j$ correspond to elements in a basis of $e_{j}Je_{i}$.
Here, $e_{i}$ denotes the corresponding primitive idempotent of the vertex
$i$. In other words, we identify $J$ with $\mathbb{K}\widetilde{Q}_{1}$, the
vector space spanned by the arrow set $\widetilde{Q}_{1}$ of $\widetilde{Q}$.
The multiplication on $J$ is transferred to an associative product

$\mu\colon\mathbb{K}\widetilde{Q}_{1}\otimes_{\mathbb{K}\widetilde{Q}_{0}}\mathbb{K}\widetilde{Q}_{1}\longrightarrow\mathbb{K}\widetilde{Q}_{1}.$

In this way, $\Lambda$ is viewed as a deformation of the radical-square-zero
algebra
$\widetilde{\Lambda}:=\mathbb{K}\widetilde{Q}_{0}\oplus\mathbb{K}\widetilde{Q}_{1}$,
since the algebra $\Lambda$ may be recovered from $\widetilde{\Lambda}$ using
the product $\mu$.

It is well known that such an associative product $\mu$ gives rise to a
differential on the graded path algebra of the opposite quiver of
$\widetilde{Q}$; see [8]. In the same manner, it gives rise to a differential
$\partial$ on the Leavitt path algebra $L(\widetilde{Q}^{\circ})$. Here,
$\widetilde{Q}^{\circ}$ is the quiver without sinks that is obtained from
$\widetilde{Q}$ by removing sinks repeatedly. The resulting dg Leavitt path
algebra is denoted by $L(\widetilde{Q}^{\circ})_{\partial}$ temporarily.

The following result describes the (dg) singularity categories of $\Lambda$ in
terms of dg Leavitt path algebras; see Theorem 10.5. It indicates that dg
Leavitt path algebras are ubiquitous in the study of singularity categories.

######  Theorem A.

Let $\Lambda=\mathbb{K}Q/I$ be a finite dimensional algebra with radical
quiver $\widetilde{Q}$. Then there is a quasi-equivalence

$\mathbf{S}_{\rm dg}(\Lambda)\simeq\mathbf{per}_{\rm
dg}(L(\widetilde{Q}^{\circ})_{\partial}).$

Consequently, there are triangle equivalences

$\mathbf{D}_{\rm
sg}(\Lambda)\simeq\mathbf{per}(L(\widetilde{Q}^{\circ})_{\partial})\quad\mbox{
and }\quad\mathbf{K}_{\rm ac}(\Lambda\mbox{-}{\rm
Inj})\simeq\mathbf{D}(L(\widetilde{Q}^{\circ})_{\partial}).$

If the algebra $\Lambda$ is radical square zero, then $\widetilde{Q}=Q$ and
the differential $\partial$ vanishes. Applying Theorem A to this situation, we
recover the mentioned results in [63] and [24, 22].

The idea behind Theorem A is illustrated by the following diagram.

(1.6)

The horizontal arrows indicate the quasi-equivalences between the relevant dg
singularity categories and perfect dg derived categories. For the vertical
arrow on the right, we mention that it is customary to deform a dg algebra by
only changing its differential, a particular $A_{\infty}$-deformation [67];
compare [45]. However, we do not know how to deduce the quasi-equivalence at
the bottom from the one at the top via the deformation theory [47, 53] of dg
categories; see Remark 10.6(2).

The proof of Theorem A is divided into two steps. We first introduce the
_singular Yoneda dg category_ $\mathcal{SY}$ of $\Lambda$, which turns out to
be quasi-equivalent to $\mathbf{S}_{\rm dg}(\Lambda)$. Secondly, using the
explicit description of $\mathcal{SY}$, we show that the endomorphism algebra
of $\mathbb{K}Q_{0}$ in $\mathcal{SY}$ is isomorphic to the dg Leavitt path
algebra $L(\widetilde{Q}^{\circ})_{\partial}$.

The construction of $\mathcal{SY}$ follows from a general operation for dg
categories described as follows. Let $\mathcal{C}$ be a dg category, $\Omega$
a dg endofunctor on $\mathcal{C}$, and $\theta\colon{\rm
Id}_{\mathcal{C}}\rightarrow\Omega$ a closed natural transformation of degree
zero satisfying $\theta\Omega=\Omega\theta$. To invert $\theta_{X}$ for all
objects $X$, we construct a new and explicit dg category $\mathcal{SC}$ with a
dg functor

$\iota\colon\mathcal{C}\longrightarrow\mathcal{SC},$

called the (strict) _dg localization_ along $\theta$. The objects of
$\mathcal{SC}$ are the same as $\mathcal{C}$, and the Hom complexes are
defined via a colimit construction which is similar to the one in defining the
singular Hochschild cochain complex [70].

To obtain $\mathcal{SY}$ from the above general operation, we consider the
_Yoneda dg category_ $\mathcal{Y}$, a natural dg enhancement of the derived
category of $\Lambda$ via the bar resolution [37]. The relevant dg endofunctor
on $\mathcal{Y}$ is induced by noncommutative differential forms [29, 69].

The quasi-equivalence between $\mathcal{SY}$ and $\mathbf{S}_{\rm
dg}(\Lambda)$ is a special case of the following general result; see Theorem
6.4. We mention that a similar idea of this result is implicitly contained in
[43, Section 7].

######  Theorem B.

Assume that $\mathcal{C}$ is pretriangulated. Then $\mathcal{SC}$ is
pretriangulated and $\iota$ induces a quasi-equivalence

$\mathcal{C}/\mathcal{N}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{SC},$

where $\mathcal{N}$ is the full dg subcategory formed by the cones of
$\theta_{X}$, and $\mathcal{C}/\mathcal{N}$ denotes the dg quotient category.

For dg quotient categories, we refer to [39, 30]. In general, the structure of
a dg quotient category is rather complicated and mysterious. Theorem B
describes certain dg quotient categories explicitly.

###  1.2. Conventions and structure

We fix a commutative ring $\mathbb{K}$ and work over $\mathbb{K}$. This means
that we require that all the categories and functors are $\mathbb{K}$-linear.
In Section 10 we will further assume that $\mathbb{K}$ is a field.

We fix two $\mathbb{K}$-algebras $E$ and $\Lambda$ together with a fixed
homomorphism $E\rightarrow\Lambda$ of $\mathbb{K}$-algebras. Denote by
$\overline{\Lambda}$ its cokernel, which is naturally an $E$-$E$-bimodule. In
most cases, we will further assume that $\Lambda$ is augmented over $E$, that
is, there is an algebra homomorphism $\pi\colon\Lambda\rightarrow E$ such that
the composition
$E\rightarrow\Lambda\stackrel{{\scriptstyle\pi}}{{\rightarrow}}E$ is the
identity. Then the $E$-$E$-bimodule $\overline{\Lambda}$ has an induced
associative product
$\mu\colon\overline{\Lambda}\otimes_{E}\overline{\Lambda}\rightarrow\overline{\Lambda}$
by identifying $\overline{\Lambda}$ with the kernel of $\pi$. We will view $E$
as a $\Lambda$-module via the homomorphism $\pi$.

In the sequel, we will work in the relative setup. For example, we will study
various $E$-relative derived categories and $E$-relative singularity
categories of $\Lambda$.

By default, a module means a left module. For example, ${\rm Hom}_{E}(-,-)$
means the Hom bifunctor on the category of left $E$-modules. A left $E$-module
$M$ is usually denoted by ${}_{E}M$, which emphasizes the $E$-action from the
left side.

Throughout, we use cohomological notation. In the dg setup, we always consider
homogeneous elements or morphisms. The translation functor on any triangulated
category is denoted by $\Sigma$.

The paper is organized as follows. In Section 2, we study the Cohn algebra
$C_{E}(M)$ and Leavitt algebra $L_{E}(M)$ associated to an $E$-$E$-bimodule
$M$. We prove that the Leavitt algebra is isomorphic to the colimit of an
explicit sequence; see Theorem 2.6.

In Section 3, we assume that the bimodule $M$ is equipped with an associative
product $\mu\colon M\otimes_{E}M\rightarrow M$. We show that $\mu$ induces a
differential on the Cohn algebra, which descends to a differential on the
Leavitt algebra. Consequently, we obtain the dg Cohn algebra and dg Leavitt
algebra associated to $(M,\mu)$. In Section 4, we work in the quiver case.
More precisely, for a finite quiver $Q$, we set $E=\mathbb{K}Q_{0}$ and
$M=\mathbb{K}Q_{1}$. Applying the results in Section 3 to this situation, we
obtain the dg Cohn path algebra and dg Leavitt path algebra; compare
Proposition 4.1.

We recall basic facts on pretriangulated dg categories in Section 5. We
introduce an explicit dg localization in Section 6. The universal property in
Proposition 6.2 justifies this terminology. Theorem 6.4 shows that the dg
localization is quasi-equivalent to a dg quotient category.

Inspired by [37], we introduce the Yoneda dg category $\mathcal{Y}$ of
$\Lambda$ in Section 7. It is quasi-equivalent to the dg derived category of
$\Lambda$; see Proposition 7.3 and Corollary 7.5. We prove that the
endomorphism algebra of $E$ in $\mathcal{Y}$ is isomorphic to a dg tensor
algebra associated to $(\overline{\Lambda},\mu)$; see Proposition 7.6.

In Section 8, we introduce noncommutative differential forms [29, 69] with
values in complexes of $\Lambda$-modules. This gives rise to a dg endofunctor
$\Omega_{\rm nc}$ on $\mathcal{Y}$, together with a natural transformation
$\theta\colon\mathrm{Id}_{\mathcal{Y}}\to\Omega_{\rm nc}$. We actually show
that the assumptions for the dg localization in Section 6 are satisfied on
$(\mathcal{Y},\Omega_{\rm nc},\theta)$.

In Section 9, we take the dg localization of $\mathcal{Y}$ along $\theta$, and
obtain the singular Yoneda dg category $\mathcal{SY}$ of $\Lambda$. This
terminology is justified by Proposition 9.1 and Corollary 9.3, that is, the
singular Yoneda dg category $\mathcal{SY}$ is quasi-equivalent to the dg
singularity category. In Theorem 9.5, we prove that the endomorphism algebra
of $E$ in $\mathcal{SY}$ is exactly isomorphic to the dg Leavitt algebra
$L_{E}(\overline{\Lambda})$ associated to $(\overline{\Lambda},\mu)$, studied
in Section 3.

In Section 10, we apply Theorem 9.5 to any finite dimensional algebra
$\Lambda$. Theorem 10.5 relates the dg singularity category of $\Lambda$ to
the dg Leavitt path algebra, which is associated to the radical quiver
$\widetilde{Q}$ of $\Lambda$ and a transferred associative product $\mu$ on
$\mathbb{K}\widetilde{Q}_{1}$. In the end, we give an explicit example of a dg
Leavitt path algebra, whose minimal $A_{\infty}$-model is explicitly
described.

In the appendix, Bernhard Keller and Yu Wang give an alternative proof of
Theorem 10.5 using Koszul-Moore duality in [42] and derived localizations in
[14].

##  2\. The Cohn and Leavitt algebras

Throughout this section, we assume that $E$ is a $\mathbb{K}$-algebra and that
$M$ is an $E$-$E$-bimodule on which $\mathbb{K}$ acts centrally. We study the
Cohn algebra and Leavitt algebra associated to $M$.

Denote by $M^{*}={\rm Hom}_{E}(M,E)$ the left dual $E$-$E$-bimodule whose
bimodule structure is given by

(2.1) $\displaystyle(afb)(m)=f(ma)b\quad\quad\text{for $a,b\in E$, $m\in M$
and $f\in M^{*}$.}$

Denote by

$T_{E}(M^{*})=E\oplus M^{*}\oplus(M^{*})^{\otimes_{E}2}\oplus\cdots$

the tensor algebra. Its typical element
$f_{1}\otimes_{E}f_{2}\otimes_{E}\cdots\otimes_{E}f_{q}$, with $f_{i}\in
M^{*}$ for each $1\leq i\leq q$, will be abbreviated as $f_{1,q}$. If $q=0$,
the notation $f_{1,q}$ usually means the unit element $1_{E}$.

Inspired by [25, § 8] and [2, Definition 1.5.1], we will define the _Cohn
algebra_ $C_{E}(M)$ associated to $M$ as follows. As a $\mathbb{K}$-module, we
have

$C_{E}(M):=T_{E}(M^{*})\otimes_{E}T_{E}(M)=\bigoplus_{p\geq
0}T_{E}(M^{*})\otimes_{E}{M^{\otimes_{E}p}}.$

Its typical element

(2.2) $\displaystyle
f_{1}\otimes_{E}f_{2}\otimes_{E}\cdots\otimes_{E}f_{q}\otimes_{E}x_{1}\otimes_{E}x_{2}\otimes_{E}\cdots\otimes_{E}x_{p}$

will be abbreviated as $f_{1,q}\otimes_{E}x_{1,p}$ for $f_{i}\in M^{*}$,
$x_{j}\in M$ and $p,q\geq 0$. Take another typical element
$g_{1,s}\otimes_{E}y_{1,t}$. The multiplication of $C_{E}(M)$ is determined by
the following rule:

(2.3)
$\displaystyle(f_{1,q}\otimes_{E}x_{1,p})\bullet(g_{1,s}\otimes_{E}y_{1,t})=f_{1,q}\otimes_{E}Z\otimes_{E}y_{1,t}$

where the middle tensor $Z$ is equal to

$\displaystyle\begin{cases}g_{p}(x_{1}g_{p-1}(x_{2}g_{p-2}(\dotsb(x_{p-1}g_{1}(x_{p}))\dotsb)))\,g_{p+1,s}\in(M^{*})^{\otimes_{E}s-p},&\text{
if $p<s$};\\\
g_{p}(x_{1}g_{p-1}(x_{2}g_{p-2}(\dotsb(x_{p-1}g_{1}(x_{p}))\dotsb)))\quad\in
E,&\text{ if $p=s$};\\\
x_{1,p-s}\,g_{s}(x_{p-s+1}g_{s-1}(x_{p-s+2}g_{s-2}(\dotsb(x_{p-1}g_{1}(x_{p}))\dotsb)))\in
M^{\otimes_{E}p-s},&\text{ if $p>s$}.\end{cases}$

It is routine to verify that the above multiplication makes $C_{E}(M)$ into an
associative $\mathbb{K}$-algebra and that its unit is given by $1_{E}\in E$.
We observe that $T_{E}(M^{*})$ and $T_{E}(M)$ are naturally subalgebras of
$C_{E}(M)$.

The following example illustrates the multiplication $\bullet$ of $C_{E}(M)$
in more detail.

######  Example 2.1.

We have

$(f_{1,q}\otimes_{E}x_{1,3})\bullet(g_{1,4}\otimes_{E}y_{1,t})=f_{1,q}\otimes_{E}g_{3}(x_{1}g_{2}(x_{2}g_{1}(x_{3})))g_{4}\otimes_{E}y_{1,t},$

which lies in $(M^{*})^{\otimes_{E}(q+1)}\otimes_{E}M^{\otimes_{E}t}$. The
element $g_{3}(x_{1}g_{2}(x_{2}g_{1}(x_{3})))$ lies in $E$, and the expression
$g_{3}(x_{1}g_{2}(x_{2}g_{1}(x_{3})))g_{4}$ means the left $E$-action of
$g_{3}(x_{1}g_{2}(x_{2}g_{1}(x_{3})))$ on the element $g_{4}\in M^{*}$; see
(2.1). Similarly, we have

$(f_{1,q}\otimes_{E}x_{1,4})\bullet(g_{1,3}\otimes_{E}y_{1,t})=f_{1,q}\otimes_{E}x_{1}g_{3}(x_{2}g_{2}(x_{3}g_{1}(x_{4})))\otimes_{E}y_{1,t},$

which lies in $(M^{*})^{\otimes_{E}q}\otimes_{E}M^{\otimes_{E}(t+1)}$. The
expression $x_{1}g_{3}(x_{2}g_{2}(x_{3}g_{1}(x_{4})))$ means the right
$E$-action of $g_{3}(x_{2}g_{2}(x_{3}g_{1}(x_{4})))\in E$ on the element
$x_{1}\in M$.

We observe that $x\bullet g=g(x)\in E$ for $x\in M$ and $g\in M^{*}$.
Therefore, the inclusion $M^{*}\oplus M\subseteq C_{E}(M)$ induces a well-
defined $\mathbb{K}$-algebra homomorphism

$\Phi\colon T_{E}(M^{*}\oplus M)/{(x\otimes_{E}g-g(x)\;|\;x\in M,g\in
M^{*})}\longrightarrow C_{E}(M).$

######  Proposition 2.2.

The above algebra homomorphism $\Phi$ is an isomorphism.

###### Proof.

Denote the domain of $\Phi$ by $R$. In $C_{E}(M)$, we have

$f_{1,q}\otimes_{E}x_{1,p}=f_{1}\bullet\cdots\bullet f_{q}\bullet
x_{1}\bullet\cdots\bullet x_{p}.$

It follows that $E\oplus(M^{*}\oplus M)$ generates $C_{E}(M)$ and thus $\Phi$
is surjective.

We define a $\mathbb{K}$-linear map

$\Phi^{\prime}\colon C_{E}(M)\longrightarrow R,$

which sends a typical element $f_{1,q}\otimes_{E}x_{1,p}\in C_{E}(M)$ to the
image of the corresponding tensor $f_{1,q}\otimes_{E}x_{1,p}\in
T_{E}(M^{*}\oplus M)$ in $R$. Using (2.3), we verify that $\Phi^{\prime}$ is
an algebra homomorphism. We deduce $\Phi^{\prime}\circ\Phi={\rm Id}_{R}$ by
evaluating the both sides on $E\oplus(M^{*}\oplus M)$. Then $\Phi$ is
injective, proving the required statement. ∎

######  Remark 2.3.

The following evaluation map

${\rm ev}\colon M\otimes_{E}M^{*}\longrightarrow E,\quad x\otimes_{E}g\mapsto
g(x)$

is an $E$-$E$-bimodule map. Then $(M,M^{*},{\rm ev})$ is an $R$-system in the
sense of [18, Definition 1.1]. By the above isomorphism, we observe that the
Cohn algebra $C_{E}(M)$ is isomorphic to the Toeplitz ring of $(M,M^{*},{\rm
ev})$; see [18, Theorem 1.7].

Assume that the underlying left $E$-module of $M$ is finitely generated
projective. We have the canonical isomorphism of $E$-$E$-bimodules

$M^{*}\otimes_{E}M\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm
Hom}_{E}(M,M),\quad\quad f\otimes x\mapsto(m\mapsto f(m)x).$

We denote by $c\in M^{*}\otimes_{E}M$ the preimage of ${\rm Id}_{M}$, which is
called the _Casimir element_ of $M$. We observe that $ac=ca$ for any $a\in E$.

Write $c=\sum_{i\in S}\alpha_{i}^{*}\otimes_{E}\alpha_{i}\in
M^{*}\otimes_{E}M$. Then $\\{\alpha_{i}\\}_{i\in S}$ and
$\\{\alpha_{i}^{*}\\}_{i\in S}$ form a dual basis of $M$, i.e. we have

(2.4) $\displaystyle x=\sum_{i\in
S}\alpha_{i}^{*}(x)\alpha_{i}\quad\quad\mbox{and}\quad\quad f=\sum_{i\in
S}\alpha_{i}^{*}f(\alpha_{i})$

for any $x\in M$ and $f\in M^{*}$. We will view $c$ as an element in
$C_{E}(M)$.

The following definition is inspired by [50, § 3] and [1, 4, 5].

######  Definition 2.4.

Let $M$ be an $E$-$E$-bimodule with ${}_{E}M$ finitely generated projective.
The _Leavitt algebra_ $L_{E}(M)$ associated to $M$ is defined to be the
quotient algebra

$L_{E}(M)=C_{E}(M)/{(1_{E}-c)}.$

By the above isomorphism $\Phi$, we infer an isomorphism of algebras

$L_{E}(M)\simeq T_{E}(M^{*}\oplus M)/{(m\otimes_{E}g-g(m),\ 1_{E}-c\mid m\in
M,g\in M^{*})}.$

Similar to Remark 2.3, the Leavitt algebra $L_{E}(M)$ is isomorphic to the
Cuntz-Pimsner ring of $(M,M^{*},{\rm ev})$ relative to the whole algebra $E$;
see [18, Definition 3.16] and compare [18, Example 5.8].

