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# Deformed Mirror Symmetry for Punctured Surfaces

Raf Bocklandt and Jasper van de Kreeke

(15 March 2023)

###### Abstract

Mirror symmetry originally envisions a correspondence between deformations of
the A-side and deformations of the B-side. In this paper, we achieve an
explicit correspondence in the case of punctured surfaces.

The starting point is the noncommutative mirror equivalence
$\operatorname{Gtl}Q\cong\operatorname{mf}(\operatorname{Jac}\check{Q},\ell)$
for a punctured surface $Q$. We pick a deformation $\operatorname{Gtl}_{q}Q$
which captures a large part of the deformation theory and includes the
relative Fukaya category. To find the corresponding deformation of
$\operatorname{mf}(\operatorname{Jac}\check{Q},\ell)$, we deform work of Cho-
Hong-Lau which interprets mirror symmetry as Koszul duality. As result we
explicitly obtain the corresponding deformation
$\operatorname{mf}(\operatorname{Jac}_{q}\check{Q},\ell_{q})$ together with a
deformed mirror functor
$\operatorname{Gtl}_{q}Q\xrightarrow{\sim}\operatorname{mf}(\operatorname{Jac}_{q}\check{Q},\ell_{q})$.

The bottleneck is to verify that the algebra $\operatorname{Jac}_{q}\check{Q}$
is indeed a (flat) deformation of $\operatorname{Jac}\check{Q}$. We achieve
this by deploying a result of Berger-Ginzburg-Taillefer on deformations of CY3
algebras, which however requires the relations to be homogeneous. We show how
to replace this homogeneity requirement by a simple boundedness condition and
obtain flatness of $\operatorname{Jac}_{q}\check{Q}$ for almost all $Q$.

We finish the paper with examples, including a full treatment of the
3-punctured sphere and 4-punctured torus. With the help of our computations in
[23], we describe $\operatorname{Jac}_{q}\check{Q}$ explicitly. It turns out
that the deformed potential $\ell_{q}$ is still central in
$\operatorname{Jac}_{q}\check{Q}$, in contrast to the popular slogan that
central elements do not survive under deformation.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on $A_{\infty}$-categories
    1. 2.1 $A_{\infty}$-categories
    2. 2.2 Deformations of $A_{\infty}$-categories
    3. 2.3 Submodules of completed tensor products
    4. 2.4 $\mathfrak{m}$-adically free modules
    5. 2.5 On the quasi-flatness condition
  3. 3 Preliminaries on Koszul duality
    1. 3.1 Modules
    2. 3.2 Koszul duality
    3. 3.3 Classical Koszual duals
    4. 3.4 Calabi-Yau algebras
    5. 3.5 Van den Bergh and Serre duality
    6. 3.6 Jacobi algebras
    7. 3.7 Cyclic $A_{\infty}$-algebras
    8. 3.8 Cho-Hong-Lau roadmap
  4. 4 Preliminaries on mirror symmetry
    1. 4.1 Gentle algebras
    2. 4.2 Zigzag paths
    3. 4.3 Matrix factorizations
    4. 4.4 Jacobi algebras of dimers
    5. 4.5 Mirror symmetry for punctured surfaces
  5. 5 Preliminaries on the category of zigzag paths
    1. 5.1 Category of zigzag paths
    2. 5.2 Deformed category of zigzag paths
    3. 5.3 Minimal model structure
    4. 5.4 Preparation for mirror objects
  6. 6 Flatness of superpotential deformations
    1. 6.1 Flatness and quasi-flatness
    2. 6.2 Berger-Ginzburg inclusion
    3. 6.3 Notation and conventions
    4. 6.4 Relations of bounded type
    5. 6.5 Ideals tailored to boundedness
    6. 6.6 Bounded strong Berger-Ginzburg inclusion
    7. 6.7 Quasi-flatness in the completed path algebra
    8. 6.8 Quasi-flatness in the path algebra
    9. 6.9 Closedness results
    10. 6.10 Dimers of bounded type
    11. 6.11 Main theorems on flatness
  7. 7 A deformed Cho-Hong-Lau construction
    1. 7.1 Perspective from Koszul duality
    2. 7.2 The Cho-Hong-Lau construction
    3. 7.3 Deformed Landau-Ginzburg model
    4. 7.4 Projectives of deformed algebras
    5. 7.5 Deformed matrix factorizations
    6. 7.6 Deformed mirror functor
  8. 8 Deformed mirror symmetry
    1. 8.1 Mirror symmetry by Cho-Hong-Lau
    2. 8.2 Midpoint polygons
    3. 8.3 Cyclicity and slow growth
    4. 8.4 Deformed superpotential
    5. 8.5 Deformed potential
    6. 8.6 Deformed mirror objects
    7. 8.7 Main result
  9. A Examples
    1. A.1 3-punctured sphere
    2. A.2 4-punctured torus

##  1 Introduction

Mirror symmetry is the quest for equivalences between Fukaya categories and
categories of coherent sheaves. Noncommutative mirror symmetry is the quest
for equivalences between Fukaya categories and categories of sheaves on
noncommutative spaces. In the present paper, our starting point is
noncommutative mirror symmetry for punctured surfaces:

Gentle algebra$\operatorname{Gtl}Q$Matrix
factorizations$\operatorname{mf}(\operatorname{Jac}\check{Q},\ell)$[12]

$\sim$

In the present paper, we pick one specific deformation
$\operatorname{Gtl}_{q}Q$ of $\operatorname{Gtl}Q$ and find the corresponding
deformation of $\operatorname{mf}(\operatorname{Jac}\check{Q},\ell)$. The
result is a deformed mirror equivalence:

Deformed gentle algebra$\operatorname{Gtl}_{q}Q$Deformed matrix
factorizations$\operatorname{mf}(\operatorname{Jac}_{q}\check{Q},\ell_{q})$

$\sim$

In what follows, we explain our quest from different perspectives. We explain
the philosophy of the specific deformation $\operatorname{Gtl}_{q}Q$, comment
on the source of mirror functors from the construction of Cho, Hong and Lau
[17], and explain a bottleneck concerning the question whether the deformed
Jacobi algebra $\operatorname{Jac}_{q}\check{Q}$ is indeed a deformation of
$\operatorname{Jac}\check{Q}$.

#### Deformation theory

In $A_{\infty}$-deformation theory one studies possible modifications of a
given $A_{\infty}$-structure which keep the $A_{\infty}$-relations intact. One
possible line of study consists of formal (infinitesimally) curved
$A_{\infty}$-deformations. The base ring for such deformations is a local
algebra $B$ with a few additional properties.

An interesting question arises when one is given two equivalent
$A_{\infty}$-categories $\mathcal{C}$, $\mathcal{D}$ and tries to transfer a
deformation from $\mathcal{C}$ to $\mathcal{D}$. As a starting point, one is
given a quasi-equivalence $F:\mathcal{C}\to\mathcal{D}$ and a deformation
$\mathcal{D}_{q}$. Transferring the deformation $\mathcal{C}_{q}$ via $F$ then
entails finding a deformation $\mathcal{D}_{q}$ of $\mathcal{D}$ together with
a deformation $F_{q}$ of $F$ such that
$F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$ is a functor of deformed
$A_{\infty}$-categories.

The difficulty in transferring $A_{\infty}$-deformations lies in the character
of $A_{\infty}$-theory. Indeed, both the $A_{\infty}$-products of
$\mathcal{C}$ and $\mathcal{D}$ and the functor $F$ have higher components,
which make it impossible to quickly to write down the corresponding
deformation $\mathcal{D}_{q}$.

Nevertheless, it is known that a transfer of deformations along quasi-
equivalences always exists. The clue is to interpret $A_{\infty}$-deformations
of $\mathcal{C}$ as Maurer-Cartan elements of the Hochschild DGLA
$\operatorname{HC}(\mathcal{C})$. The quasi-equivalence
$F:\mathcal{C}\to\mathcal{D}$ then gives rise to a non-canonical
$L_{\infty}$-morphism
$F_{*}:\operatorname{HC}(\mathcal{C})\to\operatorname{HC}(\mathcal{D})$. By
applying $F_{*}$ to a given deformation $\mathcal{C}_{q}$, viewed as Maurer-
Cartan element, one obtains the corresponding deformation $\mathcal{D}_{q}$.
While this abstract interpretation does make $\mathcal{D}_{q}$ computable, it
sets the stage for the systematic quest of deformed mirror symmetry.

#### Gentle algebras

A popular way of modeling wrapped Fukaya categories of punctured surfaces
deploys gentle algebras[12]. In this framework, one starts from an oriented
closed surface $S$ with a finite set of punctures $M\subseteq S$. One chooses
a system $\mathcal{A}$ of arcs which connect the punctures and divide the
surface into polygons. To an arc system, one can associate a so-called gentle
algebra $\operatorname{Gtl}\mathcal{A}$, reminiscent of the classical
associative gentle algebras of [2]. The gentle algebra
$\operatorname{Gtl}\mathcal{A}$ is actually an $A_{\infty}$-category whose
higher products detect the topology of the punctured surface $S\setminus M$.
It was shown in [12] that it accurately models the wrapped Fukaya category of
$(S,M)$.

Dimers are specific kinds of arc systems which suit the purposes of mirror
symmetry. We shall consider the specific mirror symmetry of punctured
surfaces, built in [12]. This statement of mirror symmetry entails a quasi-
isomorphism
$F:\operatorname{Gtl}Q\to\operatorname{mf}(\operatorname{Jac}\check{Q},\ell)$.
The dimer $\check{Q}$ is the so-called dual dimer of $Q$ and can be built from
$Q$ in a combinatorical way. In contrast, the mirror functor $F$ itself is
only given non-constructively and built in an inductive way by solving cocycle
equations. Whenever we are given a deformation $\operatorname{Gtl}_{q}Q$, it
would be very hard to explicitly find the corresponding deformation of
$\operatorname{mf}(\operatorname{Jac}\check{Q},\ell)$.

#### Deformed Fukaya categories

Seidel [32] has introduced the idea of deforming Fukaya categories relative to
a divisor. The idea is to introduce a formal parameter $q$ and weight every
pseudoholomorphic disk by $q^{s}$ where $s$ counts the number of intersections
of the disk with the divisor. In [13], we have transported this concept to the
world of gentle algebras. The result is a deformation
$\operatorname{Gtl}_{q}Q$, in which every puncture comes with its own
deformation parameter. Whenever a disk covers the punctures
$q_{1},\ldots,q_{k}$, the contribution of this disk is weighted by the product
$q_{1}\ldots q_{k}$.

We raised the hope that our candidate deformation $\operatorname{Gtl}_{q}Q$
would be the correct way to implement Seidel’s idea on the side of gentle
algebras. In [23], we examined this expectation and computed a part of the
$A_{\infty}$-structure of the derived category
$\operatorname{H}\operatorname{Tw}\operatorname{Gtl}_{q}Q$. Very specifically,
it concerns the subcategory
$\operatorname{H}\mathbb{L}_{q}\subseteq\operatorname{H}\operatorname{Tw}\operatorname{Gtl}_{q}Q$
given by the zigzag paths in $Q$. The result is an explicit description of the
$A_{\infty}$-structure of $\operatorname{H}\mathbb{L}_{q}$ in terms of certain
types of immersed disks. While it is hard to determine values for products of
non-transversal sequences in the relative Fukaya category, our description of
$\operatorname{H}\mathbb{L}_{q}$ determines their values very accurately.
Although our calculation is limited to the zigzag paths, we consider [23] a
crude verification that $\operatorname{Gtl}_{q}Q$ is the correct transport of
Seidel’s vision to gentle algebras and can be considered a “relative wrapped
Fukaya category”.