######  Lemma 2.5.

The principal ideal $(1_{E}-c)$ of $C_{E}(M)$ is spanned, as an
$E$-$E$-bimodule, by elements of the form
$f_{1,q}\otimes_{E}x_{1,p}-f_{1,q}\otimes_{E}c\otimes_{E}x_{1,p}$ for $p,q\geq
0$.

###### Proof.

Denote by $I$ the $E$-$E$-subbimodule of $C_{E}(M)$ spanned by elements of the
form $f_{1,q}\otimes_{E}x_{1,p}-f_{1,q}\otimes_{E}c\otimes_{E}x_{1,p}.$ Since

$f_{1,q}\otimes_{E}x_{1,p}-f_{1,q}\otimes_{E}c\otimes_{E}x_{1,p}=f_{1,q}\bullet(1_{E}-c)\bullet
x_{1,p},$

it follows that $I\subseteq(1_{E}-c)$. We claim that $I$ is a two-sided ideal
of $C_{E}(M)$. Then the required statement follows.

We only prove that $I$ is a left ideal, since similarly one proves that it is
also a right ideal. It is clear that $I$ is a left $T_{E}(M^{*})$-submodule of
$C_{E}(M)$. Hence, it suffices to prove that for any $x\in M$, the element
$w:=x\bullet(f_{1,q}\otimes_{E}x_{1,p}-f_{1,q}\otimes_{E}c\otimes_{E}x_{1,p})$
still lies in $I$.

There are two cases. If $q\geq 1$, then
$w=f_{1}(x)(f_{2,q}\otimes_{E}x_{1,p}-f_{2,q}\otimes_{E}c\otimes_{E}x_{1,p})$,
which clearly lies in $I$. If $q=0$, we have

$w=x\otimes_{E}x_{1,p}-\sum_{i\in
S}\alpha_{i}^{*}(x)\alpha_{i}\otimes_{E}x_{1,p}=0,$

where the right equality follows from (2.4). Then $w$ trivially lies in $I$. ∎

For each $p\geq 0$, we have a natural morphism of $T_{E}(M^{*})$-$E$-bimodules

(2.5) $\displaystyle T_{E}(M^{*})\otimes_{E}M^{\otimes_{E}p}$
$\displaystyle\longrightarrow T_{E}(M^{*})\otimes_{E}M^{\otimes_{E}(p+1)}$
$\displaystyle f_{1,q}\otimes_{E}x_{1,p}$ $\displaystyle\longmapsto
f_{1,q}\otimes_{E}c\otimes_{E}x_{1,p}.$

Letting $p$ vary, we obtain a sequence of morphisms.

We have the following structure theorem on Leavitt algebras; compare [63,
Subsections 1.2 and 5.5].

######  Theorem 2.6.

Let $M$ be an $E$-$E$-bimodule with ${}_{E}M$ finitely generated projective.
Then as a $T_{E}(M^{*})$-$E$-bimodule, the Leavitt algebra $L_{E}(M)$ is
isomorphic to the colimit of the above sequence.

###### Proof.

By the construction of colimits, the mentioned colimit is isomorphic to the
following quotient bimodule

$(\bigoplus_{p\geq 0}T_{E}(M^{*})\otimes_{E}M^{\otimes_{E}p})/I=C_{E}(M)/I,$

where $I$ is the $E$-$E$-subbimodule spanned by elements of the form
$f_{1,q}\otimes_{E}x_{1,p}-f_{1,q}\otimes_{E}c\otimes_{E}x_{1,p}$. By Lemma
2.5, $I$ coincides with the principal ideal $(1_{E}-c)$ of $C_{E}(M)$. Then we
are done. ∎

##  3\. The dg Cohn and Leavitt algebras

As in the previous section, let $E$ be a $\mathbb{K}$-algebra and $M$ be an
$E$-$E$-bimodule on which $\mathbb{K}$ acts centrally. Throughout this
section, we further assume that ${}_{E}M$ is finitely generated projective. We
will introduce the dg Cohn algebra and dg Leavitt algebra associated to a pair
$(M,\mu)$, where $\mu$ is an associative bilinear map on $M$.

Recall that $V^{*}={\rm Hom}_{E}(V,E)$ for any $E$-$E$-bimodule $V$. We
observe that the following canonical map of $E$-$E$-bimodules

$\begin{array}[]{rccc}{\rm
can}\colon&M^{*}\otimes_{E}M^{*}&\longrightarrow&(M\otimes_{E}M)^{*}\\\
&f_{1}\otimes_{E}f_{2}&\longmapsto&(x_{1}\otimes_{E}x_{2}\mapsto
f_{2}(x_{1}f_{1}(x_{2}))\in E)\end{array}$

is an isomorphism.

We fix an $E$-$E$-bimodule homomorphism

$\mu\colon M\otimes_{E}M\longrightarrow M$

which is associative, that is,

$\mu\circ(\mu\otimes_{E}{\rm Id}_{M})=\mu\circ({\rm Id}_{M}\otimes_{E}\mu).$

Then we have two induced maps of $E$-$E$-bimodules:

(3.1) $\displaystyle\partial_{+}\colon
M^{*}\stackrel{{\scriptstyle\mu^{*}}}{{\longrightarrow}}(M\otimes_{E}M)^{*}\xrightarrow{{\rm
can}^{-1}}M^{*}\otimes_{E}M^{*}$

and

(3.2) $\displaystyle\partial_{-}\colon M\longrightarrow
M^{*}\otimes_{E}M\otimes_{E}M\xrightarrow{{\rm
Id}_{M^{*}}\otimes_{E}\mu}M^{*}\otimes_{E}M.$

Here, the unnamed arrow sends $x$ to $c\otimes_{E}x$ with $c$ the Casimir
element of $M$.

The following elementary facts are well known; compare [64, 3.7]. We mention
that the first statement is somehow dual to [49, Remark 4.17] in the bocs
theory.

######  Lemma 3.1.

Keep the notation as above. Then the following statements hold.

  1. (1)

The map $\partial_{+}$ is coassociative, that is,
$(\partial_{+}\otimes_{E}{\rm Id}_{M^{*}})\circ\partial_{+}=({\rm
Id}_{M^{*}}\otimes_{E}\partial_{+})\circ\partial_{+}$.

  2. (2)

The map $\partial_{-}$ makes $M$ into a left $(M^{*},\partial_{+})$-comodule,
that is, $(\partial_{+}\otimes_{E}{\rm Id}_{M})\circ\partial_{-}=({\rm
Id}_{M^{*}}\otimes_{E}\partial_{-})\circ\partial_{-}$.

###### Proof.

(1) follows from the associativity of $\mu$ by duality. We observe that

$(\partial_{+}\otimes_{E}{\rm Id}_{M^{*}})\circ\partial_{+}=({\rm
can}_{1})^{-1}\circ(\mu\otimes_{E}{\rm Id}_{M})^{*}\circ\mu^{*}=({\rm
can}_{1})^{-1}\circ(\mu\circ(\mu\otimes_{E}{\rm Id}_{M}))^{*}.$

Here, ${\rm can}_{1}\colon
M^{*}\otimes_{E}M^{*}\otimes_{E}M^{*}\rightarrow(M\otimes_{E}M\otimes_{E}M)^{*}$
is the canonical isomorphism sending $f_{1,3}$ to the map $(x_{1,3}\mapsto
f_{3}(x_{1}f_{2}(x_{2}f_{1}(x_{3})))\in E)$. Similarly, we have

$({\rm Id}_{M^{*}}\otimes_{E}\partial_{+})\circ\partial_{+}=({\rm
can}_{1})^{-1}\circ(\mu\circ({\rm Id}_{M}\otimes_{E}\mu))^{*}.$

(2) Recall that $c=\sum_{i\in S}\alpha_{i}^{*}\otimes_{E}\alpha_{i}\in
M^{*}\otimes_{E}M$. For any $x\in M$, we have

$(\partial_{+}\otimes_{E}{\rm Id}_{M})(\partial_{-}(x))=\sum_{i\in
S}\partial_{+}(\alpha_{i}^{*})\otimes_{E}\mu(\alpha_{i}\otimes x)$

and

$\displaystyle({\rm Id}_{M^{*}}\otimes_{E}\partial_{-})(\partial_{-}(x))$
$\displaystyle=\sum_{i,j\in
S}\alpha_{i}^{*}\otimes_{E}\alpha_{j}^{*}\otimes_{E}\mu(\alpha_{j}\otimes_{E}\mu(\alpha_{i}\otimes_{E}x))$
$\displaystyle=\sum_{i,j\in
S}\alpha_{i}^{*}\otimes_{E}\alpha_{j}^{*}\otimes_{E}\mu(\mu(\alpha_{j}\otimes_{E}\alpha_{i})\otimes_{E}x)).$

The associativity of $\mu$ is used in the last equality. Therefore, it
suffices to verify the following identity in
$M^{*}\otimes_{E}M^{*}\otimes_{E}M$.

(3.3) $\displaystyle\sum_{i\in
S}\partial_{+}(\alpha_{i}^{*})\otimes_{E}\alpha_{i}=\sum_{i,j\in
S}\alpha_{i}^{*}\otimes_{E}\alpha_{j}^{*}\otimes_{E}\mu(\alpha_{j}\otimes_{E}\alpha_{i})$

There is a canonical isomorphism ${\rm can}_{2}\colon
M^{*}\otimes_{E}M^{*}\otimes_{E}M\rightarrow{\rm Hom}_{E}(M\otimes_{E}M,M)$
sending $f_{1,2}\otimes_{E}y$ to the map $(x_{1,2}\mapsto
f_{2}(x_{1}f_{1}(x_{2}))y\in M)$. On one hand, we have

$\displaystyle{\rm can}_{2}\Big{(}\sum_{i\in
S}\partial_{+}(\alpha_{i}^{*})\otimes_{E}\alpha_{i}\Big{)}(x_{1}\otimes_{E}x_{2})=\sum_{i\in
S}\alpha_{i}^{*}(\mu(x_{1}\otimes_{E}x_{2}))\alpha_{i}=\mu(x_{1}\otimes_{E}x_{2}).$

Here, the left equality uses the definition of $\partial_{+}$ and the right
one uses (2.4). On the other hand, we have

$\displaystyle{\rm can}_{2}\Big{(}\sum_{i,j\in
S}\alpha_{i}^{*}\otimes_{E}\alpha_{j}^{*}\otimes_{E}\mu(\alpha_{j}\otimes_{E}\alpha_{i})\Big{)}(x_{1}\otimes_{E}x_{2})$
$\displaystyle=\sum_{i,j\in
S}\alpha_{j}^{*}(x_{1}\alpha_{i}^{*}(x_{2}))\mu(\alpha_{j}\otimes_{E}\alpha_{i})$
$\displaystyle=\sum_{i\in S}\mu\Big{(}\sum_{j\in
S}\alpha_{j}^{*}(x_{1}\alpha_{i}^{*}(x_{2}))\alpha_{j}\otimes_{E}\alpha_{i}\Big{)}$
$\displaystyle=\sum_{i\in
S}\mu(x_{1}\alpha_{i}^{*}(x_{2})\otimes_{E}\alpha_{i})$
$\displaystyle=\mu\Big{(}x_{1}\otimes_{E}\sum_{i\in
S}\alpha_{i}^{*}(x_{2})\alpha_{i}\Big{)}$
$\displaystyle=\mu(x_{1}\otimes_{E}x_{2}).$

Here, both the third and fifth equalities use (2.4). Then we are done with
(3.3). ∎

We consider the tensor algebra $T_{E}(M^{*}\oplus M)$. It is
$\mathbb{Z}$-graded by means of ${\rm deg}\;E=0$, ${\rm deg}\;M^{*}=1$ and
${\rm deg}\;M=-1$. By [8, Lemma 1.8], the bimodule morphisms $\partial_{+}$
and $\partial_{-}$ induce a unique $E$-derivation

$\partial\colon T_{E}(M^{*}\oplus M)\longrightarrow T_{E}(M^{*}\oplus M)$

of degree one such that $\partial(x)=\partial_{-}(x)$ and
$\partial(f)=\partial_{+}(f)$ for any $x\in M$ and $f\in M^{*}$. This means
that $\partial$ satisfies the following graded Leibniz rule

(3.4)
$\displaystyle\partial(u\otimes_{E}v)=\partial(u)\otimes_{E}v+(-1)^{|u|}u\otimes_{E}\partial(v)$

for any homogenous elements $u,v\in T_{E}(M^{*}\oplus M)$, and $\partial(a)=0$
for any $a\in E$.

We observe that $\partial^{2}|_{M^{*}}=0$ by Lemma 3.1(1) and that
$\partial^{2}|_{M}=0$ by Lemma 3.1(2); here, we use the minus sign in the
graded Leibniz rule. By [8, Remark 1.7(3)], we infer that $\partial^{2}=0$. In
other word, $(T_{E}(M^{*}\oplus M),\partial)$ is a dg tensor algebra; compare
[8, Section 1].

######  Remark 3.2.

We mention the following asymmetry in the dg tensor algebra
$(T_{E}(M^{*}\oplus M),\partial)$: the subalgebra $T_{E}(M)$ is not closed
under $\partial$, while the subalgebra $T_{E}(M^{*})$ is closed under
$\partial$ and thus becomes a dg subalgebra.

######  Lemma 3.3.

The Casimir element $c$, viewed as an element in $T_{E}(M\oplus M^{*})$, is
closed, that is, $\partial(c)=0$.

###### Proof.

By the graded Leibniz rule, we have

$\displaystyle\partial(c)=$ $\displaystyle\sum_{i\in
S}\partial_{+}(\alpha_{i}^{*})\otimes_{E}\alpha_{i}-\sum_{i\in
S}\alpha_{i}^{*}\otimes_{E}\partial_{-}(\alpha_{i})$ $\displaystyle=\sum_{i\in
S}\partial_{+}(\alpha_{i}^{*})\otimes_{E}\alpha_{i}-\sum_{i,j\in
S}\alpha_{i}^{*}\otimes_{E}\alpha_{j}^{*}\otimes_{E}\mu(\alpha_{j}\otimes_{E}\alpha_{i})=0.$

Here, the second equality uses the construction (3.2) of $\partial_{-}$ and
the last equality is precisely (3.3). ∎

######  Lemma 3.4.

The two-sided ideal $(x\otimes_{E}g-g(x)\;|\;x\in M,g\in M^{*})$ of
$T_{E}(M^{*}\oplus M)$ is a dg ideal, that is, it is closed under $\partial$.

###### Proof.

Denote the above ideal by $J$. By the graded Leibniz rule, it suffices to
prove that $\partial(x\otimes_{E}g-g(x))$ still lies in $J$. Since $g(x)\in E$
and thus $\partial(g(x))=0$, we have

$\partial(x\otimes_{E}g-g(x))=\partial_{-}(x)\otimes_{E}g-x\otimes_{E}\partial_{+}(g).$

Define an element $\phi\in M^{*}$ by

$\phi(y)=g(\mu(y\otimes_{E}x))\;\mbox{ for any }y\in M.$

By the definition of $\partial_{-}$, we have

$\partial_{-}(x)\otimes_{E}g=\sum_{i\in
S}\alpha_{i}^{*}\otimes_{E}\mu(\alpha_{i}\otimes x)\otimes_{E}g.$

Therefore, we infer that the element
$w_{1}:=\partial_{-}(x)\otimes_{E}g-\sum_{i\in
S}\alpha_{i}^{*}g(\mu(\alpha_{i}\otimes_{E}x))$ lies in the two-sided ideal
$J$. Moreover, we have

$w_{1}=\partial_{-}(x)\otimes_{E}g-\sum_{i\in
S}\alpha_{i}^{*}\phi(\alpha_{i})=\partial_{-}(x)\otimes_{E}g-\phi,$

where the right equality uses (2.4).

Write $\partial_{+}(g)=\sum_{j\in T}h_{j}\otimes_{E}f_{j}$. Then the following
element

$w_{2}:=x\otimes_{E}\partial_{+}(g)-\sum_{j\in
T}h_{j}(x)f_{j}=\Big{(}\sum_{j\in
T}(x\otimes_{E}h_{j}-h_{j}(x))\Big{)}\otimes_{E}f_{j}$

lies in $J$. By the definition (3.1) of $\partial_{+}$, we have
$\mathrm{can}\circ\partial_{+}(g)(y\otimes x)=\mu^{*}(g)(y\otimes x)$ for any
$y\in M$. Namely, we have

$\sum_{j\in T}f_{j}(yh_{j}(x))=g(\mu(y\otimes_{E}x))=\phi(y).$

We infer that $w_{2}=x\otimes_{E}\partial_{+}(g)-\phi$. Since $w_{1},w_{2}$
lie in $J$, so does their difference
$w_{1}-w_{2}=\partial_{-}(x)\otimes_{E}g-x\otimes_{E}\partial_{+}(g)$. We
deduce the required statement. ∎

Combining Proposition 2.2 with Lemma 3.4, we infer that the differential
$\partial$ on $T_{E}(M^{*}\oplus M)$ descends to the Cohn algebra $C_{E}(M)$.
By Lemma 3.3, the Casimir element $c$ is closed in $C_{E}(M)$. Therefore, the
differential $\partial$ descends further to the Leavitt algebra $L_{E}(M)$.

By abuse of notation, we will use $\partial$ to denote the induced
differentials on both $C_{E}(M)$ and $L_{E}(M)$. We emphasize that $\partial$
is uniquely determined by $\partial_{+}$ in (3.1) and $\partial_{-}$ in (3.2)
via the graded Leibniz rule (3.4).

######  Definition 3.5.

Let $M$ be an $E$-$E$-bimodule with ${}_{E}M$ finitely generated projective.
Assume that $\mu\colon M\otimes_{E}M\rightarrow M$ is an associative morphism
of $E$-$E$-bimodules. The resulting dg algebras $(C_{E}(M),\partial)$ and
$(L_{E}(M),\partial)$ are called the _dg Cohn algebra_ and _dg Leavitt
algebra_ associated to $(M,\mu)$, respectively.

######  Remark 3.6.

We claim that the differential $\partial$ on $L_{E}(M)$ is completely
determined by $\partial_{+}$. To be more precise, assume that
$\partial^{\prime}$ is any $E$-derivation on $L_{E}(M)$ whose restriction to
$M^{*}$ is $\partial_{+}$. We will show that the restriction of
$\partial^{\prime}$ to $M$ necessarily coincides with $\partial_{-}$, which
particularly yields $\partial^{\prime}=\partial$.

Recall the Casimir element $c=\sum_{i\in S}\alpha_{i}^{*}\otimes_{E}\alpha$.
For any $x\in M$, we have $x\otimes_{E}\alpha_{i}^{*}=\alpha_{i}^{*}(x)\in E$.
Therefore, we have

$0=\partial^{\prime}(x\otimes_{E}\alpha_{i}^{*})=\partial^{\prime}(x)\otimes_{E}\alpha_{i}^{*}-x\otimes_{E}\partial_{+}(\alpha_{i}^{*}).$

By the relation $1_{E}=c$, we have

(3.5) $\displaystyle\partial^{\prime}(x)=\sum_{i\in
S}\partial^{\prime}(x)\otimes_{E}\alpha_{i}^{*}\otimes_{E}\alpha_{i}=\sum_{i\in
S}x\otimes_{E}\partial_{+}(\alpha_{i}^{*})\otimes_{E}\alpha_{i}.$

The above identity together with the graded Leibniz rule (3.4) already
confirms the claim. Moreover, we have

$\displaystyle\sum_{i\in
S}x\otimes_{E}\partial_{+}(\alpha_{i}^{*})\otimes_{E}\alpha_{i}$
$\displaystyle=\sum_{i,j\in
S}\alpha_{i}^{*}(x)\alpha_{j}^{*}\otimes_{E}\mu(\alpha_{j}\otimes_{E}\alpha_{i})$
$\displaystyle=\sum_{i,j\in
S}\alpha_{j}^{*}\otimes_{E}\mu(\alpha_{j}\alpha_{i}^{*}(x)\otimes_{E}\alpha_{i})$
$\displaystyle=\sum_{i,j\in
S}\alpha_{j}^{*}\otimes_{E}\mu(\alpha_{j}\otimes_{E}\alpha_{i}^{*}(x)\alpha_{i})$
$\displaystyle=\sum_{j\in
S}\alpha_{j}^{*}\otimes_{E}\mu(\alpha_{j}\otimes_{E}x)=\partial_{-}(x).$

Here, the first equality uses (3.3) and the relation
$x\otimes_{E}\alpha_{i}^{*}=\alpha_{i}^{*}(x)$, the second one uses the fact
$\alpha_{i}^{*}(x)c=c\alpha_{i}^{*}(x)\in M^{*}\otimes_{E}M$, and the fourth
one uses (2.4). This proves $\partial^{\prime}(x)=\partial_{-}(x)$. Since any
$E$-derivation on $L_{E}(M)$ is uniquely determined by its values at the
generating $E$-$E$-bimodule $M^{*}\oplus M$, it follows that
$\partial^{\prime}=\partial.$

##  4\. Quivers and path algebras

In this section, we study the dg Cohn algebra and dg Leavitt algebra in the
quiver situation, namely the dg Cohn path algebra and dg Leavitt path algebra,
respectively. The differentials can be described explicitly.

A quiver is a directed graph. Formally, it is a quadruple
$Q=(Q_{0},Q_{1};s,t)$ consisting of a set $Q_{0}$ of vertices, a set $Q_{1}$
of arrows and two maps $s,t\colon Q_{1}\xrightarrow[]{}Q_{0}$, which associate
to each arrow $\alpha$ its starting vertex $s(\alpha)$ and its terminating
vertex $t(\alpha)$, respectively. We visualize an arrow $\alpha$ as
$\alpha\colon s(\alpha)\rightarrow t(\alpha)$. A vertex is called a sink if no
arrow starts in this vertex. The quiver $Q$ is _finite_ provided that both
$Q_{0}$ and $Q_{1}$ are finite sets.

We fix a finite quiver $Q$. A path of length $n$ is a sequence
$p=\alpha_{n}\dotsb\alpha_{2}\alpha_{1}$ of arrows with
$t(\alpha_{j})=s(\alpha_{j+1})$ for $1\leq j\leq n-1$. Denote by $l(p)=n$. The
starting vertex of $p$, denoted by $s(p)$, is defined to be $s(\alpha_{1})$.
The terminating vertex of $p$, denoted by $t(p)$, is defined to be
$t(\alpha_{n})$. We identify an arrow with a path of length one. We associate
to each vertex $i\in Q_{0}$ a _trivial_ path $e_{i}$ of length zero, and set
$s(e_{i})=i=t(e_{i})$. Denote by $Q_{n}$ the set of paths of length $n$.

The _path algebra_ $\mathbb{K}Q=\bigoplus_{n\geq 0}\mathbb{K}Q_{n}$ is a free
$\mathbb{K}$-module with a basis given by all the paths in $Q$, whose
multiplication is given as follows: for two paths $p$ and $q$ satisfying
$s(p)=t(q)$, the product $pq$ is their concatenation; otherwise, the product
$pq$ is defined to be zero. Here, we write the concatenation of paths from
right to left. For example, we have $e_{t(p)}p=p=pe_{s(p)}$ for each path $p$.

We denote by $\overline{Q}$ the _double quiver_ of $Q$, which is obtained from
$Q$ by adding for each arrow $\alpha\in Q_{1}$ a new arrow $\alpha^{*}$ in the
opposite direction, that is, we have $s(\alpha^{*})=t(\alpha)$ and
$t(\alpha^{*})=s(\alpha)$. The added arrows $\alpha^{*}$ are called the
_ghost_ arrows. Denote by $Q_{1}^{*}$ the set formed by the ghost arrows. More
generally, we denote by $Q_{n}^{*}$ the set formed by all paths of length $n$
which consist entirely of ghost arrows. For a path
$p=\alpha_{n}\cdots\alpha_{2}\alpha_{1}\in Q_{n}$, we set
$p^{*}=\alpha_{1}^{*}\alpha_{2}^{*}\cdots\alpha_{n}^{*}\in Q^{*}_{n}$.

Set $E=\mathbb{K}Q_{0}=\bigoplus_{i\in Q_{0}}\mathbb{K}e_{i}$ and
$M=\mathbb{K}Q_{1}$. Then $E$ is a subalgebra of $\mathbb{K}Q$ and $M$ is
naturally an $E$-$E$-bimodule. Recall that $M^{*}={\rm Hom}_{E}(M,E)$. Each
ghost arrow $\alpha^{*}$ gives rise to an element
$(\beta\mapsto\delta_{\alpha,\beta}e_{t(\alpha)})$ in $M^{*}$. Here, $\delta$
is the Kronecker symbol. In this way, we have an identification of
$E$-$E$-bimodules.

$M^{*}=\mathbb{K}Q_{1}^{*}$

It is well known that the inclusions $E\subseteq\mathbb{K}Q$ and
$M\subseteq\mathbb{K}Q$ induce a canonical isomorphism

$T_{E}(M)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{K}Q$

of algebras. In more detail, a tensor
$\alpha_{n}\otimes_{E}\cdots\otimes_{E}\alpha_{2}\otimes_{E}\alpha_{1}$ of
arrows is sent to the corresponding path
$p=\alpha_{n}\cdots\alpha_{2}\alpha_{1}$. Similarly, we use the above
identification $M^{*}=\mathbb{K}Q_{1}^{*}$ and embed $M^{*}$ into
$\mathbb{K}\overline{Q}$. Then we obtain a canonical isomorphism

(4.1) $\displaystyle T_{E}(M^{*}\oplus
M)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{K}\overline{Q}$

of algebras.

As in [2, Definition 1.5.1], the _Cohn path algebra_ $C(Q)$ is defined as the
following quotient algebra.

$C(Q)=\mathbb{K}\overline{Q}/{(\alpha\beta^{*}-\delta_{\alpha,\beta}e_{t(\alpha)}\;|\;\alpha,\beta\in
Q_{1})}$

Following [1, 4, 5], the _Leavitt path algebra_ $L(Q)$ is defined as the
further quotient algebra.

$L(Q)=C(Q)/{(e_{i}-\sum_{\\{\alpha\in
Q_{1}\;|\;s(\alpha)=i\\}}\alpha^{*}\alpha\;|\;i\in Q_{0}\mbox{ are non-sinks
in $Q$})}$

The relations $\alpha\beta^{*}-\delta_{\alpha,\beta}e_{t(\alpha)}$ and
$e_{i}-\sum_{\\{\alpha\in Q_{1}\;|\;s(\alpha)=i\\}}\alpha^{*}\alpha$ are known
as the first and second _Cuntz-Krieger relations_ , respectively. For
relations between Cohn path algebras and Leavitt path algebras, we refer to
[2, Theorem 1.5.18].

Denote by $Q^{\circ}$ the quiver without sinks, that is obtained from $Q$ by
removing sinks repeatedly.

######  Proposition 4.1.

Keep the notation as above. Then the following statements hold.

  1. (1)

There is a canonical isomorphism $C_{E}(M)\simeq C(Q)$ of algebras.

  2. (2)

There is a canonical isomorphism $L_{E}(M)\simeq L(Q^{\circ})$ of algebras.

###### Proof.

(1) follows from Proposition 2.2 and the isomorphism (4.1). Here, we observe
that the element $\alpha\otimes_{E}\beta^{*}-\beta^{*}(\alpha)\in
T_{E}(M^{*}\oplus M)$ corresponds to
$\alpha\beta^{*}-\delta_{\alpha,\beta}e_{t(\alpha)}\in\mathbb{K}\overline{Q}$.

(2) Recall that $1_{E}=\sum_{i\in Q_{0}}e_{i}$. Using the above identification
$M^{*}=\mathbb{K}Q_{1}^{*}$, we infer that the Casimir element $c\in
M^{*}\otimes_{E}M$ corresponds to $\sum_{\alpha\in Q_{1}}\alpha^{*}\alpha\in
C(Q)$. Therefore, using (1), we infer that $L_{E}(M)$ is isomorphic to

$\displaystyle C(Q)\big{/}{\big{(}\sum_{i\in Q_{0}}e_{i}-\sum_{\alpha\in
Q_{1}}\alpha^{*}\alpha\big{)}}$ $\displaystyle=$ $\displaystyle
C(Q)\Big{/}{\Big{(}e_{i}-\sum_{\\{\alpha\in
Q_{1}\;|\;s(\alpha)=i\\}}\alpha^{*}\alpha,e_{j}\;|\;i\in Q_{0}\mbox{ non-
sinks, }j\in Q_{0}\mbox{ sinks}\Big{)}}.$

It follows that $L_{E}(M)$ is isomorphic to $L(Q)/{(e_{j}\;|\;j\in Q_{0}\mbox{
sinks})}$.

We observe by the second Cuntz-Krieger relation, the ideal ${(e_{j}\;|\;j\in
Q_{0}\mbox{ sinks})}$ coincides with the following ideal

$(e_{j}\;|\;j\in Q_{0}\backslash Q^{\circ}_{0}).$

It follows that the quotient algebra $L(Q)/{(e_{j}\;|\;j\in Q_{0}\mbox{
sinks})}$ is isomorphic to $L(Q^{\circ})$. Then the required isomorphism
follows readily. ∎

By Proposition 4.1(1) and the explicit construction of $C_{E}(M)$, we infer
that $C(Q)$ is a free $\mathbb{K}$-module with a basis given by the following
set

$\\{p^{*}q\;|\;p\mbox{ and }q\mbox{ are paths in }Q\mbox{ satisfying
}t(p)=t(q)\\}.$

We observe that both $C(Q)$ and $L(Q)$ is naturally $\mathbb{Z}$-graded such
that $|e_{i}|=0$, $|\alpha|=-1$ and $|\alpha^{*}|=1$.

######  Remark 4.2.

The chosen grading of $L(Q)$ here is different from the one in [63, 24, 22],
where the degrees of $e_{i}$, $\alpha$ and $\alpha^{*}$ equal $0$, $1$ and
$-1$, respectively. There is an involution $(-)^{*}\colon L(Q)\to L(Q)$ of
algebras given by $(e_{i})^{*}=e_{i},(\alpha)^{*}=\alpha^{*}$ and
$(\alpha^{*})^{*}=\alpha$. We observe that the involution identifies the two
gradings on $L(Q)$. Therefore, there is no essential difference between these
two gradings. In the following consideration of dg Leavitt path algebras, one
sees that the grading here is more reasonable.

Recall that $E=\mathbb{K}Q_{0}$ and $M=\mathbb{K}Q_{1}$. We fix an associative
morphism $\mu\colon M\otimes_{E}M\rightarrow M$ of $E$-$E$-bimodules.
Identifying $M\otimes_{E}M$ with $\mathbb{K}Q_{2}$, we might view $\mu$ as an
$E$-$E$-bimodule map

$\mu\colon\mathbb{K}Q_{2}\longrightarrow\mathbb{K}Q_{1}.$

Consequently, it is uniquely determined by the following formula: for each
path $p$ of length two in $Q$, we have

(4.2) $\displaystyle\mu(p)=\sum_{\\{\alpha\in
Q_{1}\;|\;\alpha//p\\}}\lambda_{p,\alpha}\alpha$

for some structure coefficients $\lambda_{p,\alpha}\in\mathbb{K}$. Here
$\alpha//p$ indicates that $\alpha$ is _parallel_ to $p$, i.e.
$s(\alpha)=s(p)$ and $t(\alpha)=t(p)$. The associativity of $\mu$ is
equivalent to the following condition: for each path
$q=\alpha_{3}\alpha_{2}\alpha_{1}$ of length three and each arrow $\alpha$
parallel to $q$, we have

$\sum_{\\{\beta\in
Q_{1}\;|\;\beta//\alpha_{3}\alpha_{2}\\}}\lambda_{\alpha_{3}\alpha_{2},\beta}\lambda_{\beta\alpha_{1},\alpha}=\sum_{\\{\beta^{\prime}\in
Q_{1}\;|\;\beta^{\prime}//\alpha_{2}\alpha_{1}\\}}\lambda_{\alpha_{2}\alpha_{1},\beta^{\prime}}\lambda_{\alpha_{3}\beta^{\prime},\alpha}.$

Recall from the previous section that $C_{E}(M)$ is naturally a dg algebra. By
the canonical isomorphism in Proposition 4.1(1) and transferring the
structures, we infer that $C(Q)$ is a dg algebra with differential $\partial$,
called the _dg Cohn path algebra_ associated to $(Q,\mu)$. As the differential
descends to $L(Q^{\circ})$, we obtain the dg algebra
$(L(Q^{\circ}),\partial)$, called the _dg Leavitt path algebra_ associated to
$(Q,\mu)$.

######  Remark 4.3.

By Lemma 3.3 and its proof, we observe that the differential $\partial$ on
$C(Q)$ descends to $L(Q)$. Then the differential of $L(Q^{\circ})$ is
inherited from the one of $L(Q)$ via the isomorphism

$L(Q)/{(e_{j}\;|\;j\in Q_{0}\mbox{ sinks})}\simeq L(Q^{\circ}).$

By Proposition 4.1(2), the dg Leavitt path algebra $(L(Q^{\circ}),\partial)$,
rather than $(L(Q),\partial)$, is more relevant to us.

Recall that the differential $\partial$ of $C(Q)$ is completely determined by
$\partial_{+}$ and $\partial_{-}$; see (3.1) and (3.2). Both maps are uniquely
determined by the structure coefficients $\lambda_{p,\alpha}$ in (4.2).

To make them explicit, we use the identification $M^{*}=\mathbb{K}Q^{*}_{1}$.
Moreover, we identify $M^{*}\otimes_{E}M^{*}$ with $\mathbb{K}Q_{2}^{*}$,
sending a typical tensor $\alpha^{*}\otimes_{E}\beta^{*}$ of ghost arrows to
$(\beta\alpha)^{*}=\alpha^{*}\beta^{*}\in Q_{2}^{*}$. Then we have

(4.5)
$\displaystyle\begin{array}[]{cccc}\partial_{+}\colon&\mathbb{K}Q_{1}^{*}&\longrightarrow&\mathbb{K}Q^{*}_{1}\otimes_{E}\mathbb{K}Q^{*}_{1}=\mathbb{K}Q_{2}^{*},\\\
&\alpha^{*}&\mapsto&\displaystyle\sum_{\\{p\in
Q_{2}\;|\;\alpha//p\\}}\lambda_{p,\alpha}p^{*}\end{array}$

and

(4.8)
$\displaystyle\begin{array}[]{cccc}\partial_{-}\colon&\mathbb{K}Q_{1}&\longrightarrow&\mathbb{K}Q_{1}^{*}\otimes_{E}\mathbb{K}Q_{1}\subseteq
C(Q),\\\ &\alpha&\mapsto&\displaystyle\sum_{\\{\beta\in
Q_{1}\;|\;s(\beta)=t(\alpha)\\}}\beta^{*}\mu(\beta\alpha)=\sum\lambda_{\beta\alpha,\beta^{\prime}}\;\beta^{*}\beta^{\prime}.\end{array}$

where the last sum without subscript runs over ${\\{\beta,\beta^{\prime}\in
Q_{1}\;|\;s(\beta)=t(\alpha),\;\beta^{\prime}//\beta\alpha\\}}$.

We now give a concrete example.

######  Example 4.4.

Let $n\geq 1$ and $R_{n}$ be the rose quiver with one vertex and $n$ loops.

$\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{1}}$$\scriptstyle{x_{2}}$$\scriptstyle{x_{n}}$$\scriptstyle{\cdots}$

Then $E=\mathbb{K}$ and $M=\bigoplus_{i=1}^{n}\mathbb{K}x_{i}$. Define a
$\mathbb{K}$-linear product $\mu\colon M\otimes_{\mathbb{K}}M\rightarrow M$
according to the following rule:

$\displaystyle\mu(x_{i}\otimes x_{j})=\begin{cases}x_{i+j},&\text{ if $i+j\leq
n$};\\\ 0,&\text{otherwise.}\end{cases}$

We observe that $\mu$ is associative.

The dg Cohn path algebra and dg Leavitt path algebra associated to
$(R_{n},\mu)$ are described as follows:

$C(R_{n})=\mathbb{K}\langle
x_{1},\dotsc,x_{n},y_{1},\dotsc,y_{n}\rangle/{(x_{i}y_{j}-\delta_{i,j}\;|\;1\leq
i,j\leq n)}$

and

$L(R_{n})=\mathbb{K}\langle
x_{1},\dotsc,x_{n},y_{1},\dotsc,y_{n}\rangle/{(x_{i}y_{j}-\delta_{i,j},\
1-\sum_{k=1}^{n}y_{k}x_{k}\;|\;1\leq i,j\leq n)}.$

Both algebras are graded such that $|x_{i}|=-1$ and $|y_{i}|=1$. Here, we
write $y_{i}$ for the ghost arrow $x_{i}^{*}$. The differential $\partial$ on
both algebras is uniquely determined by the following formula: for each $1\leq
i\leq n$, we have

$\partial(y_{i})=\sum_{1\leq j\leq i-1}y_{j}y_{i-j}\quad\mbox{ and
}\quad\partial(x_{i})=\sum_{i<j\leq n}y_{j-i}x_{j}.$

In particular, we have $\partial(y_{1})=0=\partial(x_{n})$.

The algebras $C(R_{n})$ and $L(R_{n})$ are known as the (classical) Cohn
algebra and Leavitt algebra, respectively. We mention that $C(R_{1})$ is also
called the Toeplitz-Jacobson algebra; see [2, Proposition 1.3.7] and [36].
Moreover, $L(R_{1})$ is isomorphic to the Laurent polynomial algebra
$\mathbb{K}[y,y^{-1}]$ in one variable.

##  5\. Pretriangulated dg categories

In this section, we recall some basic facts on dg categories. We are mainly
concerned with pretriangulated dg categories, dg quotient categories and the
perfect dg derived categories of dg algebras. The main references are [30,
44].

###  5.1. DG categories and functors

Let $\mathcal{C}$ be a dg category. For two objects $X$ and $Y$, its Hom
complex is usually denoted by
$\mathcal{C}(X,Y)=(\bigoplus_{p\in\mathbb{Z}}\mathcal{C}(X,Y)^{p},d_{\mathcal{C}})$.
Morphisms in $\mathcal{C}(X,Y)^{p}$ are said to be homogeneous of degree $p$.
A morphism $f\colon X\rightarrow Y$ is said to be closed, if
$d_{\mathcal{C}}(f)=0$.

We denote by $Z^{0}(\mathcal{C})$ the ordinary category of $\mathcal{C}$,
which has the same objects as $\mathcal{C}$ and whose morphisms are precisely
closed morphisms in $\mathcal{C}$ of degree zero, that is, its Hom modules are
the zeroth cocycles of the corresponding Hom complexes. Similarly, the
_homotopy category_ $H^{0}(\mathcal{C})$ has the same objects and its Hom
modules are give by the zeroth cohomology of the corresponding Hom complexes.
An object $X$ is _contractible_ in $\mathcal{C}$ if ${\rm Id}_{X}$ is a
coboundary, or equivalently, $X$ is isomorphic to the zero object in
$H^{0}(\mathcal{C})$.

We denote by $\mathcal{C}^{\rm op}$ the _opposite_ dg category of
$\mathcal{C}$, whose composition $\circ^{\rm op}$ is given by $g\circ^{\rm
op}f=(-1)^{|g||f|}f\circ g$.

A closed morphism $f\colon X\rightarrow Y$ of degree zero is called a _dg-
isomorphism_ , if it is an isomorphism in $Z^{0}(\mathcal{C})$, or
equivalently in $\mathcal{C}$; it is called a _homotopy equivalence_ , if its
image in $H^{0}(\mathcal{C})$ is an isomorphism.

In the following examples, we fix the notation which will be used later. For a
$\mathbb{K}$-algebra $\Lambda$, we denote by $\Lambda\mbox{-Mod}$ the abelian
category of left $\Lambda$-modules.