Relative Fukaya categories have already served as A-side of mirror symmetry
before. For instance, Lekili and Perutz [24] find a commutative mirror for the
relative Fukaya category of the $1$-punctured torus, apparently the first use
of a relative Fukaya category in mirror symmetry. In [25], Lekili and
Polishchuk generalize this result to the case of the $n$-punctured torus. They
depart from a finite collection of split-generators of the Fukaya category and
compute part of their deformed products in the relative Fukaya category. Their
mirror is then obtained by guessing the correct deformation on the B-side.
Complete knowledge of the products in the relative Fukaya category or even a
relative wrapped Fukaya category are not required in their approach.

#### Mirror functors

A rich source of mirror functors is the recent construction of Cho, Hong and
Lau [17]. Their construction associates to a given $A_{\infty}$-category
$\mathcal{C}$ with a suitable subcategory $\mathbb{L}\subseteq\mathcal{C}$ a
Landau-Ginzburg model $(\operatorname{Jac}Q^{\mathbb{L}},\ell)$ together with
an $A_{\infty}$-functor

$F:\mathcal{C}\to\operatorname{MF}(\operatorname{Jac}Q^{\mathbb{L}},\ell).$

The Cho-Hong-Lau construction can be applied to the category
$\mathcal{C}=\operatorname{H}\operatorname{Tw}\operatorname{Gtl}Q$ by choosing
$\mathbb{L}$ to be the subcategory given by so-called zigzag paths. This
application yields back the original mirror symmetry for punctured surfaces
$\operatorname{Gtl}Q\cong\operatorname{mf}(\operatorname{Jac}\check{Q},\ell)$
from [12].

In the present paper, it is our aim to produce a deformation of
$\operatorname{mf}(\operatorname{Jac}Q,\ell)$ which corresponds to
$\operatorname{Gtl}_{q}Q$. Thanks to the Cho-Hong-Lau construction, this task
becomes straightforward: The first step is to deform the Cho-Hong-Lau
construction. The result is a procedure which generates mirror functors of the
kind
$\mathcal{C}_{q}\to\operatorname{MF}(\operatorname{Jac}_{q}Q^{\mathbb{L}},\ell_{q})$.
The second step is to apply this deformed construction to the case of
$\mathcal{C}_{q}=\operatorname{H}\operatorname{Tw}\operatorname{Gtl}_{q}Q$.
The result is a quasi-isomorphism of deformed $A_{\infty}$-categories

$F_{q}:\operatorname{Gtl}_{q}Q\xrightarrow{\sim}\operatorname{mf}(\operatorname{Jac}_{q}\check{Q},\ell_{q}).$

In particular, the deformation
$\operatorname{mf}(\operatorname{Jac}_{q}\check{Q},\ell_{q})$ is the desired
deformation of $\operatorname{mf}(\operatorname{Jac}\check{Q},\ell)$
corresponding to $\operatorname{Gtl}_{q}Q$. It is possible to describe the
algebra $\operatorname{Jac}_{q}\check{Q}$, the potential $\ell_{q}$ and the
mirror objects $F_{q}(a)$ explicitly. This requires heavy computations in the
minimal model $\operatorname{H}\operatorname{Tw}\operatorname{Gtl}_{q}Q$ which
we have performed in [23]. Thanks to these earlier computations, we offer in
the present paper an explicit description of
$\operatorname{Jac}_{q}\check{Q}$, $\ell_{q}$ and $F_{q}(a)$ in terms of
combinatorics in $Q$.

#### Flatness of superpotential deformations

A bottleneck in this paper is the question whether
$\operatorname{Jac}_{q}\check{Q}$ is a (flat) deformation of
$\operatorname{Jac}\check{Q}$ as an algebra. Indeed, our deformed Cho-Hong-Lau
construction leads to a definition of $\operatorname{Jac}_{q}\check{Q}$ as a
mere quotient of $\mathbb{C}\check{Q}\llbracket Q_{0}\rrbracket$ by certain
deformed relations. A quotient by deformed relations however need not be an
algebra deformation in general. The question is whether the specific case of
$\operatorname{Jac}_{q}\check{Q}$ is a deformation of
$\operatorname{Jac}\check{Q}$ nevertheless. To resolve this question, we prove
a flatness result for superpotential deformations of CY3 algebras.

Our flatness result is a culmination of a long sequence of improvements in the
literature. Our starting point is the work of Berger, Ginzburg and Taillefer
[6, 7] which concerns PBW deformations of CY3 algebras. Like all previous
results, their work requires the superpotential $W$ to be homogeneous. We
translate their work to the setting of formal deformations and show that the
homogeneity condition is superfluous and can be replaced by a mild boundedness
condition. We obtain a flatness result for formal deformations of CY3 algebras
with nonhomogeneous superpotential. In particular, it follows from this result
that $\operatorname{Jac}_{q}\check{Q}$ is a flat deformation of
$\operatorname{Jac}\check{Q}$ for almost all dimers $\check{Q}$.

Ultimately, our flatness result renders the category
$\operatorname{mf}(\operatorname{Jac}_{q}\check{Q},\ell_{q})$ a deformation of
$\operatorname{mf}(\operatorname{Jac}\check{Q},\ell)$ and $F_{q}$ an
equivalence of deformations. This proves noncommutative mirror symmetry for
punctured surfaces.

#### Assembling deformed mirror symmetry

The present paper is the final one in a series of three. We explain here the
purpose of this series, which results have been obtained in the first papers
and how we build on them in the present paper.

Our original motivation was to transport Seidel’s idea of relative Fukaya
categories to the world of gentle algebras and to use it as A-side in a
deformed mirror symmetry for punctured surfaces. We realized that an effective
way of constructing the deformation of
$\operatorname{mf}(\operatorname{Jac}\check{Q},\ell)$ was to follow the Cho-
Hong-Lau construction, inserting the deformation $\operatorname{Gtl}_{q}Q$ as
an input instead of $\operatorname{Gtl}Q$. This approach requires an explicit
and lengthy minimal model calculation, which is the reason we distributed the
material into a series of three papers.

The first paper in the series is [13] and concerns the deformation theory of
the gentle algebras $\operatorname{Gtl}\mathcal{A}$ under certain assumptions
on the arc system $\mathcal{A}$. One of the main results is a complete
classification of the deformations of $\operatorname{Gtl}\mathcal{A}$.

The second paper in the series is [23] and conducts all the necessary
computations for applying the Cho-Hong-Lau construction. It focuses on the
case of the specific deformation $\operatorname{Gtl}_{q}Q$ and defines the
category of deformed zigzag paths $\mathbb{L}_{q}$. By means of a deformed
Kadeishvili construction, it builds a minimal model
$\operatorname{H}\mathbb{L}_{q}$ for $\mathbb{L}_{q}$. The main result is an
explicit description of the deformed $A_{\infty}$-structure of
$\operatorname{H}\mathbb{L}_{q}$ in terms of four types of immersed disks.
These four types are labeled CR, ID, DS and DW disks and they agree precisely
with the immersed disks one expects form the relative Fukaya category.

In the present paper, we tie the previous calculations together. We start by
deforming the Cho-Hong-Lau construction in general. Then, we apply this
deformed construction to the special case of $\operatorname{Gtl}_{q}Q$ and
obtain a deformed Jacobi algebra $\operatorname{Jac}_{q}\check{Q}$ and a
deformed central element $\ell_{q}$. Thanks to the second paper, we have an
explicit description of the $A_{\infty}$-structure on
$\operatorname{H}\mathbb{L}_{q}$, giving an explicit description of
$\operatorname{Jac}_{q}\check{Q}$ and $\ell_{q}$. This description is
theoretically given in terms of the CR, ID, DS and DW disks from [23], but
simplifies a bit in the present paper because mostly products of transversal
sequences are regarded. Apart from proving that
$\operatorname{Jac}_{q}\check{Q}$ is indeed a deformation of
$\operatorname{Jac}\check{Q}$, simply plugging in the results of [23] already
finishes deformed mirror symmetry.

#### Structure of the paper

In section 2, we review $A_{\infty}$-categories and their deformations. We
also introduce notation and terminology for treating algebra deformations,
including the $\mathfrak{m}$-adic topology and flatness conditions. In section
3, we present Koszul duality and the relationship between cyclic
$A_{\infty}$-algebras and Calabi-Yau dg algebras. We show how to tweak Koszul
duality in order to obtain $A_{\infty}$-functors similar to the Cho-Hong-Lau
construction. In section 4, we review dimers, gentle algebras and mirror
symmetry of punctured surfaces. In section 5, we review the definition of the
category of deformed zigzag paths $\mathbb{L}_{q}$ and description of its
minimal model $\operatorname{H}\mathbb{L}_{q}$ from [23]. In section 6, we
investigate deformations of Jacobi algebras given by deformations of the
superpotential. We also consider the specific case
$\operatorname{Jac}\check{Q}$ of Jacobi algebras of dimers. In section 7, we
motivate and review the Cho-Hong-Lau construction. We provide an explicit
deformed construction and resolve a few technicalities. In section 8, we apply
the deformed Cho-Hong-Lau construction to the specific case of
$\operatorname{Gtl}_{q}Q$. We provide explicit descriptions of the deformed
Jacobi algebra $\operatorname{Jac}_{q}\check{Q}$, the central element
$\ell_{q}$ and the deformed matrix factorizations $F_{q}(a)$. In Appendix A,
we work out deformed mirror symmetry for the examples of the 3-punctured
sphere and a 4-punctured torus.

### Acknowledgements

This project was supervised by the first author and is part of the second
author’s PhD thesis. The second author thanks the participants of his 2020
Summer Camp on Derived Categories, Stability Conditions and Deformations. The
project was supported by the NWO grant “Algebraic methods and structures in
the theory of Frobenius manifolds and their applications” (TOP1.17.012).

##  2 Preliminaries on $A_{\infty}$-categories

In this section, we recollect background material on $A_{\infty}$-categories
and fix notation. In section 2.1, we recall $A_{\infty}$-categories, their
functors, twisted completion and minimal models. In section 2.2, we recall
completed tensor products, deformations of $A_{\infty}$-categories and their
functors. We very briefly comment on the construction of twisted completion
and minimal model for $A_{\infty}$-deformations from [23]. In section 2.3, we
introduce specific terminology and properties for submodules of
$B\widehat{\otimes}X$ as preparation for section 6. In section 2.4, we recall
$\mathfrak{m}$-adically free modules. In section 2.5, we examine variants of
our flatness condition for ideals.

###  2.1 $A_{\infty}$-categories

In this section we recall $A_{\infty}$-categories, completed tensor products,
$A_{\infty}$-deformations and functors between $A_{\infty}$-deformations. The
material is standard and can for instance be found in [10]. Throughout we work
over an algebraically closed field of characteristic zero and write
$\mathbb{C}$.

######  Definition 2.1.

A ($\mathbb{Z}$\- or $\mathbb{Z}/2\mathbb{Z}$-graded, strictly unital)
$A_{\infty}$-category $\mathcal{C}$ consists of a collection of objects
together with $\mathbb{Z}$\- or $\mathbb{Z}/2\mathbb{Z}$-graded hom spaces
$\operatorname{Hom}(X,Y)$, distinguished identity morphisms
$\mathrm{id}_{X}\in\operatorname{Hom}^{0}(X,X)$ for all $X\in\mathcal{C}$,
together with multilinear higher products

$\mu^{k}:\operatorname{Hom}(X_{k},X_{k+1})\otimes\ldots\otimes\operatorname{Hom}(X_{1},X_{2})\to\operatorname{Hom}(X_{1},X_{k+1}),\quad
k\geq 1$

of degree $2-k$ such that the $A_{\infty}$-relations and strict unitality
axioms hold: For every compatible morphisms $a_{1},\ldots,a_{k}$ we have

$\displaystyle\sum_{0\leq j<i\leq
k}(-1)^{\|a_{n}\|+\ldots+\|a_{1}\|}\mu(a_{k},\ldots,\mu(a_{i},\ldots,a_{j+1}),a_{j},\ldots,a_{1})=0,$
$\displaystyle\mu^{2}(a,\mathrm{id}_{X})=a,~{}\mu^{2}(\mathrm{id}_{Y},a)=(-1)^{|a|}a,~{}\mu^{\geq
3}(\ldots,\mathrm{id}_{X},\ldots)=0.$

Next we recall the additive completion $\operatorname{Add}\mathcal{C}$ of an
$A_{\infty}$-category $\mathcal{C}$. This category consists of formal sums of
shifted objects. The hom space between two objects consists of matrices of
morphisms between the summands.