######  Example 5.1.

Let $\Lambda$ be a $\mathbb{K}$-algebra. For two complexes $X$ and $Y$ of
$\Lambda$-modules, we denote by ${\rm Hom}_{\Lambda}(X,Y)$ the Hom complex
given as follows: its $p$-th homogeneous component is given by an infinite
product

${\rm Hom}_{\Lambda}(X,Y)^{p}=\prod_{n\in\mathbb{Z}}{\rm
Hom}_{\Lambda\mbox{-}{\rm Mod}}(X^{n},Y^{n+p}),$

whose elements will be denoted by $f=\\{f^{n}\\}_{n\in\mathbb{Z}}$ with
$f^{n}\in{\rm Hom}_{\Lambda\mbox{-}{\rm Mod}}(X^{n},Y^{n+p})$; the
differential $d$ acts on $f$ via

$d(f)^{n}=d_{Y}^{n+p}\circ f^{n}-(-1)^{|f|}f^{n+1}\circ
d_{X}^{n},\quad\quad\text{for each $n\in\mathbb{Z}.$}$

The collection of all complexes of $\Lambda$-modules with these Hom complexes
yields a dg category, denoted by $C_{\rm dg}(\Lambda\mbox{-}{\rm Mod})$. We
observe that $Z^{0}C_{\rm dg}(\Lambda\mbox{-}{\rm Mod})=C(\Lambda\mbox{-{\rm
Mod}})$ is the category of complexes of $\Lambda$-modules and that
$H^{0}C_{\rm dg}(\Lambda\mbox{-}{\rm Mod})=\mathbf{K}(\Lambda\mbox{-{\rm
Mod}})$ is the classical homotopy category of complexes of $\Lambda$-modules.

Let $\mathfrak{a}$ be an additive category. Slightly generalizing the above
construction, we obtain the dg category $C_{\rm dg}(\mathfrak{a})$ of
complexes in $\mathfrak{a}$. The homotopy category $H^{0}C_{\rm
dg}(\mathfrak{a})$ equals $\mathbf{K}(\mathfrak{a})$, the classical homotopy
category of complexes in $\mathfrak{a}$.

The dg category $C_{\rm dg}(\mathbb{K}\mbox{-}{\rm Mod})$ is usually denoted
by $C_{\rm dg}(\mathbb{K})$.

######  Example 5.2.

Let $\mathcal{C}$ and $\mathcal{D}$ be two dg categories. Assume that
$\mathcal{C}$ is small. For two dg functors
$F,G\colon\mathcal{C}\rightarrow\mathcal{D}$, a natural transformation
$\eta=(\eta_{X})_{X\in{\rm Obj}(\mathcal{C})}\colon F\rightarrow G$ of degree
$p$ consists of morphisms $\eta_{X}\colon F(X)\rightarrow G(X)$ of degree $p$
in $\mathcal{D}$ satisfying the following graded naturality property: for any
morphism $a\colon X\rightarrow X^{\prime}$ in $\mathcal{C}$, we have

$G(a)\circ\eta_{X}=(-1)^{p|a|}\eta_{X^{\prime}}\circ F(a)\colon F(X)\to
G(X^{\prime}).$

We now define the Hom complex ${\rm Hom}(F,G)$ such that its $p$-th component
is formed by natural transformations of degree $p$ from $F$ to $G$ and that
its differential is given by $d(\eta)_{X}=d_{\mathcal{D}}(\eta_{X})$ for any
object $X\in\mathcal{C}$.

The collection of all dg functors from $\mathcal{C}$ to $\mathcal{D}$ together
with the Hom complexes yields a dg category, denoted by ${\rm Fun}_{\rm
dg}(\mathcal{C},\mathcal{D}).$ In particular, a natural transformation
$\eta=(\eta_{X})_{X\in{\rm Obj(\mathcal{C})}}$ is closed if
$d_{\mathcal{D}}(\eta_{X})=0$ for any object $X\in\mathcal{C}$.

By a left dg $\mathcal{C}$-module, we mean a dg functor
$M\colon\mathcal{C}\rightarrow C_{\rm dg}(\mathbb{K})$. Write

$\mathcal{C}\mbox{-{\rm DGMod}}={\rm Fun}_{\rm dg}(\mathcal{C},C_{\rm
dg}(\mathbb{K}))$

for the dg category formed by left dg $\mathcal{C}$-modules. Write
$\mathbf{K}(\mathcal{C})=H^{0}(\mathcal{C}\mbox{-{\rm DGMod}})$ and
$\mathbf{D}(\mathcal{C})$ for the homotopy category and derived category of dg
$\mathcal{C}$-modules, respectively.

For a left dg $\mathcal{C}$-module $M$ and each $i\in\mathbb{Z}$, the shifted
$\mathcal{C}$-module $\Sigma^{i}(M)$ is defined as follows: as a complex

$\Sigma^{i}(M)(X)=\Sigma^{i}(MX),\quad\quad\text{for each object $X$ in
$\mathcal{C}$}$

and for any morphism $f\colon X\rightarrow X^{\prime}$ in $\mathcal{C}$, the
induced map
$\Sigma^{i}(M)(f)\colon\Sigma^{i}(MX)\rightarrow\Sigma^{i}(MX^{\prime})$ sends
$m$ to $(-1)^{i|f|}M(f)(m)$. Here, for simplicity, we write $MX=M(X)$ and
$MX^{\prime}=M(X^{\prime})$. For a closed morphism $\eta\colon M\rightarrow N$
of degree zero between dg modules, its _cone_ ${\rm Cone}(\eta)$ is a dg
module defined as follows:

${\rm Cone}(\eta)(X):={\rm Cone}(\eta_{X})=NX\oplus\Sigma(MX)$

is the mapping cone of $\eta_{X}\colon MX\rightarrow NX$, and ${\rm
Cone}(\eta)(f)$ is given by $\left(\begin{smallmatrix}N(f)&0\\\
0&\Sigma(M)(f)\end{smallmatrix}\right)$.

A dg functor $F\colon\mathcal{C}\rightarrow\mathcal{D}$ is said to be quasi-
fully faithful if for any objects $X$ and $Y$, the induced cochain map

$\mathcal{C}(X,Y)\longrightarrow\mathcal{D}(FX,FY)$

is a quasi-isomorphism. Consequently, $H^{0}(F)\colon
H^{0}(\mathcal{C})\rightarrow H^{0}(\mathcal{D})$ is fully faithful. The dg
functor $F$ is called a _quasi-equivalence_ , provided that it is quasi-fully
faithful and $H^{0}(F)$ is dense.

We denote by $\mathbf{dgcat}$ the category of small dg categories, whose
morphisms are dg functors. The _homotopy category_ $\mathbf{Hodgcat}$ is the
localization of $\mathbf{dgcat}$ with respect to all the quasi-equivalences.
In other words, $\mathbf{Hodgcat}$ is obtained from $\mathbf{dgcat}$ by
formally inverting quasi-equivalences. By the model structure [65] on
$\mathbf{dgcat}$, the morphisms between two objects in $\mathbf{Hodgcat}$ form
a set; compare [30, Appendix B.4-6].

For dg categories $\mathcal{C}$ and $\mathcal{D}$, a morphism in
$\mathbf{Hodgcat}$ between them is sometimes called a _dg quasi-functor_. It
can be realised as a roof

$\mathcal{C}\stackrel{{\scriptstyle
F}}{{\longleftarrow}}\mathcal{C}^{\prime}\stackrel{{\scriptstyle
F^{\prime}}}{{\longrightarrow}}\mathcal{D}$

of dg functors, where $F$ is a quasi-equivalence; moreover, $F$ can be taken
as a semi-free resolution of $\mathcal{C}$; moreover, the dg quasi-functor is
an isomorphism if and only if the dg functor $F^{\prime}$ is a quasi-
equivalence. For details, we refer to [65] and [30, Appendix B.5].

In the sequel, we abuse a dg quasi-functor with a genuine dg functor, and
abuse isomorphisms in $\mathbf{Hodgcat}$ with quasi-equivalences. In practice,
we will relax the smallness assumption by the following remark; compare [56,
Remark 1.22 and Appendix A].

######  Remark 5.3.

We usually encounter non-small dg categories, when we consider unbounded
complexes or arbitrary dg modules. Then we have to choose a universe
$\mathbb{U}$ and restrict ourselves to $\mathbb{U}$-small complexes and
modules; compare [68, Section 2] and [44, Subsection 4.4, p.172]. This allows
us to treat unbounded complexes equally in the framework of the homotopy
category $\mathbf{Hodgcat}$.

###  5.2. Exact and pretriangulated dg categories

Let $\mathcal{C}$ be a small dg category. Consider the Yoneda embedding

$\mathbf{Y}_{\mathcal{C}}\colon\mathcal{C}\longrightarrow\mathcal{C}^{\rm
op}\mbox{-DGMod},\;X\mapsto\mathcal{C}(-,X).$

It is a fully faithful dg functor, which induces a fully faithful functor

$H^{0}(\mathbf{Y}_{\mathcal{C}})\colon
H^{0}(\mathcal{C})\longrightarrow\mathbf{K}(\mathcal{C}^{\rm op}).$

Recall from [39, Section 2] that the dg category $\mathcal{C}$ is _exact_ (=
strongly pretriangulated in the sense of [12]) if the essential image of
$\mathbf{Y}_{\mathcal{C}}$ is closed under shifts and cones. In other words,
all the shifted modules $\Sigma^{i}\mathcal{C}(-,X)$ and cones ${\rm
Cone}(\mathcal{C}(-,f))$ are dg-representable for any object $X$ and any
closed morphism $f$ of degree zero.

The following internal characterization of exact dg categories is well known;
see [13, Section 3].

######  Lemma 5.4.

Let $\mathcal{C}$ be a small dg category. Then $\mathcal{C}$ is exact if and
only if the following two conditions are satisfied:

  1. (1)

the internal shifts of objects exist, that is, for each object $X$, there
exist two objects $X_{1}$ and $X_{2}$ with two closed isomorphisms
$X\rightarrow X_{1}$ and $X_{2}\rightarrow X$ in $\mathcal{C}$ of degree one;

  2. (2)

the internal cones of morphisms exist, that is, for each closed morphism
$f\colon X\rightarrow X^{\prime}$ of degree zero, there is a diagram in
$\mathcal{C}$

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{p}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$

with $|j|=0=|t|$, $|p|=1$ and $|s|=-1$ subject to the following identities:

$p\circ j=0=t\circ s,\ \ {\rm Id}_{Z}=s\circ p+j\circ t,\ \ {\rm
Id}_{X^{\prime}}=t\circ j,\ \ {\rm Id}_{X}=p\circ s$

and

$d_{\mathcal{C}}(j)=0=d_{\mathcal{C}}(p),\ \ f=t\circ d_{\mathcal{C}}(s).$

The dg category $\mathcal{C}$ is _pretriangulated_ [12] if all the shifted
modules $\Sigma^{i}\mathcal{C}(-,X)$ and cones ${\rm Cone}(\mathcal{C}(-,f))$
are homotopy equivalent to representable functors, or equivalently, the
essential image of $H^{0}(\mathbf{Y}_{\mathcal{C}})$ is a triangulated
subcategory of $\mathbf{K}(\mathcal{C}^{\rm op})$. Consequently, for a
pretriangulated dg category $\mathcal{C}$, its homotopy category
$H^{0}(\mathcal{C})$ has a canonical triangulated structure in the following
sense: for any dg functor $F\colon\mathcal{C}\rightarrow\mathcal{D}$ between
pretriangulated dg categories, the induced functor $H^{0}(F)\colon
H^{0}(\mathcal{C})\rightarrow H^{0}(\mathcal{D})$ is naturally a triangle
functor. We mention that an exact dg category is clearly pretriangulated.

For any dg category $\mathcal{C}$, its _exact hull_ means an exact dg category
$\mathcal{C}^{\rm ex}$ with a fully faithful dg functor

${\rm can}_{\mathcal{C}}\colon\mathcal{C}\longrightarrow\mathcal{C}^{\rm ex},$

which induces a dg-equivalence

(5.1) $\displaystyle{\rm Fun}_{\rm dg}(\mathcal{C}^{\rm
ex},\mathcal{D})\longrightarrow{\rm Fun}_{\rm
dg}(\mathcal{C},\mathcal{D}),\;F\mapsto F\circ{\rm can}_{\mathcal{C}}$

for any exact dg category $\mathcal{D}$. Indeed, one might take
$\mathcal{C}^{\rm ex}$ to be the smallest full dg subcategory of
$\mathcal{C}^{\rm op}\mbox{-DGMod}$ containing the representable functors and
closed under shifts and cones; then ${\rm can}_{\mathcal{C}}$ is given by the
Yoneda embedding. For an explicit construction of the exact hull, we refer to
[12] and [30, Subsection 2.4].

The following facts are standard. The first statement implies that
pretriangulated dg categories are invariant under quasi-equivalences. In
contrast, exact dg categories usually are not invariant under quasi-
equivalences.

######  Lemma 5.5.

Let $\mathcal{C}$ and $\mathcal{D}$ be two small dg categories. Then the
following hold.

  1. (1)

Let $F\colon\mathcal{C}\rightarrow\mathcal{D}$ be a quasi-equivalence. Then
$\mathcal{C}$ is pretriangulated if and only if so is $\mathcal{D}$.

  2. (2)

Assume that $\mathcal{D}$ is exact and that
$F\colon\mathcal{C}\rightarrow\mathcal{D}$ is a fully-faithful dg functor with
$H^{0}(F)$ dense. Then $\mathcal{C}$ is pretriangulated.

###### Proof.

For (1), we consider the following diagram, which is commutative up to
isomorphism.

$\textstyle{H^{0}(\mathcal{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H^{0}(\mathbf{Y}_{\mathcal{C}})}$$\scriptstyle{H^{0}(F)}$$\textstyle{H^{0}(\mathcal{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H^{0}(\mathbf{Y}_{\mathcal{D}})}$$\textstyle{\mathbf{K}(\mathcal{C}^{\rm
op})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\otimes_{\mathcal{C}}\mathcal{D}}$$\textstyle{\mathbf{K}(\mathcal{D}^{\rm
op})}$

Here, $\mathcal{D}$ is viewed a dg $\mathcal{C}$-$\mathcal{D}$-bimodule. The
vertical arrows are fully faithful and the bottom arrow is a triangle functor.
As $H^{0}(F)$ is an equivalence, it follows immediately that
$H^{0}(\mathcal{C})$ is a triangulated subcategory of
$\mathbf{K}(\mathcal{C}^{\rm op})$ if and only if the same holds for
$\mathcal{D}$. Then (1) follows immediately.

(2) is a very special case of (1), once we observe that $\mathcal{D}$ is
pretriangulated and that $F$ is a quasi-equivalence. ∎

The following elementary fact will be used often; see [57, Lemma 2.5] and [19,
Lemma 3.1].

######  Lemma 5.6.

Let $F\colon\mathcal{C}\rightarrow\mathcal{D}$ be a dg functor between two
pretriangulated dg categories. Assume that $H^{0}(F)$ is a triangle
equivalence. Then $F$ is a quasi-equivalence.

Let $\mathcal{C}$ be a small dg category. For a full dg subcategory
$\mathcal{D}$, we denote by $\mathcal{C}/\mathcal{D}$ the corresponding dg
quotient. For an explicit construction of $\mathcal{C}/\mathcal{D}$, we refer
to [30, Subsection 3.1]; compare [39, Section 4]. Denote by
$q\colon\mathcal{C}\rightarrow\mathcal{C}/\mathcal{D}$ the quotient functor,
which acts on objects by the identity.

Strictly speaking, $q\colon\mathcal{C}\rightarrow\mathcal{C}/\mathcal{D}$ is a
dg quasi-functor, which is not necessarily a genuine dg functor; see [30,
Subsections 3.5 and 3.6]. In other words, to study dg quotient categories, one
has to work in the homotopy category $\mathbf{Hodgcat}$.

The following universal property of $q$ is due to [30, Theorem 1.6.2(ii)];
compare [44, Theorem 4.8].

######  Lemma 5.7.

Assume that $F\colon\mathcal{C}\rightarrow\mathcal{C}^{\prime}$ is a dg
functor such that $F(D)$ is contractible for any object $D$ in $\mathcal{D}$.
Then there is a unique morphism
$\overline{F}\colon\mathcal{C}/\mathcal{D}\rightarrow\mathcal{C}^{\prime}$ in
$\mathbf{Hodgcat}$ satisfying $F=\overline{F}\circ q$.

The following fundamental fact will be used frequently; see [30, Theorem 3.4]
and [56, Theorem 1.3(i) and Lemma 1.5].

######  Lemma 5.8.

Assume that both $\mathcal{C}$ and $\mathcal{D}$ are pretriangulated. Then
$\mathcal{C}/\mathcal{D}$ is pretriangulated. Moreover, the quotient functor
$q$ induces an isomorphism of triangulated categories

$H^{0}(\mathcal{C})/H^{0}(\mathcal{D})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{0}(\mathcal{C}/\mathcal{D}),$

where $H^{0}(\mathcal{C})/H^{0}(\mathcal{D})$ denotes the corresponding
Verdier quotient.

###  5.3. The perfect dg derived categories

Recall that an additive category $\mathfrak{a}$ is _idempotent-split_ provided
that each idempotent morphism $e\colon X\rightarrow X$ admits a factorization
$X\stackrel{{\scriptstyle u}}{{\rightarrow}}Y\stackrel{{\scriptstyle
v}}{{\rightarrow}}X$ satisfying $u\circ v={\rm Id}_{Y}$.

Let $\mathcal{T}$ be a triangulated category. For any set $S$ of objects, we
denote by ${\rm thick}\langle S\rangle$ its _thick hull_ , that is, the
smallest triangulated subcategory of $\mathcal{T}$ containing $S$ and closed
under direct summands. An object $X$ is said to be a _generator_ of
$\mathcal{T}$, provided that $\mathcal{T}={\rm thick}\langle X\rangle$.

Let $A$ be a dg algebra. We will always view $A$ as a dg category with a
single object. Then we have the dg category $A\mbox{-DGMod}$ of left dg
$A$-modules and the homotpy category $\mathbf{K}(A)=H^{0}(A\mbox{-DGMod})$.
The thick hull ${\rm thick}\langle A\rangle$ of $A$ is usually denoted by
$\mathbf{per}(A)$, whose objects are called _perfect modules_. Denote by
$\mathbf{per}_{\rm dg}(A)$ the full dg subcategory of $A\mbox{-DGMod}$ formed
by perfect $A$-modules, called the _perfect dg derived category_ of $A$.

The following result is implicitly contained in [37, Subsection 4.2]. Recall
that $A^{\rm op}$ denotes the opposite dg algebra of $A$.

######  Proposition 5.9.

Let $\mathcal{C}$ be a pretriangulated dg category. Assume that
$H^{0}(\mathcal{C})$ is idempotent-split and that $X$ is a generator of
$H^{0}(\mathcal{C})$. Then there is a quasi-equivalence

$\mathcal{C}(X,-)\colon\mathcal{C}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbf{per}_{\rm
dg}(\mathcal{C}(X,X)^{\rm op}).$

###### Proof.

Write $A=\mathcal{C}(X,X)^{\rm op}$ and
$F=\mathcal{C}(X,-)\colon\mathcal{C}\rightarrow A\mbox{-DGMod}$. We observe
that $F(X)=A$.

Consider the triangle functor $H^{0}(F)\colon
H^{0}(\mathcal{C})\rightarrow\mathbf{K}(A)$. Since $X$ generates
$H^{0}(\mathcal{C})$, we infer that the essential image of $H^{0}(F)$ lies in
$\mathbf{per}(A)$. By the Yoneda embedding, $F$ induces a quasi-isomorphism

$\mathcal{C}(X,X)\longrightarrow A\mbox{-DGMod}(F(X),F(X)).$

The above complexes compute $H^{0}(\mathcal{C})(X,\Sigma^{n}(X))$ and ${\rm
Hom}_{\mathbf{K}(A)}(F(X),\Sigma^{n}F(X))$, respectively. We conclude that
$H^{0}(F)$ induces an isomorphism

$H^{0}(\mathcal{C})(X,\Sigma^{n}(X))\simeq{\rm
Hom}_{\mathbf{K}(A)}(F(X),\Sigma^{n}F(X)),\quad\text{for each
$n\in\mathbb{Z}$}.$

Since $X$ generates $H^{0}(\mathcal{C})$, we infer from [9, Lemma 1] that
$H^{0}(F)$ is fully faithful. Since $H^{0}(\mathcal{C})$ is idempotent-split,
we infer that

$H^{0}(F)\colon H^{0}(\mathcal{C})\longrightarrow\mathbf{per}(A)$

is a triangle equivalence. Then the required quasi-equivalence follows
immediately from Lemma 5.6. ∎

Let $\mathcal{T}$ be a triangulated category with arbitrary coproducts. An
object $X$ is _compact_ if ${\rm Hom}_{\mathcal{T}}(X,-)$ commutes with
arbitrary coproducts. Denote by $\mathcal{T}^{c}$ the full subcategory formed
by compact objects; it is a thick triangulated subcategory. In particular,
$\mathcal{T}^{c}$ is always idempotent-split. The triangulated category
$\mathcal{T}$ is _compactly generated_ , provided that there is a set
$\mathcal{S}$ of compact objects such that each nonzero object $X$ satisfies
${\rm Hom}_{\mathcal{T}}(\Sigma^{i}(S),X)\neq 0$ for some $S\in\mathcal{S}$
and $i\in\mathbb{Z}$. As a typical example, the derived category
$\mathbf{D}(\mathcal{C})$ of dg modules over a small dg category $\mathcal{C}$
is compactly generated; moreover, we have

(5.2) $\displaystyle\mathbf{D}(\mathcal{C})^{c}={\rm
thick}\langle\mathcal{C}(X,-)\mid X\in{\rm Obj}(\mathcal{C})\rangle.$

For details, we refer to [37, Subsection 5.3].

For each dg algebra $A$, we have an inclusion
$A\hookrightarrow\mathbf{per}_{\rm dg}(A)^{\rm op}$ of dg categories, which
sends the unique object to the dg $A$-module $A$ itself. We have the
restriction

${\rm res}\colon\mathbf{D}(\mathbf{per}_{\rm dg}(A)^{\rm
op})\longrightarrow\mathbf{D}(A),\quad M\mapsto M(A)$

along the above inclusion.

The following result is a special case of [37, Theorem 8.1]. We sketch a proof
for the convenience of the reader.

######  Lemma 5.10.

The above restriction functor is a triangle equivalence.

###### Proof.

Write $\mathcal{C}=\mathbf{per}_{\rm dg}(A)$. As $\mathcal{C}$ is exact and
$H^{0}(\mathcal{C})$ is idempotent-split, the Yoneda embedding
$\mathbf{Y}_{\mathcal{C}}$ allows us to identify $H^{0}(\mathcal{C})$ with
$\mathbf{D}(\mathcal{C}^{\rm op})^{c}$; see (5.2).

The restriction functor ‘$\rm res$’ preserves infinite coproducts. Therefore,
it suffices to prove that it preserves compact objects and restricts to an
equivalence between the full subcategories formed by compact objects.

Since ‘${\rm res}$’ sends a representable functor $\mathcal{C}(-,P)$ to
$\mathcal{C}(A,P)=P$, it follows that it preserves compact objects. Moreover,
the following composition is the identity functor.

$H^{0}(\mathcal{C})\xrightarrow{\sim}\mathbf{D}(\mathcal{C}^{\rm
op})^{c}\xrightarrow{\rm
res}\mathbf{D}(A)^{c}\xrightarrow{\sim}\mathbf{per}(A)=H^{0}(\mathcal{C})$

We infer that ‘${\rm res}$’ restricts to an equivalence between
$\mathbf{D}(\mathcal{C}^{\rm op})^{c}$ and $\mathbf{D}(A)^{c}$. ∎

##  6\. An explicit dg localization

We introduce an explicit dg localization. Throughout this section, we fix a
triple $(\mathcal{C},\Omega,\theta)$. Here, $\mathcal{C}$ is a dg category,
$\Omega\colon\mathcal{C}\rightarrow\mathcal{C}$ is a dg endofunctor, and
$\theta\colon{\rm Id}_{\mathcal{C}}\rightarrow\Omega$ is a closed natural
transformation of degree zero satisfying $\theta\Omega=\Omega\theta$.

In the setup, for any object $X$, we have $d_{\mathcal{C}}(\theta_{X})=0$,
$|\theta_{X}|=0$ and $\theta_{\Omega
X}=\Omega(\theta_{X})\in{\mathcal{C}}(\Omega X,\Omega^{2}(X))$. Moreover,
$\theta_{X}$ is natural in $X$. Here, for each $p\geq 1$, we denote by
$\Omega^{p}$ the $p$-th iterated composition of $\Omega$. Set $\Omega^{0}={\rm
Id}_{\mathcal{C}}$.

We will define a new dg category $\mathcal{SC}$ as follows: its objects are
the same as $\mathcal{C}$ and the Hom complexes are given by

(6.1) $\displaystyle\mathcal{SC}(X,Y)=\varinjlim_{p\geq
0}\;\mathcal{C}(X,\Omega^{p}(Y)),$

where $\mathcal{C}(X,\Omega^{p}(Y))\rightarrow\mathcal{C}(X,\Omega^{p+1}(Y))$
sends $f$ to $\theta_{\Omega^{p}(Y)}\circ f$. For each
$f\in\mathcal{C}(X,\Omega^{p}(Y))$, its image in $\mathcal{SC}(X,Y)$ is
denoted by $[f;p]$. Therefore, we have

(6.2) $\displaystyle[f;p]=[\theta_{\Omega^{p}(Y)}\circ f;p+1].$

The degree of $[f;p]$ equals the one of $f$. The differential of
$\mathcal{SC}(X,Y)$ is given such that $d([f;p])=[d_{\mathcal{C}}(f);p]$. For
two morphisms $[f;p]\colon X\rightarrow Y$ and $[g;q]\colon Y\rightarrow Z$,
their composition is given by

$[g;q]\circ[f;p]:=[\Omega^{p}(g)\circ f;p+q].$

One verifies that the composition is well defined, and that $\mathcal{SC}$ is
a dg category. In particular, $[{\rm Id}_{X};0]$ is the identity of $X$ in
$\mathcal{SC}$. We mention that the above construction resembles the one in
[43, Subsection 5.1].

######  Lemma 6.1.

Keep the notation as above. Then the following statements hold.

  1. (1)

For each object $X$, the morphism $[\theta_{X};0]\colon X\rightarrow\Omega(X)$
is a dg-isomorphism in $\mathcal{SC}$ with $[\theta_{X};0]^{-1}=[{\rm
Id}_{\Omega(X)};1]$.

  2. (2)

For any $p\geq 0$ and morphism $f\colon X\rightarrow\Omega^{p}(Y)$ in
$\mathcal{C}$, we have

$[f;p]=[\theta_{Y};0]^{-1}\circ[\theta_{\Omega(Y)};0]^{-1}\circ\cdots\circ[\theta_{\Omega^{p-1}(Y)};0]^{-1}\circ[f;0].$

  3. (3)

An object $X$ is contractible in $\mathcal{SC}$ if and only if the morphism
$\theta_{\Omega^{n}(X)}\circ\cdots\circ\theta_{\Omega(X)}\circ\theta_{X}\in\mathcal{C}(X,\Omega^{n+1}(X))$
is a coboundary for some $n$.

###### Proof.

(1) follows from the following direct computations:

$[{\rm Id}_{\Omega(X)};1]\circ[\theta_{X};0]=[\theta_{X};1]=[{\rm Id}_{X};0],$

and

$[\theta_{X};0]\circ[{\rm
Id}_{\Omega(X)};1]=[\Omega(\theta_{X});1]=[\theta_{\Omega(X)};1]=[{\rm
Id}_{\Omega(X)};0].$

Here, in both identities we use (6.2); moreover, in the second identity, we
use the assumption $\Omega\theta=\theta\Omega$.

(2) By (1), the right hand side of the required identity equals

$[{\rm Id}_{\Omega(Y)};1]\circ[{\rm
Id}_{\Omega^{2}(Y)};1]\circ\cdots\circ[{\rm Id}_{\Omega^{p}(Y)};1]\circ[f;0].$

This composition equals $[f;p]$.

(3) Assume that $X$ is contractible in $\mathcal{SC}$, that is, there is a
morphism $[f;p]$ of degree $-1$ such that

$[{\rm Id}_{X};0]=d([f;p])=[d_{\mathcal{C}}(f);p],$

where $f\colon X\rightarrow\Omega^{p}(X)$ is of degree $-1$. From the colimit
construction (6.1), the identity $[{\rm Id}_{X};0]=[d_{\mathcal{C}}(f);p]$
means that there is a sufficiently large $n$ such that the following identity
holds in $\mathcal{C}$

$\displaystyle\theta_{\Omega^{n}(X)}\circ\cdots\circ\theta_{\Omega(X)}\circ\theta_{X}\circ{\rm
Id}_{X}$
$\displaystyle=\theta_{\Omega^{n}(X)}\circ\cdots\circ\theta_{\Omega^{p}(X)}\circ
d_{\mathcal{C}}(f)$
$\displaystyle=d_{\mathcal{C}}(\theta_{\Omega^{n}(X)}\circ\cdots\circ\theta_{\Omega^{p}(X)}\circ
f).$

Here, the second equality uses the assumption that $\theta$ is closed and of
degree zero. This implies the “only if” part. Similarly, we may prove the “if”
part. ∎

There is a canonical dg functor

$\iota\colon\mathcal{C}\longrightarrow\mathcal{SC}$

given by $\iota(X)=X$ and $\iota(f)=[f;0]$. By Lemma 6.1(1), each morphism
$\iota(\theta_{X})$ is a dg-isomorphism. By the following universal property,
we might call $\iota$ a (strict) _dg localization_ of $\mathcal{C}$ along
$\theta$; compare [68, Subsection 8.2] and [45, Subsection 3.9].

######  Proposition 6.2.

Let $F\colon\mathcal{C}\rightarrow\mathcal{D}$ be a dg functor such that for
each object $X$ in $\mathcal{C}$, $F(\theta_{X})$ is a dg-isomorphism in
$\mathcal{D}$. Then there is a unique dg functor
$F^{\prime}\colon\mathcal{SC}\rightarrow\mathcal{D}$ satisfying
$F=F^{\prime}\circ\iota$.

###### Proof.

Let us first prove the uniqueness. By $F=F^{\prime}\circ\iota$, we have that
$F^{\prime}(X)=F(X)$ and $F^{\prime}([f;0])=F(f)$ for any object $X$ and
morphism $f$ in $\mathcal{C}$. For a general morphism $[f;p]$ in
$\mathcal{SC}$, we apply Lemma 6.1(2) to deduce

(6.3) $\displaystyle F^{\prime}([f;p])=F(\theta_{Y})^{-1}\circ
F(\theta_{\Omega(Y)})^{-1}\circ\cdots\circ
F(\theta_{\Omega^{p-1}(Y)})^{-1}\circ F(f).$

This identity implies the uniqueness of $F^{\prime}$.

To construct such a dg functor, we set $F^{\prime}(X)=F(X)$ and use (6.3) to
define the action of $F^{\prime}$ on morphisms. It is routine to verify that
$F^{\prime}$ is a well-defined dg functor and is the required one. ∎

######  Lemma 6.3.

Let $\iota\colon\mathcal{C}\rightarrow\mathcal{SC}$ be as above. If
$\mathcal{C}$ is exact (resp. pretriangulated), then so is $\mathcal{SC}$.

###### Proof.

(1) Assume first that $\mathcal{C}$ is exact. To show that $\mathcal{SC}$ is
exact, it suffices to verify the two conditions in Lemma 5.4. The first
condition is clear, as $\mathcal{C}$ satisfies the same condition.

Let us verify the second condition. We first observe that $\iota(f)=[f;0]$ has
an internal cone, which is given by the internal cone of $f$ in $\mathcal{C}$.
For a general morphism $[f;p]$, we just combine Lemma 6.1(2) with the
following general fact: given a dg-isomorphism $h\colon X\rightarrow Y$ in a
dg category $\mathcal{D}$, a closed morphism $g\colon X^{\prime}\rightarrow X$
of degree zero has an internal cone if and only if so does $h\circ g$.

(2) Assume that $\mathcal{C}$ is pretriangulated. Consider its exact hull
${\rm can}_{\mathcal{C}}\colon\mathcal{C}\rightarrow\mathcal{C}^{\rm ex}$. By
the universal property (5.1) of the exact hull, the dg endofunctor $\Omega$
extends uniquely to a dg endofunctor $\Omega^{\rm ex}$ on $\mathcal{C}^{\rm
ex}$; moreover, $\theta\colon{\rm Id}_{\mathcal{C}}\rightarrow\Omega$ extends
to $\theta^{\rm ex}\colon{\rm Id}_{\mathcal{C}^{\rm ex}}\rightarrow\Omega^{\rm
ex}$, which is also closed of degree zero. Therefore, we can form the dg
localization $\iota^{\rm ex}\colon\mathcal{C}^{\rm
ex}\rightarrow\mathcal{S}(\mathcal{C}^{\rm ex})$. We have the following
commutative diagram.

$\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{{\rm
can}_{\mathcal{C}}}$$\textstyle{\mathcal{C}^{\rm
ex}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\rm
ex}}$$\textstyle{\mathcal{SC}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{S}{\rm
can}_{\mathcal{C}}}$$\textstyle{\mathcal{S}(\mathcal{C}^{\rm ex})}$

The bottom dg functor is induced from ${\rm can}_{\mathcal{C}}$ and thus is
also fully faithful. Since $\mathcal{C}$ is pretriangulated, $H^{0}({\rm
can}_{\mathcal{C}})$ is an equivalence. Applying $H^{0}$ to the commutative
diagram, we infer that $H^{0}(\mathcal{S}{\rm can}_{\mathcal{C}})$ is dense.
By (1), we know that $\mathcal{S}(\mathcal{C}^{\rm ex})$ is exact. Applying
Lemma 5.5(2) to $\mathcal{S}{\rm can}_{\mathcal{C}}$, we infer that
$\mathcal{SC}$ is pretriangulated. ∎

We assume now that the dg category $\mathcal{C}$ is pretriangulated. For each
object $X$, we denote by ${\rm Cone}(\theta_{X})$ the cone of the image of
$\theta_{X}$ in $H^{0}(\mathcal{C})$. In other words, ${\rm Cone}(\theta_{X})$
is determined by the following exact triangle in $H^{0}(\mathcal{C})$.

(6.4) $\displaystyle
X\stackrel{{\scriptstyle\theta_{X}}}{{\longrightarrow}}\Omega(X)\longrightarrow{\rm
Cone}(\theta_{X})\longrightarrow\Sigma(X)$

Here, we confuse $\theta_{X}$ with its image in $H^{0}(\mathcal{C})$. Denote
by

${\rm thick}\langle{\rm Cone}(\theta_{X})\;|\;X\in{\rm
Obj}(\mathcal{C})\rangle$

the thick hull of these cones in $H^{0}(\mathcal{C})$. The full dg subcategory
of $\mathcal{C}$ formed by the objects in ${\rm thick}\langle{\rm
Cone}(\theta_{X})\;|\;X\in{\rm Obj}(\mathcal{C})\rangle$ is denoted by
$\mathcal{N}$.

The following general result shows that the dg localization is quasi-
equivalent to a dg quotient. A similar idea appears implicitly in [43, Section
7, the proof of Theorem 2], which relates the dg orbit category in [43,
Subsection 5.1] to a dg quotient. The precise relationship between the
following result and the mentioned one in [43] will be explored elsewhere.

######  Theorem 6.4.

Let $\mathcal{C}$ be a pretriangulated dg category, and let
$\iota\colon\mathcal{C}\rightarrow\mathcal{SC}$ be the dg localization along
$\theta$. Then $\iota$ induces an isomorphism in $\mathbf{Hodgcat}$

$\mathcal{C}/\mathcal{N}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{SC}$

which yields an isomorphism of triangulated categories

$H^{0}(\mathcal{C})/{{\rm thick}\langle{\rm Cone}(\theta_{X})\;|\;X\in{\rm
Obj}(\mathcal{C})\rangle}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{0}(\mathcal{SC}).$

###### Proof.

Recall that $\iota(\theta_{X})$ is a dg-isomorphism in $\mathcal{SC}$.
Therefore, its image in $H^{0}(\mathcal{SC})$ is an isomorphism. Applying
$H^{0}(\iota)$ to (6.4), we infer that ${\rm Cone}(\theta_{X})$ is annihilated
by $H^{0}(\iota)$, that is, $\iota({\rm Cone}(\theta_{X}))$ is contractible in
$\mathcal{SC}$. It follows that $\iota$ sends any object in $\mathcal{N}$ to a
contractible object in $\mathcal{SC}$. Therefore, by the universal property in
Lemma 5.7, $\iota$ induces a morphism

$\overline{\iota}\colon\mathcal{C}/\mathcal{N}\longrightarrow\mathcal{SC}$

in $\mathbf{Hodgcat}$. Moreover, it induces a triangle functor

$\Phi:=H^{0}(\overline{\iota})\colon
H^{0}(\mathcal{C})/{H^{0}(\mathcal{N})}\longrightarrow H^{0}(\mathcal{SC}).$

We claim that the induced triangle functor $\Phi$ is fully faithful. As $\Phi$
acts on objects by the identity and thus it is dense, this claim implies that
$\Phi$ is an isomorphism of triangulated categories. By Lemma 5.6,
$\overline{\iota}$ is an isomorphism in $\mathbf{Hodgcat}$.

Let us prove the above claim. We observe that $\theta_{X}$ becomes an
isomorphism in $H^{0}(\mathcal{C})/{H^{0}(\mathcal{N})}$. Therefore, for any
object $X$, $\iota(\theta_{X})^{-1}\in H^{0}(\mathcal{SC})$ belongs to the
image of $\Phi$. By Lemma 6.1(2), any general morphism $[f;p]\colon
X\rightarrow Y$ in $H^{0}(\mathcal{SC})$ is of the following form

$[f;p]=\iota(\theta_{Y})^{-1}\circ\iota(\theta_{\Omega(Y)})^{-1}\circ\cdots\circ\iota(\theta_{\Omega^{p-1}(Y)})^{-1}\circ\iota(f).$

It follows that $[f;p]$ necessarily lies in the image of $\Phi$, that is,
$\Phi$ is full.

We assume that $\Phi(X)\simeq 0$, that is, $X$ is contractible in
$\mathcal{SC}$. By Lemma 6.1(3), the morphism
$\theta_{\Omega^{n}(X)}\circ\cdots\circ\theta_{\Omega(X)}\circ\theta_{X}$ is a
coboundary. Consider the following exact triangle in $H^{0}(\mathcal{C})$.

$X\xrightarrow{\theta_{\Omega^{n}(X)}\circ\cdots\circ\theta_{\Omega(X)}\circ\theta_{X}}\Omega^{n+1}(X)\longrightarrow
C\longrightarrow\Sigma(X)$

It follows that $\Sigma(X)$ is isomorphic to a direct summand of $C$ in
$H^{0}(\mathcal{C})$. On the other hand, as the cone of a composite morphism,
$C$ is an iterated extension of ${\rm Cone}(\theta_{\Omega^{i}(X)})$ for
$0\leq i\leq n$. We conclude that $C$ and thus $X$ lie in $H^{0}(\mathcal{N})$
and are isomorphic to zero in $H^{0}(\mathcal{C})/H^{0}(\mathcal{N})$. This
proves that $\Phi$ is faithful on objects. The claim follows from the
following general fact in [61, the proof of Theorem 3.5, p.446]: a full
triangle functor which is faithful on objects is necessarily faithful. ∎

##  7\. The Yoneda dg category

We introduce, using the bar resolution, the Yoneda dg category that is a
natural dg enhancement of the derived category. We prove that the dg tensor
algebra studied in Section 3 is isomorphic to the endomorphism algebra of a
specific object in the Yoneda dg category; see Proposition 7.6. Throughout, we
work in the relative situation.

###  7.1. The bar and Yoneda dg categories

Let $E\rightarrow\Lambda$ be an algebra homomorphism. Its cokernel is denoted
by $\overline{\Lambda}$, which has a natural $E$-$E$-bimodule structure. We
denote by $s\overline{\Lambda}$ the graded $E$-$E$-bimodule concentrated in
degree $-1$. Its element is usually written as $s\overline{a}$.