######  Definition 2.2.

Let $\mathcal{C}$ be an $A_{\infty}$ category with product
$\mu_{\mathcal{C}}$. The additive completion $\operatorname{Add}\mathcal{C}$
of $\mathcal{C}$ is the category of formal sums of shifted objects of
$\mathcal{C}$:

$A_{1}[k_{1}]\oplus\ldots\oplus A_{n}[k_{n}].$

The hom space between two such objects $X=\bigoplus A_{i}[k_{i}]$ and
$Y=\bigoplus B_{i}[m_{i}]$ is

$\operatorname{Hom}_{\operatorname{Add}\mathcal{C}}(X,Y)=\bigoplus_{i,j}\operatorname{Hom}_{\mathcal{C}}(A_{i},B_{j})[m_{j}-k_{i}].$

Here $[-]$ denotes the right-shift. The products on
$\operatorname{Add}\mathcal{C}$ are given by multilinear extensions of

$\mu_{\operatorname{Add}\mathcal{C}}^{k}(a_{k},\ldots,a_{1})=(-1)^{\sum_{j<i}\|a_{i}\|l_{j}}\mu_{\mathcal{C}}^{k}(a_{k},\ldots,a_{1}).$

Here each $a_{i}$ lies in some
$\operatorname{Hom}(X_{i}[k_{i}],X_{i+1}[k_{i+1}])$. The integer $l_{i}$
denotes the difference $k_{i+1}-k_{i}$ between the shifts and the degree
$\|a_{i}\|$ is the degree of $a_{i}$ as element of
$\operatorname{Hom}_{\mathcal{C}}(X_{i},X_{i+1})$.

Next we recall the twisted completion $\operatorname{Tw}\mathcal{C}$ of an
$A_{\infty}$-category $\mathcal{C}$. The objects of this category are virtual
chain complexes of objects of $\mathcal{C}$:

######  Definition 2.3.

A twisted complex in $\mathcal{C}$ is an object
$X\in\operatorname{Add}\mathcal{C}$ together with a morphism
$\delta\in\operatorname{Hom}^{1}_{\operatorname{Add}\mathcal{C}}(X,X)$ of
degree $1$ such that $\delta$ is strictly upper triangular and satisfies the
Maurer-Cartan equation:

$\operatorname{MC}(\delta)\coloneqq\mu^{1}(\delta)+\mu^{2}(\delta,\delta)+\ldots=0.$

We may refer to the morphism $\delta$ as the twisted differential. Note that
the upper triangularity ensures that this sum is well-defined. The twisted
completion of $\mathcal{C}$ is the $A_{\infty}$-category
$\operatorname{Tw}\mathcal{C}$ whose objects are twisted complexes. Its hom
spaces are the same as for the additive completion:

$\operatorname{Hom}_{\operatorname{Tw}\mathcal{C}}(X,Y)=\operatorname{Hom}_{\operatorname{Add}\mathcal{C}}(X,Y).$

The products on $\operatorname{Tw}\mathcal{C}$ of $\mathcal{C}$ are given by
embracing with $\delta$’s:

$\mu_{\operatorname{Tw}\mathcal{C}}^{k}(a_{k},\ldots,a_{1})=\sum_{n_{0},\ldots,n_{k}\geq
0}\mu_{\operatorname{Add}\mathcal{C}}(\underbrace{\delta,\ldots,\delta}_{n_{k}},a_{k},\ldots,a_{1},\underbrace{\delta,\ldots,\delta}_{n_{0}}).$

A functor between two $A_{\infty}$-categories is a mapping which matches the
products of the two categories:

######  Definition 2.4.

Let $\mathcal{C}$ and $\mathcal{D}$ be $A_{\infty}$-categories. Then a functor
$F:\mathcal{C}\to\mathcal{D}$ of $A_{\infty}$-categories consists of a map
$F:\operatorname{Ob}(\mathcal{C})\to\operatorname{Ob}(\mathcal{D})$ together
with for every $k\geq 1$ a degree $1-k$ multilinear map

$F^{k}:\operatorname{Hom}_{\mathcal{C}}(X_{k},X_{k+1})\otimes\ldots\otimes\operatorname{Hom}_{\mathcal{C}}(X_{1},X_{2})\to\operatorname{Hom}_{\mathcal{C}}(FX_{1},FX_{k+1})$

such that the $A_{\infty}$-functor relations hold:

$\sum_{0\leq j<i\leq
k}(-1)^{\|a_{j}\|+\ldots+\|a_{1}\|}F(a_{k},\ldots,a_{i+1},\mu(a_{i},\ldots,a_{j+1}),a_{j},\ldots,a_{1})\\\
=\sum_{\begin{subarray}{c}l\geq 0\\\ 1=j_{1}<\ldots<j_{l}\leq
k\end{subarray}}\mu(F(a_{k},\ldots,a_{j_{l}}),\ldots,F(\ldots,a_{j_{2}}),F(\ldots,a_{j_{1}})).$

The functor $F$ is an isomorphism if
$F:\operatorname{Ob}(\mathcal{C})\to\operatorname{Ob}(\mathcal{D})$ is a
bijection and
$F^{1}:\operatorname{Hom}_{\mathcal{C}}(X,Y)\to\operatorname{Hom}_{\mathcal{D}}(FX,FY)$
is an isomorphism for all $X,Y\in\mathcal{C}$. The functor $F$ is a quasi-
isomorphism if
$F:\operatorname{Ob}(\mathcal{C})\to\operatorname{Ob}(\mathcal{D})$ is a
bijection and
$F^{1}:\operatorname{Hom}_{\mathcal{C}}(X,Y)\to\operatorname{Hom}_{\mathcal{D}}(FX,FY)$
is a quasi-isomorphism of complexes for every $X,Y\in\mathcal{C}$.

######  Definition 2.5.

When $F:\mathcal{C}\to\mathcal{D}$ and $G:\mathcal{D}\to\mathcal{E}$ are
$A_{\infty}$-functors, then their composition is given by
$GF:\operatorname{Ob}(\mathcal{C})\to\operatorname{Ob}(\mathcal{E})$ on
objects and

$(GF)(a_{k},\ldots,a_{1})=\sum G(F(a_{k},\ldots),\ldots,F(\ldots,a_{1})).$

Let us recall minimal models and their notation as follows:

######  Definition 2.6.

An $A_{\infty}$-category $\mathcal{C}$ is minimal if
$\mu^{1}_{\mathcal{C}}=0$. A minimal model of $\mathcal{C}$ is any minimal
$A_{\infty}$-category $\mathcal{D}$ together with a quasi-isomorphism
$F:\mathcal{D}\to\mathcal{C}$. A minimal model of $\mathcal{C}$ is generically
denoted $\operatorname{H}\mathcal{C}$.

By the famous Kadeishvili theorem, every $A_{\infty}$-category has a minimal
model. In fact, a minimal model can be constructed semi-explicitly by sums
over trees.

###  2.2 Deformations of $A_{\infty}$-categories

In this section, we recall deformations of $A_{\infty}$-categories. We follow
[23] where also more detail can be found. We start by recalling completed
tensor products. Then we recall $A_{\infty}$-deformations and their functors.
We comment very briefly on the construction of the twisted completion and
minimal models for $A_{\infty}$-deformations from [23].

We recall now completed tensor products $B\widehat{\otimes}X$ with $B$ a local
ring and $X$ a vector space. The letter $B$ will always denote a local ring
with extra properties. We have decided to give this a name:

######  Definition 2.7.

A deformation base is a complete local Noetherian unital $\mathbb{C}$-algebra
$B$ with residue field $B/\mathfrak{m}=\mathbb{C}$. The maximal ideal is
always denoted $\mathfrak{m}$.

######  Remark 2.1.

By the Cohen structure theorem, every deformation base is of the form
$\mathbb{C}\llbracket x_{1},\ldots,x_{n}\rrbracket/I$ with $I$ denoting some
ideal.

If $X$ is a vector spaces, then
$B\widehat{\otimes}X=\lim(B/\mathfrak{m}^{k}\otimes X)$ denotes the completed
tensor product over $\mathbb{C}$. For simplicity, we write $\mathfrak{m}^{k}X$
to denote the infinitesimal part
$\mathfrak{m}^{k}X=\mathfrak{m}^{k}\widehat{\otimes}X\subseteq
B\widehat{\otimes}X$. Recall that $B\widehat{\otimes}X$ is a $B$-module and
comes with the $\mathfrak{m}$-adic topology, which turns $B\widehat{\otimes}X$
into a sequential Hausdorff space. For convenience, we may from time to time
use expressions like $x=\mathcal{O}(\mathfrak{m}^{k})$ to indicate
$x\in\mathfrak{m}^{k}X$.

######  Definition 2.8.

A map $\varphi:B\widehat{\otimes}X\to B\widehat{\otimes}Y$ is continuous if it
is continuous with respect to the $\mathfrak{m}$-adic topologies. A map
$\varphi:(B\widehat{\otimes}X_{k})\otimes\ldots\otimes(B\widehat{\otimes}X_{1})\to
B\widehat{\otimes}Y$ is continuous if for every $1\leq i\leq k$ and every
sequence of elements $x_{1},\ldots,\hat{x}_{i},\ldots,x_{k}$ the map

$\mu(x_{k},\ldots,-,\ldots,x_{1}):B\widehat{\otimes}X_{i}\to
B\widehat{\otimes}Y$

is continuous.

######  Remark 2.2.

Every element in $B\widehat{\otimes}X$ can be written as a series
$\sum_{i=0}^{\infty}m_{i}x_{i}$. Here $m_{i}$ is a sequence of elements
$m_{i}\in\mathfrak{m}^{\to\infty}$ and $x_{i}$ is a sequence of elements
$x_{i}\in X$. We have used the notation $m_{i}\in\mathfrak{m}^{\to\infty}$ to
indicate that $m_{i}\in\mathfrak{m}^{k_{i}}$ for some sequence
$(k_{i})\subseteq\mathbb{N}$ with $k_{i}\to\infty$.

######  Remark 2.3.

Every $B$-linear map $B\widehat{\otimes}X\to B\widehat{\otimes}Y$ is
automatically continuous (see [23] for the argument), so is every every
$B$-multilinear map
$(B\widehat{\otimes}X_{k})\otimes\ldots\otimes(B\widehat{\otimes}X_{1})\to
B\widehat{\otimes}Y$. Linear maps $X\to B\widehat{\otimes}Y$ can be uniquely
extended to $B$-linear maps $B\widehat{\otimes}X\to B\widehat{\otimes}Y$ and
multilinear maps $X_{k}\otimes\ldots\otimes X_{1}\to B\widehat{\otimes}Y$ can
be uniquely extended to $B$-multilinear maps
$(B\widehat{\otimes}X_{k})\otimes\ldots\otimes(B\widehat{\otimes}X_{1})\to
B\widehat{\otimes}Y$ (see [23]).

######  Remark 2.4.

The leading term of a $B$-linear map
$\varphi:B\widehat{\otimes}X\to\,B\widehat{\otimes}Y$ is the map
$\varphi_{0}:X\to Y$ given by the composition $\varphi_{0}=\pi\varphi|_{X}$,
where $\pi:B\widehat{\otimes}Y\to Y$ denotes the standard projection. If the
leading term $\varphi_{0}$ is injective or surjective, then $\varphi$ is
injective or surjective itself (see [23] for the argument).