The normalized $E$-relative bar resolution $\mathbb{B}$ of $\Lambda$ is a
complex of $\Lambda$-$\Lambda$-bimodules given as follows. As a graded
$\Lambda$-$\Lambda$-bimodule, we have

$\mathbb{B}=\Lambda\otimes_{E}T_{E}(s\overline{\Lambda})\otimes_{E}\Lambda,$

where ${\rm deg}(a_{0}\otimes_{E}s\overline{a}_{1,n}\otimes_{E}a_{n+1})=-n$.
Here, for simplicity, we write

$s\overline{a}_{1,n}:=s\overline{a}_{1}\otimes_{E}s\overline{a}_{2}\otimes_{E}\cdots\otimes_{E}s\overline{a}_{n}.$

The differential $d$ is given such that $d(a_{0}\otimes_{E}a_{1})=0$ and that

$\displaystyle d(a_{0}\otimes_{E}s\overline{a}_{1,n}\otimes_{E}a_{n+1})={}$
$\displaystyle
a_{0}a_{1}\otimes_{E}s\overline{a}_{2,n}\otimes_{E}a_{n+1}+(-1)^{n}a_{0}\otimes_{E}s\overline{a}_{1,n-1}\otimes_{E}a_{n}a_{n+1}$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{i}a_{0}\otimes_{E}s\overline{a}_{1,i-1}\otimes_{E}s\overline{a_{i}a_{i+1}}\otimes_{E}s\overline{a}_{i+2,n}\otimes_{E}a_{n+1}.$

Here and later, as usual, $s\overline{a}_{1,0}$ and $s\overline{a}_{n+1,n}$
are understood to be the empty word and should be ignored.

It is well known that $\mathbb{B}$ is a coalgebra in the monoidal category of
complexes of $\Lambda$-$\Lambda$-bimodules. To be more precise, we have a
cochain map between complexes of $\Lambda$-$\Lambda$-bimodules

$\Delta\colon\mathbb{B}\longrightarrow\mathbb{B}\otimes_{\Lambda}\mathbb{B}$

given by

$\Delta(a_{0}\otimes_{E}s\overline{a}_{1,n}\otimes_{E}a_{n+1})=\sum_{i=0}^{n}(a_{0}\otimes_{E}s\overline{a}_{1,i}\otimes_{E}1_{\Lambda})\otimes_{\Lambda}(1_{\Lambda}\otimes_{E}s\overline{a}_{i+1,n}\otimes_{E}a_{n+1}).$

The natural cochain map $\varepsilon\colon\mathbb{B}\rightarrow\Lambda$ is
induced by the multiplication of $\Lambda$. We have the following
coassociative property

$(\Delta\otimes_{\Lambda}{\rm Id}_{\mathbb{B}})\circ\Delta=({\rm
Id}_{\mathbb{B}}\otimes_{\Lambda}\Delta)\circ\Delta$

and the counital property

$(\varepsilon\otimes_{\Lambda}{\rm Id}_{\mathbb{B}})\circ\Delta={\rm
Id}_{\mathbb{B}}=({\rm
Id}_{\mathbb{B}}\otimes_{\Lambda}\varepsilon)\circ\Delta.$

Following the treatment in [37, Subsection 6.6], we define the _$E$ -relative
bar dg category_ $\mathcal{B}=\mathcal{B}_{\Lambda/E}$ as follows. The objects
are precisely all the complexes of $\Lambda$-modules, and the Hom complex
between two objects $X$ and $Y$ is given by

$\mathcal{B}(X,Y)={\rm Hom}_{\Lambda}(\mathbb{B}\otimes_{\Lambda}X,Y).$

The composition of two morphisms $f\in\mathcal{B}(X,Y)$ and
$g\in\mathcal{B}(Y,Z)$ is defined to be

$g\ast
f:=(\mathbb{B}\otimes_{\Lambda}X\xrightarrow{\Delta\otimes_{\Lambda}{\rm
Id}_{X}}\mathbb{B}\otimes_{\Lambda}\mathbb{B}\otimes_{\Lambda}X\xrightarrow{{\rm
Id}_{\mathbb{B}}\otimes_{\Lambda}f}\mathbb{B}\otimes_{\Lambda}Y\xrightarrow{g}Z).$

Moreover, the identity endomorphism in $\mathcal{B}(X,X)$ is given by

$\mathbb{B}\otimes_{\Lambda}X\xrightarrow{\varepsilon\otimes_{\Lambda}{\rm
Id}_{X}}\Lambda\otimes_{\Lambda}X=X.$

We mention that the bar dg category might be viewed as the coKleisli category
of the comonad $\mathbb{B}\otimes_{\Lambda}-$ on the dg category $C_{\rm
dg}(\Lambda\mbox{-Mod})$ of complexes of $\Lambda$-modules; see [49,
Definition 4.8(ii)].

We will unpack the above definition of $\mathcal{B}$ and obtain its
alternative form. The _$E$ -relative Yoneda dg category_
$\mathcal{Y}=\mathcal{Y}_{\Lambda/E}$ has the same objects as $\mathcal{B}$.
For two complexes $X$ and $Y$ of $\Lambda$-modules, the underlying graded
$\mathbb{K}$-module of the Hom complex $\mathcal{Y}(X,Y)$ is given by an
infinite product

$\mathcal{Y}(X,Y)=\prod_{n\geq 0}{\rm
Hom}_{E}((s\overline{\Lambda})^{\otimes_{E}n}\otimes_{E}X,Y).$

We denote

$\mathcal{Y}_{n}(X,Y):={\rm
Hom}_{E}((s\overline{\Lambda})^{\otimes_{E}n}\otimes_{E}X,Y),$

and say that elements in $\mathcal{Y}_{n}(X,Y)$ are of _filtration-degree_
$n$. Observe that $\mathcal{Y}_{0}(X,Y)={\rm Hom}_{E}(X,Y)$. The differential
$\delta$ of $\mathcal{Y}(X,Y)$ is determined by

$\begin{pmatrix}\delta_{\rm in}\\\ \delta_{\rm
ex}\end{pmatrix}\colon\mathcal{Y}_{n}(X,Y)\longrightarrow\mathcal{Y}_{n}(X,Y)\oplus\mathcal{Y}_{n+1}(X,Y),$

where

$\delta_{\rm
in}(f)(s\overline{a}_{1,n}\otimes_{E}x)=d_{Y}(f(s\overline{a}_{1,n}\otimes_{E}x))-(-1)^{|f|+n}f(s\overline{a}_{1,n}\otimes_{E}d_{X}(x))$

and

$\displaystyle\delta_{\rm ex}(f)(s\overline{a}_{1,n+1}\otimes_{E}x)=$
$\displaystyle(-1)^{|f|+1}a_{1}f(s\overline{a}_{2,n+1}\otimes_{E}x)+(-1)^{|f|+n}f(s\overline{a}_{1,n}\otimes_{E}a_{n+1}x)$
$\displaystyle+\sum_{i=1}^{n}(-1)^{|f|+i+1}f(s\overline{a}_{1,i-1}\otimes_{E}s\overline{a_{i}a_{i+1}}\otimes_{E}s\overline{a}_{i+2,n+1}\otimes_{E}x).$

The composition $\odot$ of morphisms is defined as follows: for
$f\in\mathcal{Y}_{n}(X,Y)$ and $g\in\mathcal{Y}_{m}(Y,Z)$, their composition
$g\odot f\in\mathcal{Y}_{n+m}(X,Z)$ is given such that

(7.1) $(g\odot
f)(s\overline{a}_{1,m+n}\otimes_{E}x)=(-1)^{m|f|}g(s\overline{a}_{1,m}\otimes_{E}f(s\overline{a}_{m+1,m+n}\otimes_{E}x)).$

The identity endomorphism in $\mathcal{Y}(X,X)$ is given by the genuine
identity map ${\rm Id}_{X}\in\mathcal{Y}_{0}(X,X)$.

######  Lemma 7.1.

There is an isomorphism $\mathcal{B}\simeq\mathcal{Y}$ of dg categories.

###### Proof.

We observe that $\mathbb{B}\otimes_{\Lambda}X$ is canonically isomorphic to

$\bigoplus_{n\geq
0}\Lambda\otimes_{E}(s\overline{\Lambda})^{\otimes_{E}n}\otimes_{E}X.$

Therefore, we have

(7.2) $\displaystyle\mathcal{B}(X,Y)\simeq\prod_{n\geq 0}{\rm
Hom}_{\Lambda}(\Lambda\otimes_{E}(s\overline{\Lambda})^{\otimes_{E}n}\otimes_{E}X,Y)\simeq\prod_{n\geq
0}{\rm
Hom}_{E}((s\overline{\Lambda})^{\otimes_{E}n}\otimes_{E}X,Y)=\mathcal{Y}(X,Y).$

The above isomorphism identifies $f\in\mathcal{Y}_{n}(X,Y)$ with
$\tilde{f}\colon\Lambda\otimes_{E}(s\overline{\Lambda})^{\otimes_{E}n}\otimes_{E}X\rightarrow
Y$ given by

$\tilde{f}(a_{0}\otimes_{E}s\overline{a}_{1,n}\otimes_{E}x)=a_{0}f(s\overline{a}_{1,n}\otimes_{E}x).$

The two dg categories $\mathcal{B}$ and $\mathcal{Y}$ have the same objects.
It is routine to verify that the above isomorphism of the Hom complexes
induces the required isomorphism of dg categories. ∎

###  7.2. The dg derived categories

Recall from Example 5.1 that $C_{\rm dg}(\Lambda\mbox{-Mod})$ denotes the dg
category of complexes of $\Lambda$-modules. A complex $X$ of $\Lambda$-modules
is called _$E$ -relatively acyclic_ if it is contractible as a complex of
$E$-modules, or equivalently, $X\simeq 0$ in $\mathbf{K}(E\mbox{-Mod})$; see
[38, Subsection 7.4]. In particular, an $E$-relatively acyclic complex is
acyclic. Those complexes form a full dg subcategory $C_{\rm dg}^{\rm
rel\mbox{-}ac}(\Lambda\mbox{-Mod})$. The corresponding dg quotient

$\mathbf{D}_{\rm dg}(\Lambda/E)=C_{\rm dg}(\Lambda\mbox{-Mod})/{C_{\rm
dg}^{\rm rel\mbox{-}ac}(\Lambda\mbox{-Mod})}$

is called the _$E$ -relative dg derived category_ of $\Lambda$. By Lemma 5.8,
its homotopy category $H^{0}(\mathbf{D}_{\rm dg}(\Lambda/E))$ is isomorphic to
the $E$-relative derived category

$\mathbf{D}(\Lambda/E)=\mathbf{K}(\Lambda\mbox{-Mod})/{\mathbf{K}^{\rm
rel\mbox{-}ac}(\Lambda\mbox{-Mod})}.$

A cochain map $f\colon X\rightarrow Y$ between complexes of $\Lambda$-modules
is said to be an _$E$ -relative quasi-isomorphism_ if its mapping cone is
$E$-relatively acyclic, i.e. $\mathrm{Cone}(f)\simeq 0$ in
$\mathbf{K}(E\mbox{-Mod})$.

Recall that a $\Lambda$-module $N$ is $E$-relatively projective if it is a
direct summand of $\Lambda\otimes_{E}V$ for some $E$-module $V$. A complex $P$
of $\Lambda$-modules is called _$E$ -relatively dg-projective_ provided that
each component $P^{i}$ is $E$-relatively projective and the Hom complex ${\rm
Hom}_{\Lambda}(P,X)$ is acyclic for any $E$-relatively acyclic complex $X$.

The following facts are standard.

######  Lemma 7.2.

For any complex $X$ of $\Lambda$-modules, the following statements hold.

  1. (1)

The complex $\mathbb{B}\otimes_{\Lambda}X$ of $\Lambda$-modules is
$E$-relatively dg-projective.

  2. (2)

The natural surjection $\varepsilon\otimes_{\Lambda}{\rm
Id}_{X}\colon\mathbb{B}\otimes_{\Lambda}X\twoheadrightarrow\Lambda\otimes_{\Lambda}X=X$
is an $E$-relative quasi-isomorphism.

  3. (3)

If $X$ is $E$-relatively acyclic, then $\mathbb{B}\otimes_{\Lambda}X$ is
contractible.

  4. (4)

If $X$ is $E$-relatively dg-projective, then $\varepsilon\otimes_{\Lambda}{\rm
Id}_{X}$ is a homotopy equivalence.

###### Proof.

For (1), we observe that $\mathbb{B}\otimes_{\Lambda}X$ has an ascending
filtration with factors

$\Lambda\otimes_{E}(s\overline{\Lambda})^{\otimes_{E}n}\otimes_{E}X,$

each of which is $E$-relatively dg-projective. By [38, Proposition 7.5], we
infer that $\mathbb{B}\otimes_{E}X$ is $E$-relatively dg-projective.

For (2), we recall the standard fact that
$\varepsilon\colon\mathbb{B}\rightarrow\Lambda$ is a homotopy equivalence as a
cochain map between complexes of $E$-$\Lambda$-bimodules; see for example [58,
Chapter IX, Theorem 8.1]. That is, ${\rm Cone}(\varepsilon)\simeq 0$ in
$\mathbf{K}(\text{$E\otimes\Lambda^{\rm op}$-Mod})$. This yields ${\rm
Cone}(\varepsilon\otimes_{\Lambda}{\rm Id}_{X})\simeq{\rm
Cone}(\varepsilon)\otimes_{\Lambda}{\rm Id}_{X}\simeq 0$ in
$\mathbf{K}(\text{$E$-Mod})$. We infer that $\varepsilon\otimes_{\Lambda}{\rm
Id}_{X}$ is an $E$-relative quasi-isomorphism.

For (3), it follows from (2) that $\mathbb{B}\otimes_{\Lambda}X\simeq X\simeq
0$ in $\mathbf{K}(\text{$E$-Mod})$. Then we infer (3) by the following easy
fact: any complex which is both $E$-relatively dg-projective and
$E$-relatively acyclic is necessarily contractible as a complex of
$\Lambda$-modules.

For (4), we consider the following exact triangle in
$\mathbf{K}(\Lambda\mbox{-Mod})$.

$\mathbb{B}\otimes_{\Lambda}X\xrightarrow{\varepsilon\otimes_{\Lambda}{\rm
Id}_{X}}X\longrightarrow{\rm Cone}(\varepsilon\otimes_{\Lambda}{\rm
Id}_{X})\longrightarrow\Sigma(\mathbb{B}\otimes_{\Lambda}X)$

By (1), $\mathbb{B}\otimes_{\Lambda}X$ is $E$-relatively dg-projective. Since
$X$ is $E$-relatively dg-projective, it follows that so is ${\rm
Cone}(\varepsilon\otimes_{\Lambda}{\rm Id}_{X})$. By the proof of (2), we know
that ${\rm Cone}(\varepsilon\otimes_{\Lambda}{\rm Id}_{X})$ is $E$-relatively
acyclic. Then by the above easy fact, we infer that ${\rm
Cone}(\varepsilon\otimes_{\Lambda}{\rm Id}_{X})$ is contractible as a complex
of $\Lambda$-modules. This implies (4). ∎

Consider the natural dg functor

${\rm\Theta}\colon C_{\rm
dg}(\Lambda\mbox{-Mod})\longrightarrow\mathcal{Y}_{\Lambda/E}=\mathcal{Y}$

which acts on objects by the identity, and identifies $f\in{\rm
Hom}_{\Lambda}(X,Y)$ with $f\in{\rm
Hom}_{E}(X,Y)=\mathcal{Y}_{0}(X,Y)\subseteq\mathcal{Y}(X,Y)$. Indeed, $C_{\rm
dg}(\Lambda\mbox{-Mod})$ might be viewed as a non-full dg subcategory of
$\mathcal{Y}$.

The following result justifies our terminology for $\mathcal{Y}$, since for
each $\Lambda$-module $M$, the cohomology of $\mathcal{Y}(M,M)$ is isomorphic
to the _$E$ -relative Yoneda algebra_ of $M$

${\rm Ext}^{*}_{\Lambda/E}(M,M)=\bigoplus_{i\geq 0}{\rm
Hom}_{\mathbf{D}(\Lambda/E)}(M,\Sigma^{i}(M)).$

######  Proposition 7.3.

The above dg functor $\Theta$ induces an isomorphism in $\mathbf{Hodgcat}$

$\overline{\Theta}\colon\mathbf{D}_{\rm
dg}(\Lambda/E)\simeq\mathcal{Y}_{\Lambda/E}.$

Consequently, $\mathcal{Y}_{\Lambda/E}$ is pretriangulated and we have an
isomorphism

$\mathbf{D}(\Lambda/E)\simeq H^{0}(\mathcal{Y}_{\Lambda/E})$

of triangulated categories.

###### Proof.

For any $E$-relatively acyclic complex $X$, by Lemma 7.2(3),
$\mathbb{B}\otimes_{\Lambda}X$ is contractible as a complex of
$\Lambda$-modules. Recall the isomorphism in (7.2)

$\mathcal{Y}(X,X)\simeq\mathcal{B}(X,X)={\rm
Hom}_{\Lambda}(\mathbb{B}\otimes_{\Lambda}X,X).$

It follows that $\mathcal{B}(X,X)$ and thus $\mathcal{Y}(X,X)$ are acyclic. We
infer that $X$ is contractible in $\mathcal{Y}$. This shows that $\Theta(X)=X$
is contractible for any $X$ in ${C_{\rm dg}^{\rm
rel\mbox{-}ac}(\Lambda\mbox{-Mod})}$.

By the universal property in Lemma 5.7, $\Theta$ induces a morphism in
$\mathbf{Hodgcat}$

$\overline{\Theta}\colon\mathbf{D}_{\rm
dg}(\Lambda/E)\longrightarrow\mathcal{Y}.$

As $\overline{\Theta}$ acts on objects by the identity, it suffices to prove
that it is quasi-fully faithful.

We claim that for any $E$-relatively dg-projective complex $X$, the inclusion

${\rm Hom}_{\Lambda}(X,Y)\longrightarrow\mathcal{Y}(X,Y)$

is a quasi-isomorphism. Indeed, note that the inclusion equals the composition
of

${\rm Hom}_{\Lambda}(\varepsilon\otimes_{\Lambda}{\rm Id}_{X},Y)\colon{\rm
Hom}_{\Lambda}(X,Y)\longrightarrow\mathcal{B}(X,Y)$

with the isomorphism (7.2). By Lemma 7.2(4), $\varepsilon\otimes_{\Lambda}{\rm
Id}_{X}$ is a homotopy equivalence. Therefore, ${\rm
Hom}_{\Lambda}(\varepsilon\otimes_{\Lambda}{\rm Id}_{X},Y)$ is a quasi-
isomorphism, proving the claim.

Denote by $\mathcal{P}$ the full dg subcategory of $C_{\rm
dg}(\Lambda\mbox{-Mod})$ formed by $E$-relatively dg-projective complexes. The
above claim implies that the following composition

$\mathcal{P}\stackrel{{\scriptstyle\rm inc}}{{\longrightarrow}}C_{\rm
dg}(\Lambda\mbox{-Mod})\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\mathbf{D}_{\rm
dg}(\Lambda/E)\stackrel{{\scriptstyle\overline{\Theta}}}{{\longrightarrow}}\mathcal{Y}$

is quasi-fully faithful, where ‘${\rm inc}$’ and $q$ denote the inclusion and
quotient functors, respectively. By [38, Proposition 7.4], the composite dg
functor $q\circ{\rm inc}$ is a quasi-equivalence. This implies that
$\overline{\Theta}$ is quasi-fully faithful, as required.