We recall now $A_{\infty}$-deformations. When $\mathcal{C}$ is an
$A_{\infty}\,$-category, the idea is to model its $A_{\infty}$-deformations on
the collection of enlarged hom spaces
$\\{B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)\\}_{X,Y\in\mathcal{C}}$.
Any $B$-multilinear product on these hom spaces is automatically continuous.
Similarly, functors of $A_{\infty}$-deformations will be defined as maps
between tensor products of the enlarged hom spaces and will be automatically
continuous as well.

$A_{\infty}$-deformations of $\mathcal{C}$ will always be allowed to have
infinitesimal curvature. The reason is that only this way we get a
homologically sensible notion: Whenever $\mu_{q}$ is an (infinitesimally)
curved deformation, then $\nu=\mu-\mu_{q}$ is a Maurer-Cartan element of the
Hochschild DGLA $\operatorname{HC}(\mathcal{C})$. We comment on this in more
detail in [23].

######  Definition 2.9.

Let $\mathcal{C}$ be an $A_{\infty}$ category with products $\mu$ and $B$ a
deformation base. An $A_{\infty}$-deformation of $\mathcal{C}_{q}$ of
$\mathcal{C}$ consists of

  * •

The same objects as $\mathcal{C}$,

  * •

Hom spaces
$\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y)=B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)$
for $X,Y\in\mathcal{C}$,

  * •

$B$-multilinear products of degree $2-k$

$\mu_{q}^{k}:\operatorname{Hom}_{\mathcal{C}_{q}}(X_{k},X_{k+1})\otimes\ldots\otimes\operatorname{Hom}_{\mathcal{C}_{q}}(X_{1},X_{2})\to\operatorname{Hom}_{\mathcal{C}_{q}}(X_{1},X_{k+1}),~{}k\geq
1$

  * •

Curvature of degree $2$ for every object $X\in\mathcal{C}$

$\mu_{q,X}^{0}\in\mathfrak{m}\operatorname{Hom}_{\mathcal{C}_{q}}^{2}(X,X),$

such that $\mu_{q}$ reduces to $\mu$ once the maximal ideal $\mathfrak{m}$ is
divided out, and $\mu_{q}$ satisfies the curved $A_{\infty}$ ($cA_{\infty}$)
relations

$\sum_{k\geq l\geq m\geq
0}(-1)^{\|a_{m}\|+\ldots+\|a_{1}\|}\mu_{q}(a_{k},\ldots,\mu_{q}(a_{l},\ldots),a_{m},\ldots,a_{1})=0.$

The deformation is unital if the deformed higher products still satisfy the
unitality axioms

$\mu^{2}_{q}(a,\mathrm{id}_{X})=a,~{}\mu^{2}_{q}(\mathrm{id}_{Y},a)=(-1)^{|a|}a,~{}\mu^{\geq
3}_{q}(\ldots,\mathrm{id}_{X},\ldots)=0.$

It sometimes comes handy to work with deformations that include more objects
than $\mathcal{C}$ does. We fix terminology as follows:

######  Definition 2.10.

Let $\mathcal{C}$ be an $A_{\infty}$-category. Let $O$ be an arbitrary set of
objects and $F:O\to\operatorname{Ob}(\mathcal{C})$ a map. Then the object-
cloned version $F^{*}\mathcal{C}$ of $\mathcal{C}$ is the
$A_{\infty}\,$-category given by object set $O$, hom spaces

$\operatorname{Hom}_{F^{*}\mathcal{C}}(X,Y)=\operatorname{Hom}_{\mathcal{C}}(F(X),F(Y)),\quad
X,Y\in O,$

and products simply given by the same composition as in $\mathcal{C}$.

An $A_{\infty}$-deformation of $\mathcal{C}$ always gives an induced
deformation of $F^{*}\mathcal{C}$. This provides a map of Maurer-Cartan
elements
$\operatorname{MC}(\operatorname{HC}(\mathcal{C}),B)\to\operatorname{MC}(\operatorname{HC}(F^{*}\mathcal{C}),B)$.
In case $F$ is surjective, the categories $\mathcal{C}$ and $F^{*}\mathcal{C}$
are equivalent and the map of Maurer-Cartan elements becomes a bijection after
dividing out gauge equivalence. However, the map on raw Maurer-Cartan elements
is not a bijection itself. After these comments, we are ready for the
following terminology:

######  Definition 2.11.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $B$ a deformation base. Let
$O$ be an arbitrary set and $F:O\to\operatorname{Ob}\mathcal{C}$ a map. An
object-cloning deformation is a deformation $\mathcal{D}_{q}$ of
$\mathcal{D}=F^{*}\mathcal{C}$. The object-cloning deformation is essentially
surjective if $F:\operatorname{Ob}\mathcal{D}\to\operatorname{Ob}\mathcal{C}$
reaches all objects of $\mathcal{C}$ up to isomorphism.

We are now ready to explain the natural extension of $A_{\infty}$-functors to
the deformed case.

######  Definition 2.12.

Let $\mathcal{C},\mathcal{D}$ be two $A_{\infty}$-categories and
$\mathcal{C}_{q},\mathcal{D}_{q}$ deformations. A functor of deformed
$A_{\infty}$-categories consists of a map
$F_{q}:\operatorname{Ob}(\mathcal{C})\to\operatorname{Ob}(\mathcal{D})$
together with for every $k\geq 1$ a $B$-multilinear degree $1-k$ map

$F^{k}_{q}:\operatorname{Hom}_{\mathcal{C}_{q}}(X_{k},X_{k+1})\otimes\ldots\otimes\operatorname{Hom}_{\mathcal{C}_{q}}(X_{1},X_{2})\to\operatorname{Hom}_{\mathcal{D}_{q}}(F_{q}X_{1},F_{q}X_{k+1})$

and infinitesimal curvature
$F^{0}_{q,X}\in\mathfrak{m}\operatorname{Hom}^{1}_{\mathcal{D}}(F_{q}X,F_{q}X)$
for every $X\in\mathcal{C}$, such that the curved $A_{\infty}$-functor
relations hold:

$\sum_{0\leq j\leq i\leq
k}(-1)^{\|a_{j}\|+\ldots+\|a_{1}\|}F_{q}(a_{k},\ldots,a_{i+1},\mu_{q}(a_{i},\ldots,a_{j+1}),a_{j},\ldots,a_{1})\\\
=\sum_{\begin{subarray}{c}l\geq 0\\\ 1=j_{1}<\ldots<j_{l}\leq
k\end{subarray}}\mu_{q}(F_{q}(a_{k},\ldots,a_{j_{l}}),\ldots,F_{q}(\ldots,a_{j_{2}}),F_{q}(\ldots,a_{j_{1}})).$

If $\mathcal{C}_{q}$ and $\mathcal{D}_{q}$ are strictly unital, then we say
$F_{q}$ is strictly unital if
$F_{q}^{1}(\mathrm{id}_{X})=\mathrm{id}_{F_{q}X}$ for every $X\in\mathcal{C}$
and $F_{q}^{\geq 2}(\ldots,\mathrm{id}_{X},\ldots)=0$.

######  Remark 2.5.

Note that the functor $F_{q}$ itself is allowed to have a curvature component.
The first two curved $A_{\infty}$-functor relations read

$\displaystyle F^{0}_{q}+F^{1}_{q}(\mu^{0}_{\mathcal{C}_{q},X})$
$\displaystyle=\mu^{1}_{\mathcal{D}_{q}}(F^{0}_{q,X}),$ $\displaystyle
F^{1}_{q}(\mu^{1}_{\mathcal{C}_{q}}(a))+(-1)^{\|a\|}F^{2}_{q}(\mu^{0}_{\mathcal{C}_{q},Y},a)+F^{2}_{q}(a,\mu^{0}_{\mathcal{C}_{q},X})$
$\displaystyle=\mu^{1}_{\mathcal{D}_{q}}(F^{1}_{q}(a))+\mu^{2}_{\mathcal{D}_{q}}(F^{0}_{q,Y},F^{1}_{q}(a))$
$\displaystyle\quad+\mu^{2}_{\mathcal{D}_{q}}(F^{1}_{q}(a),F^{0}_{q,X}),\quad\forall~{}a:X\to
Y.$

If $F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$ is a functor of
$A_{\infty}$-deformations, then its leading term $F:\mathcal{C}\to\mathcal{D}$
is automatically a functor of $A_{\infty}$-categories.

Twisted completions of $A_{\infty}$-deformations exist. When $\mathcal{C}_{q}$
is a deformation of $\mathcal{C}$, we can form a twisted completion
$\operatorname{Tw}\mathcal{C}_{q}$, as we have elaborated in [23]. This
category $\operatorname{Tw}\mathcal{C}_{q}$ is a deformation of
$\operatorname{Tw}\mathcal{C}$. The objects of $\operatorname{Tw}\mathcal{C}$
are defined in terms of twisted differentials as well, but the twisted
differentials do not satisfy the Maurer-Cartan equation with respect to the
deformed product $\mu_{\mathcal{C}_{q}}$. Instead, the failure to satisfy the
Maurer-Cartan equation is captured in the object’s curvature. For more details
we refer to [23].

Minimal models of $A_{\infty}$-deformations exist. When $\mathcal{C}_{q}$ is a
deformation of $\mathcal{C}$, we can form a minimal model
$\operatorname{H}\mathcal{C}_{q}$. This category
$\operatorname{H}\mathcal{C}_{q}$ is a deformation of
$\operatorname{H}\mathcal{C}$. The differential and curvature of
$\operatorname{H}\mathcal{C}_{q}$ need not vanish. Instead,
$\operatorname{H}\mathcal{C}_{q}$ carries an infinitesimal residue
differential and curvature. For more details we refer to [23].

###  2.3 Submodules of completed tensor products

We introduce here the notions of pseudoclosed and quasi-flat submodules of
$B\widehat{\otimes}X$ which we use in section 6. We also comment on
intersections between submodules.

Let us start by introducing pseudoclosed submodules. The rationale is that not
all $B$-submodules of $B\widehat{\otimes}X$ are created equal: Some are closed
under taking power series with increasing powers of $\mathfrak{m}$, some are
not. As preparation, we define the following notation:

######  Definition 2.13.

Let $X$ be a vector space and $Y\subseteq B\widehat{\otimes}X$ a subspace.
Then we put

$\displaystyle BY$
$\displaystyle\coloneqq\operatorname{Im}(B\widehat{\otimes}Y\to
B\widehat{\otimes}X),$ $\displaystyle\mathfrak{m}^{k}Y$
$\displaystyle\coloneqq\operatorname{Im}(\mathfrak{m}^{k}\widehat{\otimes}Y\to
B\widehat{\otimes}X).$

Here, the maps $B\widehat{\otimes}Y\to B\widehat{\otimes}X$ and
$\mathfrak{m}^{k}\widehat{\otimes}Y\to B\widehat{\otimes}X$ denote the
multiplication maps which send for instance $b\otimes y\mapsto by$ and
$m\otimes y\mapsto my$.

######  Remark 2.6.

Explicitly, the spaces $BY$ and $\mathfrak{m}^{k}Y$ are given by elements of
$B\widehat{\otimes}X$ that can be written respectively as

$\displaystyle x$ $\displaystyle=\sum_{i=0}^{\infty}m_{i}y_{i}\in
B\widehat{\otimes}X,\quad m_{i}\in\mathfrak{m}^{\to\infty},~{}y_{i}\in Y,$
$\displaystyle x$ $\displaystyle=\sum_{i=0}^{\infty}m_{i}y_{i}\in
B\widehat{\otimes}X,\quad m_{i}\in\mathfrak{m}^{\geq k,\to\infty},~{}y_{i}\in
Y.$

Here we use the notation $m_{i}\in\mathfrak{m}^{\to\infty}$ to indicate that
there is a sequence $(k_{i})\subseteq\mathbb{N}$ converging to $\infty$ such
that $m_{i}\in\mathfrak{m}^{k_{i}}$. The notation $m_{i}\in\mathfrak{m}^{\geq
k,\to\infty}$ indicates that we shall have $k_{i}\geq k$.

######  Remark 2.7.

We warn that the notation $BY$ is suggestive but should not be misunderstood.
The rationale behind the notation is that $BY$ should contain any $B$-linear
multiples of $Y$ and power series of such elements in growing
$\mathfrak{m}$-order. We emphasize that $BY$ is not the same as the linear
span of products $b\cdot y$ for $b\in B$ and $y\in Y$. Similarly, $BY$ is not
the same as the $\mathfrak{m}$-adic closure of this span. The analogous
warning holds for $\mathfrak{m}^{k}Y$. The intention of the notation is to
provide foundations for section 6.

######  Definition 2.14.

An $B$-submodule $M\subseteq B\widehat{\otimes}X$ is pseudoclosed if
$BM\subseteq M$.

######  Example 2.1.

If $Y\subseteq X$ is a linear subspace, then $B\widehat{\otimes}Y\subseteq
B\widehat{\otimes}X$ is pseudoclosed. In contrast, the $B$-submodule $B\otimes
X\subseteq B\widehat{\otimes}X$ is not pseudoclosed if $X$ is infinite-
dimensional, since $B(B\otimes X)=B\widehat{\otimes}X$.

Denote by $B\cdot Y\subseteq B\widehat{\otimes}X$ the space (finitely) spanned
by elements of the form $by$ with $b\in B$ and $y\in Y$. Denote by
$\mathfrak{m}^{k}\cdot Y$ the space (finitely) spanned by elements of the form
$my$ with $m\in\mathfrak{m}^{k}$ and $y\in Y$. In general, the spaces $BY$ and
$\mathfrak{m}^{k}Y$ are not the same as $B\cdot Y$ and $\mathfrak{m}^{k}\cdot
Y$. Pseudoclosed submodules are an exception:

######  Lemma 2.1.

Let $Y\subseteq B\widehat{\otimes}X$ be a pseudoclosed $B$-submodule. Then
$BY=B\cdot Y$ and $\mathfrak{m}^{k}Y=\mathfrak{m}^{k}\cdot Y$.

###### Proof.

The first statement is obvious since $BY\subseteq Y\subseteq B\cdot Y$. For
the second statement, the idea is to exploit the Cohen structure theorem.
Write $B=\mathbb{C}\llbracket q_{1},\ldots,q_{n}\rrbracket/I$, and regard the
maximal ideal $\mathfrak{m}=(q_{1},\ldots,q_{n})$. With this in mind, we can
write any element $x\in BY$ as a series

$x=\sum_{i=0}^{\infty}m_{i}\tilde{m}_{i}y_{i}.$

Here $m_{i}$ is a monomial of degree $k$ in the variables
$q_{1},\ldots,q_{n}$, the letter $\tilde{m}_{i}$ denotes a sequence
$\tilde{m}_{i}\in\mathfrak{m}^{\to\infty}$, and $y_{i}\in Y$. We conclude

$x=\sum_{\begin{subarray}{c}\text{monomials }M\\\ \text{ of degree
}k\end{subarray}}M\sum_{\begin{subarray}{c}i\geq 0\\\
m_{i}=M\end{subarray}}\tilde{m}_{i}y_{i}.$

The outer sum is finite. For every monomial $M$ of degree $k$, the inner sum
is an element of $Y$ since $Y$ is pseudoclosed by assumption. In conclusion,
every summand of the outer sum lies in $\mathfrak{m}^{k}\cdot Y$, and hence
$x\in\mathfrak{m}^{k}\cdot Y$. We have shown that
$\mathfrak{m}^{k}Y\subseteq\mathfrak{m}^{k}\cdot Y$. The inverse inclusion is
obvious. We conclude $\mathfrak{m}^{k}Y=\mathfrak{m}^{k}\cdot Y$, finishing
the proof. ∎

######  Example 2.2.

A simple application is the case of the pseudoclosed submodule
$Y=BX=B\widehat{\otimes}X$. In this case, the lemma states
$\mathfrak{m}^{k}X=\mathfrak{m}^{k}(BX)=\mathfrak{m}^{k}\cdot(BX)$ as subsets
of $B\widehat{\otimes}X$.

######  Remark 2.8.

We interpret Lemma 2.1 as follows: The space $\mathfrak{m}^{k}\cdot Y$ makes
only reference to the $B$-module structure, while the space
$\mathfrak{m}^{k}Y$ references a mixture of the $B$-module structure with the
topology of the ambient space $B\widehat{\otimes}X$. For pseudoclosed modules,
the topological part is already captured by the algebraic structure.

We introduce now a flatness condition for $B$-submodules of
$B\widehat{\otimes}X$. This flatness condition is particularly relevant in
section 6. To distinguish the notion from existing notions of flatness, we
have chosen to name it quasi-flatness.

######  Definition 2.15.

A $B$-submodule $M\subseteq B\widehat{\otimes}X$ is quasi-flat if
$M\cap\mathfrak{m}X\subseteq\mathfrak{m}M$.

######  Remark 2.9.

The inverse inclusion $\mathfrak{m}M\subseteq M\cap\mathfrak{m}X$ holds
automatically if $M$ is pseudoclosed.

######  Lemma 2.2.

Let $M\subseteq B\widehat{\otimes}X$ be a $B$-submodule. Assume $M$ is quasi-
flat and pseudoclosed. Let $\varphi:\pi(M)\to M$ be a linear section of the
projection map $\pi:M\to\pi(M)$. Then the $B$-linear extension
$\varphi:B\widehat{\otimes}\pi(M)\to M$ is a $B$-linear isomorphism.

###### Proof.

Injectivity follows from Remark 2.4, since the leading term is the identity.
For surjectivity, let $x\in M$. We construct sequences $(x_{k})$ and $(y_{k})$
such that $x=x_{1}+\ldots+x_{N}+y_{N}$ for every $N\in\mathbb{N}$ and
$x_{k}\in\varphi(\mathfrak{m}^{k}\pi(M))$ and $y_{k}\in\mathfrak{m}^{k+1}\cap
M$.

To start with, write $x=\varphi(x_{1})+y_{1}$ for some $x_{1}\in\pi(M)$ and
$y_{1}\in M\cap\mathfrak{m}X$. For induction hypothesis, assume the sequences
are given until index $k$. Then note $y_{k}\in M\cap\mathfrak{m}^{k+1}X$. By
quasi-flatness, we get $y_{k}\in\mathfrak{m}^{k+1}M$. We can then write
$y_{k}=x_{k+1}+y_{k+1}$ with $x_{k+1}\in\varphi(\mathfrak{m}^{k}\pi(M))$ and
$y_{k+1}\in M\cap\mathfrak{m}^{k+1}X$. This finishes the inductive
construction of the sequences.

Finally, we have
$x=\sum_{k=1}^{\infty}x_{k}\in\varphi(B\widehat{\otimes}\pi(M))$. We have
shown that $\varphi$ is surjective. This finishes the proof. ∎

The following is a useful criterion to find quasi-flat modules:

######  Proposition 2.1.

Let $B$ be a deformation base and $X$ a vector space. Let
$M_{1},\ldots,M_{k}\subseteq B\widehat{\otimes}X$ be $B$-submodules. Then the
following are equivalent:

  * •

We have $B\widehat{\otimes}X=M_{1}\oplus\ldots\oplus M_{k}$.

  * •

The modules $M_{i}$ are all pseudoclosed and quasi-flat, and
$X=\pi(M_{1})\oplus\ldots\oplus\pi(M_{k})$.

Here $\pi:B\widehat{\otimes}X\to X$ denotes the canonical projection map.

###### Proof.

We first show that $B\widehat{\otimes}X=M_{1}\oplus\ldots\oplus M_{k}$ implies
that all $M_{i}$ are quasi-flat and
$X=\pi(M_{1})\oplus\ldots\oplus\pi(M_{k})$. After that, we show the converse.

For the first part, assume $B\widehat{\otimes}X=M_{1}\oplus\ldots\oplus
M_{k}$. Let $1\leq i\leq n$. We first show that $M_{i}$ is closed. Indeed,
regard the projection $p_{i}:B\widehat{\otimes}X\to B\widehat{\otimes}X$ to
the component $M_{i}$. This map is clearly $B$-linear, hence
$\mathrm{id}-p_{i}$ is $B$-linear. By Remark 2.3, the map $\mathrm{id}-p_{i}$
is continuous. The kernel of $\mathrm{id}-p_{i}$ is $M_{i}$ and we conclude
that $M_{i}$ is closed. In particular, $M_{i}$ is pseudoclosed.

Next, we show that every $M_{i}$ is quasi-flat. Pick any
$x\in\mathfrak{m}X\cap M_{i}$. Since $x\in\mathfrak{m}X$ and
$B\widehat{\otimes}X=M_{1}+\ldots+M_{k}$, we can write $x=y_{1}+\ldots+y_{k}$
with $y_{j}\in\mathfrak{m}M_{j}$. By pseudoclosedness we have
$y_{j}\in\,\mathfrak{m}M_{j}\subseteq M_{j}$. Since the sum
$M_{1}\oplus\ldots\oplus M_{k}$ is direct and $x\in M_{i}$, we get $y_{j}=0$
for $j\neq i$. We conclude $x=y_{i}\in\mathfrak{m}M_{i}$. This proves every
$M_{i}$ quasi-flat.

Let us show that $X=\pi(M_{1})+\ldots+\pi(M_{k})$. Pick any $x\in X$. Since
$x$ then also lies in $B\widehat{\otimes}X$, write $x=y_{1}+\ldots+y_{k}$ with
$y_{i}\in M_{i}$. We conclude $x=\pi(x)=\pi(y_{1})+\ldots+\pi(y_{k})$,
therefore $x\in\pi(M_{1})+\ldots+\pi(M_{k})$. Since $x$ was arbitrary, this
shows $X=\pi(M_{1})+\ldots+\pi(M_{k})$.

Let us show that the sum $\pi(M_{1})+\ldots+\pi(M_{k})$ is direct. Assume by
contradiction there is a sequence $y_{1},\ldots,y_{k}$ with $y_{i}\in M_{i}$
and $\pi(y_{1})+\ldots+\pi(y_{k})=0$. Then $\pi(\sum y_{i})=0$, in other words
$\sum y_{i}\in\mathfrak{m}X$. By assumption we have
$B\widehat{\otimes}X=M_{1}+\ldots+M_{k}$, in particuar
$\mathfrak{m}X\subseteq\mathfrak{m}(M_{1}+\ldots+M_{k})$. Therefore we can
write $\sum y_{i}=\sum z_{i}$ with $z_{i}\in\mathfrak{m}M_{i}$. Since the sum
$M_{1}\oplus\ldots\oplus M_{k}$ is direct, we have $y_{i}=z_{i}$ for all $i$.
We conclude $y_{i}=z_{i}\in\mathfrak{m}X$ and therefore $\pi(y_{i})=0$ for all
$i$. This shows that the sum $\pi(M_{1})+\ldots+\pi(M_{k})$ is direct. The
first implication is proven, finishing the first part of the proof.