For the second statement, we recall that the dg derived category
$\mathbf{D}_{\rm dg}(\Lambda/E)$ is pretriangulated; see Lemma 5.8. It follows
from Lemma 5.5(1) and the isomorphism $\overline{\Theta}$ that $\mathcal{Y}$
is also pretriangulated. ∎

######  Remark 7.4.

(1) Denote by $\mathcal{Y}_{\Lambda/E}^{b}$ the full dg subcategory of
$\mathcal{Y}_{\Lambda/E}$ consisting of bounded complexes. We have the _$E$
-relative bounded dg derived category_

$\mathbf{D}_{\rm dg}^{b}(\Lambda/E)=C^{b}_{\rm dg}(\Lambda\mbox{-Mod})/{C_{\rm
dg}^{\rm rel\mbox{-}ac,b}(\Lambda\mbox{-Mod})},$

where $C^{b}_{\rm dg}(\Lambda\mbox{-Mod})$ is the full dg subcategory of
$C_{\rm dg}(\Lambda\mbox{-Mod})$ consisting of bounded complexes. Its homotopy
category $H^{0}(\mathbf{D}_{\rm dg}^{b}(\Lambda/E))$ coincides with the
$E$-relative bounded derived category

$\mathbf{D}^{b}(\Lambda/E)=\mathbf{K}^{b}(\Lambda\mbox{-Mod})/{\mathbf{K}^{{\rm
rel\mbox{-}ac},b}(\Lambda\mbox{-Mod})}.$

The bounded version of Proposition 7.3 claims an isomorphism in
$\mathbf{Hodgcat}$

$\mathbf{D}^{b}_{\rm dg}(\Lambda/E)\simeq\mathcal{Y}^{b}_{\Lambda/E}$

and an isomorphism of triangulated categories

$\mathbf{D}^{b}(\Lambda/E)\simeq H^{0}(\mathcal{Y}^{b}_{\Lambda/E}).$

(2) Assume that $E$ is a semisimple ring. Then the $E$-relative derived dg
category $\mathbf{D}_{\rm dg}(\Lambda/E)$ coincides with the (absolute) dg
derived category

$\mathbf{D}_{\rm dg}(\Lambda\mbox{-Mod})=C_{\rm
dg}(\Lambda\mbox{-Mod})/{C_{\rm dg}^{\rm ac}(\Lambda\mbox{-Mod})}.$

Similarly, $\mathbf{D}(\Lambda/E)$ coincides with
$\mathbf{D}(\Lambda\mbox{-Mod})$, the derived category of
$\Lambda\mbox{-Mod}$. Therefore, we have isomorphisms in $\mathbf{Hodgcat}$

$\mathbf{D}_{\rm
dg}(\Lambda\mbox{-Mod})\simeq\mathcal{Y}_{\Lambda/E}\quad\mbox{ and
}\quad\mathbf{D}^{b}_{\rm
dg}(\Lambda\mbox{-Mod})\simeq\mathcal{Y}^{b}_{\Lambda/E}.$

They induce isomorphisms of triangulated categories:

$\mathbf{D}(\Lambda\mbox{-Mod})\simeq
H^{0}(\mathcal{Y}_{\Lambda/E})\quad\mbox{ and
}\quad\mathbf{D}^{b}(\Lambda\mbox{-Mod})\simeq
H^{0}(\mathcal{Y}^{b}_{\Lambda/E}).$

We are mostly interested in the finitely generated modules. In the given
algebra extension $E\rightarrow\Lambda$, we assume that $E$ is a semisimple
ring and that $\Lambda$ is a left noetherian ring. Denote by
$\Lambda\mbox{-mod}$ the full subcategory of $\Lambda\mbox{-Mod}$ formed by
finitely generated $\Lambda$-modules. The _bounded dg derived category_ is
defined as

$\mathbf{D}_{\rm dg}^{b}(\Lambda\mbox{-mod})=C^{b}_{\rm
dg}(\Lambda\mbox{-mod})/{C_{\rm dg}^{{\rm ac},b}(\Lambda\mbox{-mod})},$

whose homotopy category coincides with $\mathbf{D}^{b}(\Lambda\mbox{-mod})$,
the bounded derived category of $\Lambda\mbox{-mod}$. Denote by
$\mathcal{Y}_{\Lambda/E}^{f}$ the full dg subcategory of
$\mathcal{Y}_{\Lambda/E}^{b}$ consisting of bounded complexes in
$\Lambda\mbox{-mod}$.

We have the following finite version of Proposition 7.3; compare Remark
7.4(2).

######  Corollary 7.5.

Assume that $E$ is a semisimple ring and that $\Lambda$ is a left noetherian
ring. Then there is an isomorphism in $\mathbf{Hodgcat}$

$\mathbf{D}^{b}_{\rm dg}(\Lambda\mbox{-}{\rm
mod})\simeq\mathcal{Y}^{f}_{\Lambda/E},$

which induces an isomorphism of triangulated categories

$\mathbf{D}^{b}(\Lambda\mbox{-}{\rm mod})\simeq
H^{0}(\mathcal{Y}^{f}_{\Lambda/E}).$

###  7.3. The dg tensor algebra as an endomorphism algebra

Throughout this subsection, we will impose the following finiteness conditions
on the algebra extension $E\rightarrow\Lambda$.

  1. (Fin1)

The left $E$-module ${}_{E}\Lambda$ is finitely generated projective.

  2. (Fin2)

There is an algebra homomorphism $\pi\colon\Lambda\rightarrow E$ such that the
composition $E\rightarrow\Lambda\stackrel{{\scriptstyle\pi}}{{\rightarrow}}E$
is the identity map.

Recall that $\overline{\Lambda}=\Lambda/(E\cdot 1_{\Lambda})$ is the cokernel
of $E\to\Lambda$. The natural map

${\rm
Ker}(\pi)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\overline{\Lambda},\quad
a\mapsto\overline{a}$

is an isomorphism of $E$-$E$-bimodules. The multiplication on the ideal ${\rm
Ker}(\pi)$ induces an associative $E$-$E$-bimodule morphism

(7.3)
$\displaystyle\mu\colon\overline{\Lambda}\otimes_{E}\overline{\Lambda}\longrightarrow\overline{\Lambda},\quad\overline{a}\otimes_{E}\overline{b}\mapsto\overline{ab}.$

Here, we emphasize that the elements $a$ and $b$ are chosen to lie in ${\rm
Ker}(\pi)$. Associated to $(\overline{\Lambda},\mu)$, we have the dg tensor
algebra $(T_{E}(\overline{\Lambda}^{*}),\partial)$; see Remark 3.2. We mention
that such dg tensor algebras are related to normal bocses; see [8, Remark
3.4].

We will view $E$ as a $\Lambda$-module via the algebra homomorphism $\pi$.

######  Proposition 7.6.

Assume that (Fin1) and (Fin2) hold. Then there is an isomorphism
$\mathcal{Y}_{\Lambda/E}(E,E)\simeq(T_{E}(\overline{\Lambda}^{*}),\partial)^{\rm
op}$ of dg algebras.

###### Proof.

In this proof, we write $M=\overline{\Lambda}$. Recall that

$\mathcal{Y}(E,E)=\prod_{n\geq 0}{\rm
Hom}_{E}((sM)^{\otimes_{E}n},E)=\bigoplus_{n\geq 0}{\rm
Hom}_{E}((sM)^{\otimes_{E}n},E),$

whose multiplication is induced by the composition $\odot$ and the
differential is given by $\delta_{\rm ex}$; see (7.1). Here, we note that
$\delta_{\rm in}$ vanishes.

We note that $T_{E}(M^{*})$ is graded with $(M^{*})^{\otimes_{E}n}$ in degree
$n$. To emphasize the degrees, we replace $M^{*}$ by $s^{-1}M^{*}$. Its
typical element is denoted by $s^{-1}f$ with $f\in M^{*}$. So, as a graded
algebra, we have

$T_{E}(s^{-1}M^{*})=\bigoplus_{n\geq 0}(s^{-1}M^{*})^{\otimes_{E}n}.$

For each degree $n>0$, we have a natural isomorphism of $E$-$E$-bimodules.

$\begin{array}[]{cccc}\phi^{n}\colon&(s^{-1}M^{*})^{\otimes_{E}n}&\rightarrow&{\rm
Hom}_{E}((sM)^{\otimes_{E}n},E)\\\
&s^{-1}f_{1,n}&\mapsto&(s\overline{a}_{1,n}\mapsto(-1)^{n}f_{n}(\overline{a}_{1}f_{n-1}(\dotsb(\overline{a}_{n-1}f_{1}(\overline{a}_{n}))\cdots))\in
E)\end{array}$

Define $\phi^{0}\colon E\rightarrow{\rm Hom}_{E}(E,E)$ by
$\phi^{0}(x)(y)=\phi(yx)$.

These $\phi^{n}$’s yield an isomorphism

$\phi\colon T_{E}(s^{-1}M^{*})^{\rm
op}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{Y}(E,E)$

of graded $\mathbb{K}$-modules. It is direct to verify that $\phi$ is
compatible with the multiplications.

To show that $\phi$ preserves the differentials, it suffice to verify

$\phi\circ\partial=\delta_{\rm ex}\circ\phi$

on $E\oplus s^{-1}M^{*}$. The verification on $E$ is trivial, since
$\partial|_{E}=0$ and $\delta_{\rm ex}|_{\mathcal{Y}_{0}(E,E)}=0$. Recall that
the restriction of $\partial$ on $s^{-1}M^{*}$ is given by $\partial_{+}$ in
(3.1). Then we are done by verifying the commutativity of the following
square.

(7.8)

The verification is routine, as both $\partial_{+}$ and $\delta_{\rm ex}$ are
induced from $\mu$. Here, we mention that there are minus signs in both
$\phi^{1}$ and $\delta_{\rm ex}$. ∎

##  8\. Noncommutative differential forms

In this section, we study noncommutative differential forms with values in a
complex. This gives rise to a dg endofunctor $\Omega_{\rm nc}$ on the Yoneda
dg category $\mathcal{Y}$. We also obtain a closed natural transformation
$\theta\colon{\rm Id}_{\mathcal{Y}}\to\Omega_{\rm nc}$ of degree zero
satisfying $\theta\Omega_{\rm nc}=\Omega_{\rm nc}\theta$; see Section 6. As
before, we fix an algebra extension $E\rightarrow\Lambda$ and work in the
relative situation.

Let $X$ be a complex of $\Lambda$-modules. The complex of _$X$ -valued
$E$-relative noncommutative differential $1$-forms_ is defined by

$\Omega_{{\rm nc},\Lambda/E}(X)=s\overline{\Lambda}\otimes_{E}X,$

which is graded by means of ${\rm deg}(s\overline{a}_{1}\otimes_{E}x)=|x|-1$
and whose differential is given by
$d(s\overline{a}_{1}\otimes_{E}x)=-s\overline{a}_{1}\otimes_{E}d_{X}(x)$. The
$\Lambda$-action is given by the following nontrivial rule:

(8.1) $\displaystyle
a\blacktriangleright(s\overline{a}_{1}\otimes_{E}x)=s\overline{aa_{1}}\otimes_{E}x-s\overline{a}\otimes
a_{1}x.$

To justify the above terminology, we observe that

$\Omega_{{\rm nc},\Lambda/E}(\Lambda)=s\overline{\Lambda}\otimes_{E}\Lambda$

is a stalk complex of $\Lambda$-$\Lambda$-bimodules concentrated in degree
$-1$, where the right $\Lambda$-action is given by the multiplication of
$\Lambda$. This stalk complex is called the _graded bimodule of $E$-relative
right noncommutative differential $1$-forms_ [70]. Moreover, we have a
canonical isomorphism

$\Omega_{{\rm nc},\Lambda/E}(\Lambda)\otimes_{\Lambda}X\simeq\Omega_{{\rm
nc},\Lambda/E}(X),$

which sends $(s\overline{a}_{0}\otimes_{E}a_{1})\otimes_{\Lambda}x$ to
$s\overline{a}_{0}\otimes_{E}a_{1}x$. We mention that the study of
noncommutative differential forms goes back to [29, Sections 1 and 2].

To avoid notational overload, we write $\mathcal{Y}=\mathcal{Y}_{\Lambda/E}$
and $\Omega_{\rm nc}(X)=\Omega_{{\rm nc},\Lambda/E}(X)$. We have a dg functor

$\Omega_{\rm nc}\colon\mathcal{Y}\longrightarrow\mathcal{Y},\quad
X\mapsto\Omega_{\rm nc}(X),$

which sends a morphism $f\in\mathcal{Y}_{n}(X,Y)$ to the morphism in
$\mathcal{Y}_{n}(\Omega_{\rm nc}(X),\Omega_{\rm nc}(Y))$:

$\textstyle{(s\overline{\Lambda})^{\otimes_{E}n}\otimes_{E}\Omega_{\rm
nc}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{(s\overline{\Lambda})^{\otimes_{E}(n+1)}\otimes_{E}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Id}_{s\overline{\Lambda}}\otimes_{E}f}$$\textstyle{s\overline{\Lambda}\otimes_{E}Y=\Omega_{\rm
nc}(Y).}$

We have a closed natural transformation of degree zero

$\theta\colon{\rm Id}_{\mathcal{Y}}\longrightarrow\Omega_{\rm nc}$

defined as follows. For any $X\in\mathcal{Y}$, $\theta_{X}$ lies in
$\mathcal{Y}_{1}(X,\Omega_{\rm nc}(X))\subseteq\mathcal{Y}(X,\Omega_{\rm
nc}(X))$ and is given by

$\theta_{X}(s\overline{a}\otimes_{E}x)=s\overline{a}\otimes_{E}x\in\Omega_{\rm
nc}(X).$

Note that $\theta_{X}$ is of degree zero and $\delta(\theta_{X})=0$ using the
nontrivial rule (8.1). For the naturalness of $\theta$, we observe that for
each $f\in\mathcal{Y}_{n}(X,Y)$, we have

$\theta_{Y}\odot f=\Omega_{\rm nc}(f)\odot\theta_{X}.$

Indeed, both sides send $s\overline{a}_{1,n+1}\otimes_{E}x$ to
$(-1)^{|f|}s\overline{a}_{1}\otimes_{E}f(s\overline{a}_{2,n}\otimes_{E}x)$.

We observe that

$\Omega_{\rm nc}(\theta_{X})=\theta_{\Omega_{\rm nc}(X)},$

since both sides lie in $\mathcal{Y}_{1}(\Omega_{\rm nc}(X),\Omega_{\rm
nc}^{2}(X))=\mathrm{Hom}_{E}(\Omega_{\rm nc}^{2}(X),\Omega_{\rm nc}^{2}(X))$
and correspond to the identity map of $\Omega_{\rm nc}^{2}(X)$. In summary, we
conclude that the triple $(\mathcal{Y}_{\Lambda/E},\Omega_{\rm nc},\theta)$
satisfies the assumptions made in the first paragraph of Section 6.

In the remaining of this section, we will analyze the cone of $\theta_{X}$ in
$H^{0}(\mathcal{Y})$. Consider the following short exact sequence of complexes
of $\Lambda$-modules.

$\xi_{X}\colon
0\longrightarrow\overline{\Lambda}\otimes_{E}X\stackrel{{\scriptstyle
i_{X}}}{{\longrightarrow}}\Lambda\otimes_{E}X\stackrel{{\scriptstyle
m_{X}}}{{\longrightarrow}}X\longrightarrow 0$

Here, $i_{X}(\overline{a}\otimes_{E}x)=a\otimes_{E}x-1_{\Lambda}\otimes ax$
and $m_{X}(a\otimes_{E}x)=ax$. Similar to (8.1), the $\Lambda$-action on
$\overline{\Lambda}\otimes_{E}X$ is given by
$b(\overline{a}\otimes_{E}x)=\overline{ba}\otimes_{E}x-\overline{b}\otimes_{E}ax$
and thus we have

$\Sigma(\overline{\Lambda}\otimes_{E}X)=\Omega_{\rm nc}(X).$

The mapping cone of $i_{X}$ is described as follows:

$C(X)=(\Lambda\otimes_{E}X)\oplus\Omega_{\rm nc}(X)$

whose differential is given by
$d_{C(X)}(a\otimes_{E}x,0)=(a\otimes_{E}d_{X}(x),0)$ and

$d_{C(X)}(0,s\overline{a}\otimes_{E}x)=(a\otimes_{E}x-1_{\Lambda}\otimes_{E}ax,\
-s\overline{a}\otimes_{E}d_{X}(x)).$

Denote the projection by

${\rm pr}\colon C(X)\rightarrow\Omega_{\rm nc}(X).$

As the underlying short exact sequence of $\xi_{X}$ over $E$ splits, the map

$(m_{X},0)\colon C(X)\longrightarrow X$

is an $E$-relative quasi-isomorphism.

We view both ${\rm pr}$ and $(m_{X},0)$ as morphisms in $\mathcal{Y}$, which
have filtration-degree zero. Namely, ${\rm
pr}\in\mathcal{Y}_{0}(C(X),\Omega_{\rm nc}(X))$ and
$(m_{X},0)\in\mathcal{Y}_{0}(C(X),X).$

######  Lemma 8.1.

Keep the notation as above. Then we have $\theta_{X}\odot(m_{X},0)={\rm pr}$
in $H^{0}(\mathcal{Y}_{\Lambda/E})$.

###### Proof.

Define a map $p\colon C(X)\rightarrow\Omega_{\rm nc}(X)$ of degree $-1$ by
$p(a\otimes_{E}x,0)=s\overline{a}\otimes_{E}x$ and
$p(0,s\overline{b}\otimes_{E}y)=0$. We view $p$ as an element in
$\mathcal{Y}_{0}(C(X),\Omega_{\rm nc}(X))$.

We observe that the differential $\delta_{\rm in}(p)\colon
C(X)\rightarrow\Omega_{\rm nc}(X)$ is determined by

$\delta_{\rm in}(p)(a\otimes_{E}x,0)=0\quad\mbox{ and }\quad\delta_{\rm
in}(p)(0,s\overline{b}\otimes_{E}y)=s\overline{b}\otimes_{E}y.$

The differential $\delta_{\rm ex}(p)$ lies in
$\mathcal{Y}_{1}(C(X),\Omega_{\rm
nc}(X))=\mathrm{Hom}_{E}(s\overline{\Lambda}\otimes_{E}C(X),\Omega_{\rm
nc}(X))$, which is determined by

$\delta_{\rm
ex}(p)(s\overline{a}_{1}\otimes_{E}(a\otimes_{E}x,0))=-s\overline{a}_{1}\otimes_{E}ax\quad\mbox{
and }\quad\delta_{\rm
ex}(p)(s\overline{a}_{1}\otimes_{E}(0,s\overline{b}\otimes_{E}y))=0.$

Note that $\delta_{\rm in}(p)=\mathrm{pr}$ and $\delta_{\rm
ex}(p)=-\theta_{X}\odot(m_{X},0)$. We conclude that

$\delta(p)=\delta_{\rm in}(p)+\delta_{\rm ex}(p)={\rm
pr}-\theta_{X}\odot(m_{X},0).$

This proves the required equality in $H^{0}(\mathcal{Y})$. ∎

Recall that the cone of $\theta_{X}$ is determined by the following exact
triangle in $H^{0}(\mathcal{Y})$.

$X\stackrel{{\scriptstyle\theta_{X}}}{{\longrightarrow}}\Omega_{\rm
nc}(X)\longrightarrow{\rm Cone}(\theta_{X})\longrightarrow\Sigma(X)$

######  Proposition 8.2.

There is an isomorphism ${\rm
Cone}(\theta_{X})\simeq\Sigma(\Lambda\otimes_{E}X)$ in
$H^{0}(\mathcal{Y}_{\Lambda/E})$.

###### Proof.

The short exact sequence $\xi_{X}$ induces an exact triangle in
$\mathbf{D}(\Lambda/E)$

(8.2) $\displaystyle\overline{\Lambda}\otimes_{E}X\stackrel{{\scriptstyle
i_{X}}}{{\longrightarrow}}\Lambda\otimes_{E}X\stackrel{{\scriptstyle
m_{X}}}{{\longrightarrow}}X\stackrel{{\scriptstyle
c}}{{\longrightarrow}}\Omega_{\rm nc}(X),$

where the connecting morphism $c$ is given by the following roof

$X\xleftarrow{(m_{X},0)}C(X)\stackrel{{\scriptstyle\rm
pr}}{{\longrightarrow}}\Omega_{\rm nc}(X).$

Thus, we have $c=\mathrm{pr}\circ(m_{X},0)^{-1}$ in $\mathbf{D}(\Lambda/E)$.

Recall from Proposition 7.3 the triangle isomorphism
$H^{0}(\overline{\Theta})\colon\mathbf{D}(\Lambda/E)\simeq
H^{0}(\mathcal{Y})$, which acts on objects by the identity. As
$H^{0}(\overline{\Theta})$ sends cochain maps identically to the corresponding
morphisms in $\mathcal{Y}$ of filtration-degree zero, we obtain

$H^{0}(\overline{\Theta})(c)=\mathrm{pr}\circ(m_{X},0)^{-1}=\theta_{X},$

where the right equality follows from Lemma 8.1. This yields

$H^{0}(\overline{\Theta})({\rm Cone}(c))={\rm Cone}(\theta_{X}).$

By (8.2), ${\rm Cone}(c)$ is isomorphic to $\Sigma(\Lambda\otimes_{E}X)$ in
$\mathbf{D}(\Lambda/E)$. Then the required statement follows since
$H^{0}(\overline{\Theta})(\Sigma(\Lambda\otimes_{E}X))=\Sigma(\Lambda\otimes_{E}X).$
∎

##  9\. The singular Yoneda dg category

In this section, we introduce the singular Yoneda dg category, which provides
dg enhancements for various singularity categories. We fix an algebra
extension $E\rightarrow\Lambda$ as before. We prove that the endomorphism
algebra of $E$ in the singular Yoneda dg category of $\Lambda$ is isomorphic
to a dg Leavitt algebra; see Theorem 9.5.