For the second part of the proof, assume every $M_{i}$ is pseudoclosed and
quasi-flat and $X=\pi(M_{1})\oplus\ldots\oplus\pi(M_{k})$. Choose a linear
section $\pi(M_{i})\to M_{i}$ of the projection $\pi:M_{i}\to\pi(M_{i})$.
According to Lemma 2.2, the $B$-linear extension
$\varphi_{i}:B\widehat{\otimes}\pi(M_{i})\to M_{i}$ is an isomorphism. Add up
all $\varphi_{i}$ to arrive at the map

$\displaystyle\varphi:B\widehat{\otimes}(\pi(M_{1})\oplus\ldots\oplus\pi(M_{k}))$
$\displaystyle\to B\widehat{\otimes}X,$ $\displaystyle(x_{1},\ldots,x_{k})$
$\displaystyle\mapsto\varphi_{1}(x_{1})+\ldots+\varphi_{k}(x_{k}).$

Note we view $\pi(M_{1})\oplus\ldots\oplus\pi(M_{k})$ simply as vector space
decomposition of $X$. The map $\varphi$ is $B$-linear and continuous. Its
leading term is by construction the identity on $X$. Therefore $\varphi$ is an
isomorphism.

Using the auxiliary map $\varphi$, we get that
$M_{1}+\ldots+M_{k}=B\widehat{\otimes}X$: By definition of $\varphi$, its
image is necessarily contained in $M_{1}+\ldots+M_{k}$ and we conclude that
$M_{1}+\ldots+M_{k}=B\widehat{\otimes}X$.

Let us now show that the sum $M_{1}+\ldots+M_{k}$ is direct. Pick any sequence
$x_{1},\ldots,x_{k}$ such that $x_{i}\in M_{i}$ and $x_{1}+\ldots+x_{k}=0$.
Since $\varphi_{i}$ surjects on $M_{i}$, write $x_{i}=\varphi_{i}(y_{i})$ with
$y_{i}\in B\widehat{\otimes}\pi(M_{i})$. We get that
$\varphi(y_{1}+\ldots+y_{k})=0$. Since $\varphi$ is injective, we get
$y_{1}+\ldots+y_{k}=0$, hence $y_{i}=0$ for all $i$. This shows
$x_{i}=\varphi_{i}(y_{i})=0$ and we conclude that the sum
$M_{1}\oplus\ldots\oplus M_{k}$ is direct. This finishes the proof. ∎

We finish this section by explaining a property regarding intersections of
$B$-submodules. Whenever $X,Y\subseteq B\widehat{\otimes}A$ are two subspaces,
we may ask: Does it hold that

$\mathfrak{m}X\cap\mathfrak{m}Y=\mathfrak{m}(X\cap Y)~{}?$

This inclusion does not hold in general, but we shall give here the best
possible variant in case one of the spaces $X,Y$ is not deformed. Let us start
with an example where the inclusion fails, as well as the example
$B=\mathbb{C}\llbracket q\rrbracket$.

######  Example 2.3.

Regard $B=\mathbb{C}\llbracket p,q\rrbracket$ and $A=\mathbb{C}[X]$. Let
$X=\operatorname{span}(px)\subseteq B\widehat{\otimes}A$ and
$Y=\operatorname{span}(qx)\subseteq B\widehat{\otimes}A$. Then $pqx$ lies in
$\mathfrak{m}X\cap\mathfrak{m}Y$, but not in $\mathfrak{m}(X\cap Y)$.

######  Example 2.4.

Regard $B=\mathbb{C}\llbracket q\rrbracket$ and arbitrary $A$. Let
$X,Y\subseteq B\widehat{\otimes}A$ be pseudoclosed. We claim that
$(q)X\cap(q)Y\subseteq(q)(X\cap Y)$. Indeed, pick $z\in(q)X\cap(q)Y$. By
definition of $(q)X$, we can write

$z=\sum q^{\geq 1,\to\infty}x_{i}=q\sum q^{\to\infty}x_{i},\quad z=\sum
q^{\geq 1,\to\infty}y_{i}=q\sum q^{\to\infty}y_{i}.$

Since $X$ and $Y$ are pseudoclosed, the two sums $\sum q^{\to\infty}x_{i}$ and
$\sum q^{\to\infty}y_{i}$ lie in $X$ and $Y$, respectively. Write
$z=qz^{\prime}$. Then both sums are equal to $z^{\prime}$, hence $z^{\prime}$
lies in the intersection of $X$ and $Y$. We conclude $z=qz^{\prime}\in q(X\cap
Y)$. This shows $(q)X\cap(q)Y\subseteq(q)(X\cap Y)$.

Let us now make a more general statement. The idea is to keep one of the
spaces $X,Y$ non-deformed. In other words, one of them is simply of the form
$B\widehat{\otimes}V$ for some subspace $V\subseteq A$. Let us make this
precise as follows:

######  Proposition 2.2.

Let $X$ be a vector space and $B$ a deformation base. Let $V\subseteq X$ be a
subspace and $M\subseteq B\widehat{\otimes}X$ a pseudoclosed $B$-submodule.
Then

$\mathfrak{m}^{k}M\cap(\mathfrak{m}^{k}V+\mathfrak{m}^{k+1}X)\subseteq\mathfrak{m}^{k}(M\cap(V+\mathfrak{m}X))+\mathfrak{m}^{k+1}M.$

###### Proof.

In the first part of the proof, we illustrate the statement in case of
$B=\mathbb{C}\llbracket q\rrbracket$. In the second part, we build a
commutative diagram which allows us to deduce the shape of elements of
$\mathfrak{m}^{k}M\cap\mathfrak{m}^{k}V$ without choice of basis for $B$. In
the third part, we conclude the desired statement.

For the first step, let us illustrate the case of $B=\mathbb{C}\llbracket
q\rrbracket$. Let $q^{k}x\in(q)^{k}M\cap((q)^{k}V+(q)^{k+1}X)$. In particular,
we have $q^{k}x\in(q)^{k}M$, hence $x\in M$ since $M$ is pseudoclosed. We also
have $q^{k}x\in(q)^{k}V+(q)^{k+1}X$, hence $x\in\mathbb{C}\llbracket
q\rrbracket V+(q)X$. Together this shows $q^{k}x\in q^{k}(M\cap(V+(q)X))$. We
conclude that in case $B=\mathbb{C}\llbracket q\rrbracket$ the claimed
statement holds.

For the second step, we build the following commutative diagram of linear
maps:

${\mathfrak{m}^{k}\widehat{\otimes}M}$${\frac{\mathfrak{m}^{k}}{\mathfrak{m}^{k+1}}\otimes\frac{X}{V}}$${\frac{\mathfrak{m}^{k}X}{\mathfrak{m}^{k+1}X+\mathfrak{m}^{k}V}}$$\scriptstyle{\varphi~{}=~{}\pi_{i}\otimes(\pi_{V}\circ\pi_{0})}$$\scriptstyle{\pi\circ
c}$$\scriptstyle{\psi}$$\scriptstyle{\sim}$

Let us explain the maps. The horizontal map $\varphi$ performs a projection to
$\mathfrak{m}^{k}/\mathfrak{m}^{k+1}$ on the first tensor factor and an
inclusion of $M$ into $B\widehat{\otimes}X$ followed by projection to $X$ and
projection to $X/V$ on the second tensor factor. The left vertical map
$\pi\circ c$ consists of the inclusion $M\subseteq B\widehat{\otimes}X$ on the
second tensor factor, followed by multiplication with the first tensor factor
and projection to the quotient by $\mathfrak{m}^{k+1}X+\mathfrak{m}^{k}V$. The
right vertical map $\psi$ is induced from the multiplication map.

We claim that the diagram is commutative and $\psi$ is an isomorphism. To see
commutativity, pick an element $m\otimes x$ with $m\in\mathfrak{m}^{k}$ and
$x\in M$. Under $\varphi$ it is sent to $[m]\otimes[x]$, which under $\psi$ is
sent to $[mx]$. Under $\pi\circ c$, the element $m\otimes x$ is also sent to
$[mx]$. This demonstrates commutativity. To see that $\psi$ is an isomorphism,
recall that $\mathfrak{m}^{k}X$ and $\mathfrak{m}^{k}\widehat{\otimes}X$ are
isomorphic by means of the splitting map
$\mathfrak{m}^{k}X\to\mathfrak{m}^{k}\widehat{\otimes}X$. This splitting map
induces a map

$\frac{\mathfrak{m}^{k}X}{\mathfrak{m}^{k+1}X+\mathfrak{m}^{k}V}\to\frac{\mathfrak{m}^{k}}{\mathfrak{m}^{k+1}}\otimes\frac{X}{V}.$

This map is an inverse of $\psi$. This shows that $\psi$ is an isomorphism.

For the third part of the proof, we conclude the desired inclusion. Let us
start with the remark that the kernel of $\varphi$ is equal to

$\operatorname{Ker}(\varphi)=\mathfrak{m}^{k+1}\widehat{\otimes}M+\mathfrak{m}^{k}\widehat{\otimes}(M\cap(V+\mathfrak{m}X)).$
(2.1)

Let now
$\sum_{i=0}^{\infty}m_{i}x_{i}\in\mathfrak{m}^{k}M\cap(\mathfrak{m}^{k}V+\mathfrak{m}^{k+1}X)$
with $m_{i}\in\mathfrak{m}^{\geq k,\to\infty}$ and $x_{i}\in M$. Then

$(\pi\circ c)\left(\sum m_{i}\otimes x_{i}\right)=\sum m_{i}\pi_{0}(x_{i})=0.$

Since $\pi\circ c=\psi\varphi$ and $\psi$ is injective, we get $\varphi(\sum
m_{i}\otimes x_{i})=0$. Therefore $\sum m_{i}\otimes x_{i}$ lies in the kernel
of $\varphi$, which explicitly reads

$\sum m_{i}\otimes
x_{i}\in\mathfrak{m}^{k+1}\widehat{\otimes}M+\mathfrak{m}^{k}\widehat{\otimes}(M\cap(V+\mathfrak{m}X)).$

Contracting the tensors gives

$\sum
m_{i}x_{i}\in\mathfrak{m}^{k+1}M+\mathfrak{m}^{k}(M\cap(V+\mathfrak{m}X)).$

Since $\sum m_{i}x_{i}$ was arbitrarily chosen, this finishes the proof. ∎

###  2.4 $\mathfrak{m}$-adically free modules

In this section, we recall the notion of $\mathfrak{m}$-adically free modules
and their use in $A_{\infty}$-deformations. The reason is that in section 7,
it comes very handy to use $A_{\infty}$-deformations modeled on $B$-modules
which are only noncanonically isomorphic to
$B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)$. In this section, we
first recall the definition of $\mathfrak{m}$-adically free modules, then tie
it back to quasi-flat and pseudoclosed modules. After that, we provide the
definition of $A_{\infty}$-deformations that makes use of
$\mathfrak{m}$-adically free modules.

We start by recalling the notion of $\mathfrak{m}$-adically free modules. We
pick the following definition, as in [34]:

######  Definition 2.16.

A $B$-module $M$ is $\mathfrak{m}$-adically free if there is a vector space
$X$ such that $M\cong B\widehat{\otimes}X$ as $B$-modules.

An abstract $B$-module $M$ enjoys an $\mathfrak{m}$-adic topology given by the
neighborhood basis $x+\mathfrak{m}^{k}\cdot M$ for every $x\in M$ and
$k\in\mathbb{N}$. We claim that when $M\cong B\widehat{\otimes}X$, then this
topology is automatically compatible with the $\mathfrak{m}$-adic topology on
$B\widehat{\otimes}X$:

######  Lemma 2.3.

Let $M$ be a $B$-module and $\varphi:M\to B\widehat{\otimes}X$ be a $B$-linear
isomorphism. Then $\varphi$ is a homeomorphism.

###### Proof.

By bijectivity and $B$-linearity of $\varphi$ we have
$\varphi(\mathfrak{m}^{k}\cdot M)=\mathfrak{m}^{k}\cdot(B\widehat{\otimes}X)$.
By Lemma 2.1, we have
$\mathfrak{m}^{k}\cdot(B\widehat{\otimes}X)=\mathfrak{m}^{k}X$. This shows
that $\varphi$ is a homeomorphism. ∎

######  Lemma 2.4.

Let $M,N$ be $\mathfrak{m}$-adically free $B$-modules and $\varphi:M\to N$ a
$B$-linear map. Then $\varphi$ is automatically continuous.

###### Proof.

Since $\varphi$ is $B$-linear, we have $\varphi(\mathfrak{m}^{k}\cdot
M)\subseteq\mathfrak{m}^{k}\cdot N$. This proves $\varphi$ continuous. ∎

The ad-hoc quasi-flatness condition $M\cap\mathfrak{m}X\subseteq\mathfrak{m}M$
is related to $\mathfrak{m}$-adic freeness. While the former is a condition
that makes explicit reference to the ambient space, the latter depends only on
the abstract $B$-module structure. Both are not equivalent, but we provide
here the closest tie we can get.

######  Proposition 2.3.

Let $M\subseteq B\widehat{\otimes}X$ be a $B$-submodule. Then the following
are equivalent:

  * •

$M$ is quasi-flat and pseudoclosed.

  * •

There is a $B$-linear isomorphism $B\widehat{\otimes}Y\to M$ with injective
leading term $Y\to X$.

###### Proof.

Assume $M$ is quasi-flat and pseudoclosed. Then according to lemma we get a
$B$-linear isomorphism $B\widehat{\otimes}\pi(M)\xrightarrow{\sim}M$ with
leading term the identity. This shows the claim, in particular $M$ is
$\mathfrak{m}$-adically free.

Conversely, assume $M$ is $\mathfrak{m}$-adically free, presented by an
injective leading term. Then we have an isomorphism
$\varphi:B\widehat{\otimes}Y\to M\subseteq\,B\widehat{\otimes}X$. The map is
automatically continuous. We show that $M$ is pseudoclosed: Let $\sum
m_{i}x_{i}$ be a series with $m_{i}\in\mathfrak{m}^{\to\infty}$ and $x_{i}\in
M$. Then write $x_{i}=\varphi(y_{i})$. We get $\sum m_{i}y_{i}\in
B\widehat{\otimes}Y$ and $\varphi(\sum m_{i}y_{i})=\sum
m_{i}\varphi(y_{i})=\sum m_{i}x_{i}$. This shows $\sum
m_{i}x_{i}\in\operatorname{Im}(\varphi)=M$. Hence $M$ is pseudoclosed.

To show that $M$ is quasi-flat, pick an element of $M\cap\mathfrak{m}X$,
written in the form $\varphi(x)\in M\cap\mathfrak{m}X$ with $x\in
B\widehat{\otimes}Y$. We claim $x\in\mathfrak{m}Y$. Write $x=y+z$ with $y\in
Y$ and $z\in\mathfrak{m}Y$. Then $\varphi(x)=\varphi(y)+\varphi(z)$. Hence
$\varphi(y)\in\mathfrak{m}X$. In particular $y$ vanishes under the leading
term $\pi\varphi:Y\to X$. By assumption, the leading term is injective and we
get $y=0$. This shows $\varphi(x)=\varphi(z)\in\mathfrak{m}M$. This shows
$M\cap\mathfrak{m}X\subseteq\mathfrak{m}M$. ∎

######  Remark 2.10.

It is not true that $M\subseteq B\widehat{\otimes}X$ is quasi-flat and
pseudoclosed if and only if it is $\mathfrak{m}$-adically free. For instance,
let $X=\operatorname{span}(x_{1},x_{2})$ and $B=\mathbb{C}\llbracket
q\rrbracket$. Regard the space $M=(q)x_{1}+\mathbb{C}\llbracket q\rrbracket
x_{2}$. The module $M$ is $\mathfrak{m}$-adically free through the isomorphism
$B\widehat{\otimes}X\to M$ given by $x_{1}\mapsto qx_{1}$ and $x_{2}\mapsto
x_{2}$. However $M$ is not quasi-flat since $qx_{1}\in((q)X\cap
M)\setminus(q)M$.

One can use $\mathfrak{m}$-adically free modules to model
$A_{\infty}$-deformations. To be more precise, we have so far defined
$A_{\infty}$-deformations as (infinitesimally curved) $A_{\infty}$-structures
on the completed tensor product
$B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)$. It is possible to
also allow arbitrary $\mathfrak{m}$-adically free $B$-modules instead, under
the condition that the quotient by $\mathfrak{m}$ is
$\operatorname{Hom}_{\mathcal{C}}(X,Y)$. We greatly profit from this variant
in section 7.

######  Definition 2.17.

Let $\mathcal{C}$ be a $\mathbb{Z}$-graded (or
$\mathbb{Z}/2\mathbb{Z}$-graded) $A_{\infty}\,$-category. A loose
$A_{\infty}\,$-deformation of $\mathcal{C}$ is a collection of
$\mathfrak{m}$-adically free $\mathbb{Z}$-graded (or
$\mathbb{Z}/2\mathbb{Z}$-graded) $B$-modules
$\\{\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y)\\}_{X,Y\in\mathcal{C}}$ together
with $B$-multilinear maps $\mu_{q}^{k\geq 0}$ of degree $2-k$ satisfying the
curved $A_{\infty}$-relations, together with linear isomorphisms
$\psi_{X,Y}:\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y)/(\mathfrak{m}\cdot\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y))\xrightarrow{\sim}\operatorname{Hom}_{\mathcal{C}}(X,Y)$
for every $X,Y$, such that $\mathcal{C}$ is obtained by dividing out
$\mathfrak{m}$ and identification via $\\{\psi_{X,Y}\\}$.

######  Definition 2.18.

Let $\mathcal{C},\mathcal{D}$ be two $A_{\infty}$-categories and
$\mathcal{C}_{q},\mathcal{D}_{q}$ be loose $A_{\infty}$-deformations. A
functor of loose $A_{\infty}$-deformations
$F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$ consists of maps
$F_{q}:\operatorname{Ob}\mathcal{C}\to\operatorname{Ob}\mathcal{D}$ together
with $B$-multilinear maps $F_{q}^{k\geq 0}$ such that $F_{q}$ satisfies the
curved $A_{\infty}$-functor relations and
$F_{q,X}^{0}\in\mathfrak{m}\cdot\operatorname{Hom}_{\mathcal{C}_{q}}(F_{q}(X),F_{q}(X))$
for every $X\in\mathcal{C}$. The leading term of $F_{q}$ is the functor
$F:\mathcal{C}\to\mathcal{D}$ obtained by dividing out $\mathfrak{m}$ and
identification via $\\{\psi_{X,Y}\\}$. We may also say that $F_{q}$ is a
deformation of $F$ in this case.

######  Remark 2.11.

In contrast to $A_{\infty}$-deformations modeled on
$B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)$, a loose
$A_{\infty}$-deformation does not directly give a Maurer-Cartan element of the
Hochschild DGLA $\operatorname{HC}(\mathcal{C})$. Instead, one first needs to
make a choice of $B$-linear identification
$\varphi_{X,Y}:\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y)\cong
B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)$ for every
$X,Y\in\mathcal{C}$. Of course, the identification needs to be compatible with
$\psi_{X,Y}$ in the sense that its leading term must be the identity when
identifying
$\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y)/(\mathfrak{m}\cdot\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y))$
via $\psi_{X,Y}$. In yet other words, the following diagram needs to commute:

${\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y)}$${B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)}$${\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y)/(\mathfrak{m}\cdot\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y))}$${\operatorname{Hom}_{\mathcal{C}}(X,Y).}$$\scriptstyle{\varphi_{X,Y}}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{\psi_{X,Y}}$

Once choices have been made, one obtains a Maurer-Cartan element
$\mu_{q,\varphi}\in\operatorname{MC}(\operatorname{HC}(\mathcal{C}),B)$. Let
us explain why two different choices $\varphi,\varphi^{\prime}$ yield gauge-
equivalent Maurer-Cartan elements: Both $\varphi_{X,Y}$ and
$\varphi^{\prime}_{X,Y}$ have leading term the identity when identified via
$\psi_{X,Y}$, hence the composition

$\varphi^{\prime}_{X,Y}\circ\varphi^{-1}_{X,Y}:B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)\to
B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)$

has leading term the identity (without any identification). This shows that
the two Maurer-Cartan elements $\mu_{q,\varphi}$ and
$\mu_{q,\varphi^{\prime}}$ are related by the strict gauge functor
$\varphi^{\prime}\circ\varphi^{-1}:(B\widehat{\otimes}\mathcal{C},\mu_{q,\varphi})\to(B\widehat{\otimes}\mathcal{C},\mu_{q,\varphi^{\prime}})$.

In analogy to Definition 2.11, we fix the following terminology:

######  Definition 2.19.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $B$ a deformation base. Let
$O$ be an arbitrary set and $F:O\to\operatorname{Ob}\mathcal{C}$ a map. A
loose object-cloning deformation is a loose deformation $\mathcal{D}_{q}$ of
$\mathcal{D}=F^{*}\mathcal{C}$. The loose object-cloning deformation is
essentially surjective if
$F:\operatorname{Ob}\mathcal{D}\to\operatorname{Ob}\mathcal{C}$ reaches all
objects of $\mathcal{C}$ up to isomorphism.

###  2.5 On the quasi-flatness condition

In this section, we present two alternative ways to formulate the quasi-
flatness condition that we studied in section 2.3. This serves as a
preparation for later use in section 6. The two alternatives for the quasi-
flatness inclusion $M\cap\mathfrak{m}A\subseteq\mathfrak{m}M$ read as follows:

######  Definition 2.20.

Let $M\subseteq B\widehat{\otimes}X$ be a $B$-submodule. Then $M$ satisfies
the

  * •

weak quasi-flatness inclusion if for every $k\geq 1$ we have
$M\cap\mathfrak{m}^{k}X\subseteq\mathfrak{m}^{k}M+\mathfrak{m}^{k+1}X$.

  * •

strong quasi-flatness inclusion if for every $k\geq 1$ we have
$M\cap\mathfrak{m}^{k}X\subseteq\mathfrak{m}^{k}M$.

We claim these alternative inclusions are indeed equivalent to quasi-flatness
and moreover that any quasi-flat $B$-submodule $M\subseteq
B\widehat{\otimes}X$ is automatically a closed subspace of
$B\widehat{\otimes}X$:

######  Proposition 2.4.

Let $X$ be a vector space and $B$ a deformation base. Let $M\subseteq
B\widehat{\otimes}X$ be a $B$-submodule. If $M$ is pseudoclosed, then the
following are equivalent:

  * •

$M$ is quasi-flat.

  * •

$M$ is quasi-flat and closed.

  * •

$M$ satisfies the weak quasi-flatness inclusion.

  * •

$M$ satisfies the strong quasi-flatness inclusion.

In the following lemmas, we provide a proof of Proposition 2.4.

######  Remark 2.12.

The proofs are easier to understand if one has the example of
$B=\mathbb{C}\llbracket q\rrbracket$ in mind. This case has the practical
property that any element $x\in\mathfrak{m}X$ can automatically be written as
$qy$ for some $y\in X$. If for example it now becomes known that
$x\in\mathfrak{m}M$, then it is immediate that $y\in M$. To see this, write
$x\in(q)M$ as a power series in elements of $M$ and divide by $q$:

$\displaystyle qy$
$\displaystyle=x=\sum_{n=0}^{\infty}q^{n+1}x_{n}=q\cdot\sum_{n=0}^{\infty}q^{\to\infty}x_{n},\quad\text{hence}\quad
y=\sum_{n=0}^{\infty}q^{n}x_{n}\in M.$

######  Lemma 2.5.

Let $M\subseteq B\widehat{\otimes}X$ be a pseudoclosed $B$-submodule. If $M$
satisfies the weak quasi-flatness inclusion, then $M$ is quasi-flat.