As we have seen, the triple $(\mathcal{Y}_{\Lambda/E},\Omega_{\rm nc},\theta)$
obtained in Section 8 satisfies the assumptions made in Section 6. We thus
form the dg localization along $\theta$

$\iota\colon\mathcal{Y}_{\Lambda/E}\longrightarrow\mathcal{SY}_{\Lambda/E}.$

The obtained dg category $\mathcal{SY}_{\Lambda/E}$ is called the _$E$
-relative singular Yoneda dg category_ of $\Lambda$.

Let us describe $\mathcal{SY}=\mathcal{SY}_{\Lambda/E}$ explicitly. Its
objects are just complexes of $\Lambda$-modules. For two objects $X$ and $Y$,
the Hom complex is defined to be the colimit of the following sequence of
cochain complexes.

$\mathcal{Y}(X,Y)\longrightarrow\mathcal{Y}(X,\Omega_{\rm
nc}(Y))\longrightarrow\cdots\longrightarrow\mathcal{Y}(X,\Omega_{\rm
nc}^{p}(Y))\longrightarrow\mathcal{Y}(X,\Omega_{\rm
nc}^{p+1}(Y))\longrightarrow\cdots$

The structure map sends $f$ to $\theta_{\Omega_{\rm nc}^{p}(Y)}\odot f$; see
(7.1). More precisely, for any $f\in\mathcal{Y}_{n}(X,\Omega_{\rm
nc}^{p}(Y))$, the map $\theta_{\Omega_{\rm nc}^{p}(Y)}\odot
f\in\mathcal{Y}_{n+1}(X,\Omega_{\rm nc}^{p+1}(Y))$ is given by

(9.1) $\displaystyle
s\overline{a}_{1,n+1}\otimes_{E}x\longmapsto(-1)^{|f|}s\overline{a}_{1}\otimes
f(s\overline{a}_{2,n+1}\otimes_{E}x).$

The image of $f\in\mathcal{Y}(X,\Omega_{\rm nc}^{p}(Y))$ in
$\mathcal{SY}(X,Y)$ is denoted by $[f;p]$. The composition of $[f;p]$ with
$[g;q]\in\mathcal{SY}(Y,Z)$ is defined by

$[g;q]\odot_{\rm sg}[f;p]=[\Omega_{\rm nc}^{p}(g)\odot f;p+q].$

By Proposition 7.3, the Yoneda dg category $\mathcal{Y}_{\Lambda/E}$ is
pretriangulated. We infer from Theorem 6.4 that $\mathcal{SY}_{\Lambda/E}$ is
also pretriangulated.

###  9.1. The dg singularity categories

Recall that $\mathbf{D}(\Lambda/E)$ denotes the $E$-relative derived category
of $\Lambda$. We define the _$E$ -relative virtual singularity category of
$\Lambda$_ to be the following quotient triangulated category

$\mathcal{V}(\Lambda/E)=\mathbf{D}(\Lambda/E)/{{\rm
thick}\langle\Lambda\otimes_{E}V\;|\;V\mbox{ complex of
}E\mbox{-modules}\rangle}.$

Denote by $\mathcal{N}$ the full dg subcategory of $\mathbf{D}_{\rm
dg}(\Lambda/E)$ formed by those objects in ${\rm
thick}\langle\Lambda\otimes_{E}V\;|\;V\mbox{ complex of
}E\mbox{-modules}\rangle$. The following dg quotient category

$\mathcal{V}_{\rm dg}(\Lambda/E)=\mathbf{D}_{\rm dg}(\Lambda/E)/\mathcal{N}$

might be called the _$E$ -relative virtual dg singularity category of
$\Lambda$_. By Lemma 5.8, we identify the homotopy category
$H^{0}(\mathcal{V}_{\rm dg}(\Lambda/E))$ with $\mathcal{V}(\Lambda/E)$.

Recall from Proposition 7.3 the isomorphism
$\overline{\Theta}\colon\mathbf{D}_{\rm
dg}(\Lambda/E)\simeq\mathcal{Y}_{\Lambda/E}$.

######  Proposition 9.1.

Keep the notation as above. Then the composite dg functor $\mathbf{D}_{\rm
dg}(\Lambda/E)\stackrel{{\scriptstyle\overline{\Theta}}}{{\rightarrow}}\mathcal{Y}_{\Lambda/E}\stackrel{{\scriptstyle\iota}}{{\rightarrow}}\mathcal{SY}_{\Lambda/E}$
induces an isomorphism in $\mathbf{Hodgcat}$

$\mathcal{V}_{\rm dg}(\Lambda/E)\simeq\mathcal{SY}_{\Lambda/E}.$

Consequently, we have an isomorphism of triangulated categories

$\mathcal{V}(\Lambda/E)\simeq H^{0}(\mathcal{SY}_{\Lambda/E}).$

###### Proof.

Consider the two thick hulls $\mathcal{T}_{1}={{\rm
thick}\langle\Lambda\otimes_{E}V\;|\;V\mbox{ complex of
}E\mbox{-modules}\rangle}$ and $\mathcal{T}_{2}={\rm thick}\langle{\rm
Cone}(\theta_{X})\;|\;X\mbox{ complex of }\Lambda\mbox{-modules}\rangle$ in
$H^{0}(\mathcal{Y}_{\Lambda/E})$. Denote by $\mathcal{N}_{1}$ the full dg
subcategory of $\mathcal{Y}_{\Lambda/E}$ formed by those objects in
$\mathcal{T}_{1}$ and similarly, denote by $\mathcal{N}_{2}$ the one formed by
those objects in $\mathcal{T}_{2}$. Clearly, we have
$\overline{\Theta}(\mathcal{N})=\mathcal{N}_{1}$.

By Proposition 8.2, we identify ${\rm Cone}(\theta_{X})$ with
$\Sigma(\Lambda\otimes_{E}X)=\Lambda\otimes_{E}\Sigma(X)$. Then we have
$\mathcal{T}_{2}\subseteq\mathcal{T}_{1}$. On the other hand, any complex $Y$
of the form $\Lambda\otimes_{E}V$ is isomorphic to a direct summand of
$\Lambda\otimes_{E}Y$ since the natural surjection $\Lambda\otimes_{E}Y\to Y$
splits in $\mathcal{Y}_{\Lambda/E}$ with a section given by
$a\otimes_{E}v\mapsto a\otimes_{E}1\otimes_{E}v$. It follows that
$\mathcal{T}_{1}\subseteq\mathcal{T}_{2}$. So we have
$\mathcal{T}_{1}=\mathcal{T}_{2}$ and thus $\mathcal{N}_{1}=\mathcal{N}_{2}$.
Now the required isomorphism in $\mathbf{Hodgcat}$ follows by combining the
isomorphisms in Theorem 6.4 and Proposition 7.3. ∎

The following remark is analogous to Remark 7.4.

######  Remark 9.2.

(1) Denote by $\mathcal{SY}^{b}_{\Lambda/E}$ the full dg subcategory of
$\mathcal{SY}_{\Lambda/E}$ consisting of bounded complexes. We view the
bounded homotopy category $\mathbf{K}^{b}(\mathcal{P}_{\Lambda/E})$ of
$E$-relatively projective $\Lambda$-modules as a triangulated subcategory of
$\mathbf{D}^{b}(\Lambda/E)$. Inspired by [46], we define the _$E$ -relative
completed singularity category_ of $\Lambda$ by the following Verdier quotient

$\widehat{\mathbf{S}}(\Lambda/E):=\mathbf{D}^{b}(\Lambda/E)/{\mathbf{K}^{b}(\mathcal{P}_{\Lambda/E})}.$

As its dg analogue, the _$E$ -relative completed dg singularity category_ of
$\Lambda$ is defined to be

$\widehat{\mathbf{S}}_{\rm dg}(\Lambda/E):=\mathbf{D}^{b}_{\rm
dg}(\Lambda/E)/{C^{b}_{\rm dg}(\mathcal{P}_{\Lambda/E})}.$

We mention that the relative dg singularity category [11, Definition 2.23] is
quite different from the ones here.

We have a bounded version of Proposition 9.1: there is an isomorphism in
$\mathbf{Hodgcat}$

$\widehat{\mathbf{S}}_{\rm dg}(\Lambda/E)\simeq\mathcal{SY}^{b}_{\Lambda/E},$

which induces an isomorphism of triangulated categories

$\widehat{\mathbf{S}}(\Lambda/E)\simeq H^{0}(\mathcal{SY}^{b}_{\Lambda/E}).$

(2) Assume that $E$ is a semisimple ring. Then $\mathcal{V}(\Lambda/E)$
coincides with the (absolute) virtual singularity category of $\Lambda$, which
is defined as

$\mathcal{V}(\Lambda):=\mathbf{D}(\Lambda\mbox{-Mod})/{{\rm
thick}\langle\oplus_{i\in\mathbb{Z}}\Sigma^{-i}(P^{i})\;|\;P^{i}\in\Lambda\mbox{-Proj}\rangle}.$

Similarly, $\widehat{\mathbf{S}}(\Lambda/E)$ coincides with the completed
singularity category [46] of $\Lambda$

$\widehat{\mathbf{S}}(\Lambda):=\mathbf{D}^{b}(\Lambda\mbox{-Mod})/{\mathbf{K}^{b}(\Lambda\mbox{-Proj})}.$

Their dg analogues are $\mathcal{V}_{\rm dg}(\Lambda/E)=\mathcal{V}_{\rm
dg}(\Lambda)$ and $\widehat{\mathbf{S}}_{\rm
dg}(\Lambda/E)=\widehat{\mathbf{S}}_{\rm dg}(\Lambda)$.

Consequently, we have isomorphisms in $\mathbf{Hodgcat}$

$\mathcal{V}_{\rm dg}(\Lambda)\simeq\mathcal{SY}_{\Lambda/E}\quad\mbox{ and
}\quad\widehat{\mathbf{S}}_{\rm
dg}(\Lambda)\simeq\mathcal{SY}^{b}_{\Lambda/E},$

which induce isomorphisms of triangulated categories:

$\mathcal{V}(\Lambda)\simeq H^{0}(\mathcal{SY}_{\Lambda/E})\quad\mbox{ and
}\quad\widehat{\mathbf{S}}(\Lambda)\simeq
H^{0}(\mathcal{SY}^{b}_{\Lambda/E}).$

In the remaining of this subsection, we further assume that $E$ is a
semisimple ring and that $\Lambda$ is a left noetherian ring. Denote by
$\mathcal{SY}^{f}_{\Lambda/E}$ the full dg subcategory of
$\mathcal{SY}^{b}_{\Lambda/E}$ consisting of bounded complexes of finitely
generated $\Lambda$-modules.

We view the bounded homotopy category $\mathbf{K}^{b}(\Lambda\mbox{-proj})$ of
finitely generated projective $\Lambda$-modules as a triangulated subcategory
of $\mathbf{D}^{b}(\Lambda\mbox{-mod})$. Following [16, 60], the _singularity
category_ of $\Lambda$ is defined as the following Verdier quotient

$\mathbf{D}_{\rm
sg}(\Lambda)=\mathbf{D}^{b}(\Lambda\mbox{-mod})/{\mathbf{K}^{b}(\Lambda\mbox{-proj})}.$

Its dg analogue is the _dg singularity category_ [46, 11, 15], defined as

$\mathbf{S}_{\rm dg}(\Lambda)=\mathbf{D}^{b}_{\rm
dg}(\Lambda\mbox{-mod})/{C_{\rm dg}^{b}(\Lambda\mbox{-proj})}.$

Here, we identify $C_{\rm dg}^{b}(\Lambda\mbox{-proj})$ with the full dg
subcategory of $\mathbf{D}^{b}_{\rm dg}(\Lambda\mbox{-mod})$ formed by bounded
complexes of finitely generated projective $\Lambda$-modules.

We have the following finite version of Proposition 9.1; compare Remark
9.2(2).

######  Corollary 9.3.

Assume that $E$ is a semisimple ring and that $\Lambda$ is a left noetherian
ring. Then there is an isomorphism in $\mathbf{Hodgcat}$

$\mathbf{S}_{\rm dg}(\Lambda)\simeq\mathcal{SY}^{f}_{\Lambda/E},$

which induces an isomorphism of triangulated categories

$\mathbf{D}_{\rm sg}(\Lambda)\simeq H^{0}(\mathcal{SY}^{f}_{\Lambda/E}).$

######  Remark 9.4.

(1) The inclusions
$\mathcal{SY}^{f}_{\Lambda/E}\subseteq\mathcal{SY}^{b}_{\Lambda/E}\subseteq\mathcal{SY}_{\Lambda/E}$
of dg categories induce fully faithful triangle functors

$\mathbf{D}_{\rm
sg}(\Lambda)\hookrightarrow\widehat{\mathbf{S}}(\Lambda)\hookrightarrow\mathcal{V}(\Lambda).$

The fully-faithfulness of the left functor is known; see [20, Remark 3.6] and
compare [60, Proposition 1.13].

(2) The Hom complexes in $\mathcal{SY}^{f}_{\Lambda/E}$ have similar flavor
with the singular Hochschild cochain complex [70]. In view of [46, Conjecture
1.3], we expect that the $B_{\infty}$-structure of the latter complex might be
related to the one of the Hochschild cochain complex of
$\mathcal{SY}^{f}_{\Lambda/E}$.

###  9.2. The dg Leavitt algebra as an endomorphism algebra

We now impose the conditions (Fin1) and (Fin2) in Subsection 7.3 on the
algebra extension $E\to\Lambda$. Associated to $(\overline{\Lambda},\mu)$ in
(7.3), we have the dg Leavitt algebra $(L_{E}(\overline{\Lambda}),\partial)$;
see Definition 3.5. We view $E$ as a $\Lambda$-module via the algebra
homomorphism $\pi$ in (Fin2). Identifying modules with stalk complexes
concentrated in degree zero, we view $E$ as an object in
$\mathcal{SY}_{\Lambda/E}$.

The isomorphism in the following theorem might be viewed as the core of this
work, which establishes a link between the singular Yoneda dg category and the
dg Leavitt algebra.

######  Theorem 9.5.

Assume that (Fin1) and (Fin2) hold. Then there is an isomorphism
$\mathcal{SY}_{\Lambda/E}(E,E)\simeq(L_{E}(\overline{\Lambda}),\partial)^{\rm
op}$ of dg algebras.

###### Proof.

For convenience, we write $M=\overline{\Lambda}$ and omit the subscript
$\Lambda/E$ in $\mathcal{Y}_{\Lambda/E}$ and $\mathcal{SY}_{\Lambda/E}$.
Observe that $\Omega_{\rm
nc}^{p}(E)=(sM)^{\otimes_{E}p}\otimes_{E}E=(sM)^{\otimes_{E}p}$ and that

$\mathcal{Y}(E,\Omega_{\rm nc}^{p}(E))=\prod_{n\geq 0}{\rm
Hom}_{E}((sM)^{\otimes_{E}n},(sM)^{\otimes_{E}p})=\bigoplus_{n\geq 0}{\rm
Hom}_{E}((sM)^{\otimes_{E}n},(sM)^{\otimes_{E}p}),$

whose differential is given by $\delta_{\rm ex}$; see Subsection 7.1. Here, we
note that $\delta_{\rm in}$ vanishes.

_Step 1._ We have a canonical isomorphism

$\begin{array}[]{ccccl}\psi_{p}\colon&\mathcal{Y}(E,E)\otimes_{E}(sM)^{\otimes_{E}p}&\longrightarrow&\mathcal{Y}(E,\Omega_{\rm
nc}^{p}(E))&\\\
&g\otimes_{E}s\overline{a}_{1,p}&\longmapsto&(s\overline{x}_{1,n}\mapsto(-1)^{pn}g(s\overline{x}_{1,n})s\overline{a}_{1,p})&\end{array}$

for any $g\in\mathcal{Y}_{n}(E,E)$ and
$s\overline{a}_{1,p}\in(sM)^{\otimes_{E}p}$. Recall from the proof of
Proposition 7.6 the isomorphism $\phi\colon
T_{E}(s^{-1}M^{*})\xrightarrow{\simeq}\mathcal{Y}(E,E)$. Then for each $p\geq
0$, we have a composite isomorphism

$\widetilde{\psi}_{p}\colon
T_{E}(s^{-1}M^{*})\otimes_{E}(sM)^{\otimes_{E}p}\xrightarrow{\phi\otimes_{E}\mathrm{Id}}\mathcal{Y}(E,E)\otimes_{E}(sM)^{\otimes_{E}p}\xrightarrow{\psi_{p}}\mathcal{Y}(E,\Omega^{p}_{\rm
nc}(E)).$

We claim that the following diagram

(9.6)

commutes, where $L$ denotes the map (2.5) and $R$ sends $g$ to
$\theta_{\Omega_{\rm nc}^{p}(Y)}\odot g$. Restricting to homogeneous
components, we need verify the following commutative diagram.

$\textstyle{(s^{-1}M^{*})^{\otimes_{E}n}\otimes_{E}(sM)^{\otimes_{E}p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L}$$\scriptstyle{\widetilde{\psi}_{p}}$$\textstyle{{\rm
Hom}_{E}((sM)^{\otimes_{E}n},(sM)^{\otimes_{E}p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{(s^{-1}M^{*})^{\otimes_{E}n+1}\otimes_{E}(sM)^{\otimes_{E}p+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\psi}_{p+1}}$$\textstyle{{\rm
Hom}_{E}((sM)^{\otimes_{E}n+1},(sM)^{\otimes_{E}p+1})}$

More explicitly, we have

$L(s^{-1}f_{1,n}\otimes_{E}s\overline{a}_{1,p})=\sum_{i\in
S}s^{-1}f_{1,n}\otimes_{E}s^{-1}\alpha_{i}^{*}\otimes_{E}s\overline{\alpha}_{i}\otimes_{E}s\overline{a}_{1,p}$

and $R(g)={\rm Id}_{sM}\otimes_{E}g$. Furthermore,
$\widetilde{\psi}_{p}(s^{-1}f_{1,n}\otimes_{E}s\overline{a}_{1,p})$ is given
by

$s\overline{x}_{1,n}\longmapsto(-1)^{n(p+1)}f_{n}(\overline{x}_{1}f_{n-1}(\overline{x}_{2}f_{n-2}(\cdots(\overline{x}_{n-1}f_{1}(\overline{x}_{n}))\cdots)))s\overline{a}_{1,p}.$

Similarly,
$\widetilde{\psi}_{p+1}(s^{-1}f^{\prime}_{1,n+1}\otimes_{E}s\overline{b}_{1,p+1})$
is given by

$s\overline{y}_{1,n+1}\longmapsto(-1)^{(n+1)(p+2)}f^{\prime}_{n+1}(\overline{y}_{1}f^{\prime}_{n}(\overline{y}_{2}f^{\prime}_{n-1}(\cdots(\overline{y}_{n}f^{\prime}_{1}(\overline{y}_{n+1}))\cdots)))s\overline{b}_{1,p+1}.$

Therefore,
$R\circ\widetilde{\psi}_{p}(s^{-1}f_{1,n}\otimes_{E}s\overline{a}_{1,p})$
equals to the following map

$s\overline{y}_{1,n+1}\longmapsto(-1)^{(n+1)p}\;s\overline{y}_{1}\otimes_{E}f_{n}(\overline{y}_{2}f_{n-1}(\overline{y}_{3}f_{n-2}(\cdots(\overline{y}_{n}f_{1}(\overline{y}_{n+1}))\cdots)))s\overline{a}_{1,p}.$

Using (2.4), we observe that $\widetilde{\psi}_{p+1}\circ
L(s^{-1}f_{1,n}\otimes_{E}s\overline{a}_{1,p})$ coincides with the above map,
proving the claim.

_Step 2._ Recall from Theorem 2.6 that $L_{E}(M)$ is isomorphic to the colimit
of the explicit sequence (2.5). Take the colimits along the maps $L$ and $R$
in (9.6). It follows that the isomorphisms $\widetilde{\psi}_{p}$ induces an
isomorphism of graded $\mathbb{K}$-modules

$\Psi\colon