###### Proof.

It is our task to show $M\cap\mathfrak{m}X\subseteq\mathfrak{m}M$. Pick $x\in
M\cap\mathfrak{m}X$. Iterating the weak quasi-flatness inclusion in
combination with pseudoclosedness gives

$x~{}\in~{}M\cap\mathfrak{m}X~{}\subseteq~{}\mathfrak{m}M+M\cap\mathfrak{m}^{2}X~{}\subseteq~{}\mathfrak{m}M+\mathfrak{m}^{2}M+M\cap\mathfrak{m}^{3}X~{}\subseteq~{}\ldots.$

More precisely, write $x=x_{1}+y_{1}$ with $x_{1}\in\mathfrak{m}M$ and
$y_{1}\in M\cap\mathfrak{m}^{2}X$. Then write $y_{1}=x_{2}+y_{2}$ with
$x_{2}\in\mathfrak{m}^{2}M$ and $y_{2}\in M\cap\mathfrak{m}^{3}X$. Continuing
this way, we obtain sequences $(x_{k})$ and $(y_{k})$ with the property that

$x=(x_{1}+\ldots+x_{N})+y_{N},\quad x_{k}\in\mathfrak{m}^{k}M,~{}y_{k}\in
M\cap\mathfrak{m}^{k+1}X.$

Letting $N\to\infty$ we get within $B\widehat{\otimes}X$ that

$x=\sum_{k=1}^{\infty}x_{k}.$

In principle, the right-hand side converges within the completion of $M$.
Since we assumed that $M$ is pseudoclosed, we can however do better: Every
summand $x_{k}$ lies in $\mathfrak{m}^{k}M$ and summation starts at $k=1$.
Therefore the infinite sum lies in $\mathfrak{m}M$. We conclude that
$x\in\mathfrak{m}M$. This shows $M\cap\mathfrak{m}X\subseteq\mathfrak{m}M$. ∎

######  Lemma 2.6.

Let $M\subseteq B\widehat{\otimes}X$ be a quasi-flat $B$-submodule. Then $M$
satisfies the strong quasi-flatness inclusion.

###### Proof.

The proof consists of two parts: We first prove the auxiliary inclusion
$\mathfrak{m}^{k}M\cap\mathfrak{m}^{k+1}X\subseteq\mathfrak{m}^{k+1}M$ for
every $k\geq 1$. Second, we derive the strong quasi-flatness inclusion by
iterating the auxiliary inclusion.

First, let us prove the auxiliary inclusion. Denoting by
$\pi:B\widehat{\otimes}X\to X$ the standard projection, regard the subspace
$\pi(M)\subseteq X$. We can choose a $B$-linear continuous map
$\varphi:B\widehat{\otimes}\pi(M)\to M$ with leading term the identity. Then
any $x\in M$ can be written as $x=\varphi(y)+z$ with $z\in
M\cap\mathfrak{m}\subseteq\mathfrak{m}M$.

Let $x\in\mathfrak{m}^{k}M\cap\mathfrak{m}^{k+1}X$. Then we can write
$x=\varphi(y)+z$ with $y\in\mathfrak{m}^{k}\pi(M)$ and
$z\in\mathfrak{m}^{k+1}M$. We get that $y-\varphi(y)\in\mathfrak{m}^{k+1}X$
and $\varphi(y)=x-z\in\mathfrak{m}^{k+1}X$. Summing up, we get
$y\in\mathfrak{m}^{k+1}X\cap\mathfrak{m}^{k}\pi(M)$, hence
$y\in\mathfrak{m}^{k+1}\pi(M)$. In consequence, we have
$\varphi(y)\in\mathfrak{m}^{k+1}M$. Finally, we get
$x=\varphi(y)+z\in\mathfrak{m}^{k+1}M$. This proves the auxiliary inclusion.

Finally, we combine quasi-flatness with iterated applications of the auxiliary
inclusion:

$\displaystyle M\cap\mathfrak{m}^{k}X$
$\displaystyle=(M\cap\mathfrak{m}X)\cap\mathfrak{m}^{k}X$
$\displaystyle\subseteq(\mathfrak{m}M\cap\mathfrak{m}^{2}X)\cap\mathfrak{m}^{k}X$
$\displaystyle\subseteq(\mathfrak{m}^{2}M\cap\mathfrak{m}^{3}X)\cap\mathfrak{m}^{k}X$
$\displaystyle\subseteq\ldots\subseteq\mathfrak{m}^{k}M.$

This finishes the proof. ∎

######  Lemma 2.7.

Let $M\subseteq B\widehat{\otimes}X$ be a pseudoclosed and quasi-flat
$B$-submodule. Then $M$ is closed.

###### Proof.

Let $\sum x_{n}$ be a series of elements $x_{n}\in M$ that converges in
$B\widehat{\otimes}X$. Then $x_{n}\in\mathfrak{m}^{\to\infty}X\cap M$. By the
strong quasi-flatness inclusion, we get $x_{n}\in\mathfrak{m}^{\to\infty}M$.
The limit of the series hence lies in $BM$. Since $M$ is pseudoclosed, the
limit lies in $M$. We conclude that $M$ is closed. ∎

The combination of the above lemmas proves Proposition 2.4. For
$B$-submodules, pseudoclosed and quasi-flat implies closed, and closed implies
pseudoclosed. However closed does not imply quasi-flat, for instance the
submodule $M=\mathbb{C}\llbracket q\rrbracket
x_{1}+(q)x_{2}\subseteq\mathbb{C}\llbracket
q\rrbracket\widehat{\otimes}\operatorname{span}(x_{1},x_{2})$ is closed but
not quasi-flat.

##  3 Preliminaries on Koszul duality

In the present section, we present Koszul duality as a preparation for section
7. Koszul duality is a phenomenon which provides a rich source of nontrivial
$A_{\infty}$-functors by matching $A_{\infty}$-algebras and dg algebras.
Classical Koszul duality involves only ordinary (associative, non-dg) algebras
and makes best-effort statements on their homological properties. Modern
Koszul duality concerns $A_{\infty}$-algebras and dg algebras and largely
recovers classical Koszul duality from a more elegant description.

A-side | B-side
---|---
Augmented $A_{\infty}$-algebra | DG algebra
$A=(\mathbb{C}\mathrm{id}\oplus\mathbb{C}X_{1}\oplus\ldots\oplus\mathbb{C}X_{n},\mu_{A})$ | $A^{!}=(\mathbb{C}\langle\\!\langle{x}^{\vee}_{1},\ldots,{x}^{\vee}_{n}\rangle\\!\rangle,{\mu}^{\vee}_{A})$
Cyclic $A_{\infty}$-structure | Calabi-Yau dg structure
Degree $n$ | Dimension $n$
$A$-coderivation | Linear map
$m:T(\bar{A}[1])\otimes M\to T(\bar{A}[1])\otimes N$ | ${m}^{\vee}:M\to N\otimes A^{!}$
$A$-module | Twisted complex
$(M,\mu_{M})$ | $F(M)=(M\otimes A^{!},{\mu}^{\vee}_{M,0})$

Technically, modern Koszul duality consists of dualizing the
$A_{\infty}$-axioms on vector space level. The results can be described both
abstractly in terms of dual vector spaces and the bar construction, as well as
concretely by means of a choice of basis and a construction of dg structure on
a power series algebra. In the present section, we follow the works of
Ginzburg [19], Van den Bergh [33, 8], Lu-Wu-Palmieri-Zhang [27] and [9].

In section 3.1, we recall $A_{\infty}$-modules and their categories. In
section 3.2, we recall Koszul duality between augmented $A_{\infty}$-algebras
and dg algebras and the Koszul duality functor. In section 3.3, we recall
several classical properties of Koszul duality and explain how they survive in
modern Koszul duality. In section 3.4, we recall Calabi-Yau dg algebras. In
section 3.5, we focus on ordinary (non-dg) Calabi-Yau algebras. In section
3.6, we focus on the special case of ordinary Calabi-Yau algebras of dimension
$n=3$. In section 3.7, we recall the notion of cyclic $A_{\infty}$-algebras
and explain the correspondence between cyclic $A_{\infty}$-algebras and dg
Calabi-Yau algebras via Koszul duality. In section 3.8, we tweak Koszul
duality statements in order to motivate the Cho-Hong-Lau construction.

######  Remark 3.1.

In this section, we partially follow the dg sign convention rather than the
$A_{\infty}$-sign convention:

$\displaystyle\mu^{1}(\mu^{2}(f,g))$
$\displaystyle=\mu^{2}(\mu^{1}(f),g)+(-1)^{|f|}\mu^{2}(f,\mu^{1}(g)),$
$\displaystyle\mu^{2}(f,\mu^{2}(g,h))$
$\displaystyle=\mu^{2}(\mu^{2}(f,g),h).$

The precise distribution is as follows: The $A_{\infty}$-algebra $A$ follows
$A_{\infty}$-signs. The dg algebra $A^{!}$, the dg categories
$\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A$ and
$\operatorname{Tw}A^{!}$ as well as the dg functor
$F:\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A\to\operatorname{Tw}A^{!}$
follow dg signs. We amend all signs to the $A_{\infty}$-setting in section 7.

###  3.1 Modules

In this section, we recall modules over $A_{\infty}$-algebras. We start by
recalling the interpretation of $A_{\infty}$-structures as coderivations on
the bar construction. Then we recall $A_{\infty}$-modules and categories of
$A_{\infty}$-modules.

We start by recalling the tensor coalgebra construction. We denote by $[1]$
the left-shift.

######  Definition 3.1.

Let $A$ be an $A_{\infty}$-algebra. Regard the tensor algebra

$T(A[1])=\bigoplus_{n\in\mathbb{N}}A[1]^{\otimes n}.$

The canonical coproduct $\Delta:T(A[1])\to T(A[1])\otimes T(A[1])$ is given by

$\Delta(a_{k}\otimes\ldots\otimes a_{1})=\sum_{0\leq i\leq
k}(a_{i}\otimes\ldots\otimes a_{1})\otimes(a_{k}\otimes\ldots\otimes
a_{i+1}).$ (3.1)

A coderivation $m:T(A[1])\to T(A[1])$ is a linear map satisfying the co-
Leibniz rule

$\Delta\circ m=(m\otimes\mathrm{id}+\mathrm{id}\otimes m)\circ\Delta.$

Here the maps $m\otimes\mathrm{id}$ and $\mathrm{id}\otimes m$ bear the sign
given by the Koszul sign rule:

$\displaystyle(m\otimes\mathrm{id})((a_{k}\otimes\ldots\otimes
a_{1})\otimes(b_{l}\otimes\ldots\otimes b_{1}))$
$\displaystyle=m(a_{k}\otimes\ldots\otimes
a_{1})\otimes(b_{l}\otimes\ldots\otimes b_{1})$
$\displaystyle(\mathrm{id}\otimes m)((a_{k}\otimes\ldots\otimes
a_{1})\otimes(b_{l}\otimes\ldots\otimes b_{1}))$
$\displaystyle=(-1)^{\|a_{1}\|+\ldots+\|a_{k}\|}(a_{k}\otimes\ldots\otimes
a_{1})\otimes m(b_{l}\otimes\ldots\otimes b_{1}).$

######  Remark 3.2.

The $A_{\infty}$-product $\mu~{}$ on $A$ can be interpreted as a coderivation
$\mu_{A}:T(A[1])\to T(A[1])$ given by

$\mu_{A}(a_{k}\otimes\ldots\otimes a_{1})=\sum_{0\leq j<i\leq
k}(-1)^{\|a_{1}\|+\ldots+\|a_{j}\|}a_{k}\otimes\ldots\otimes\mu(a_{i},\ldots,a_{j+1})\otimes\ldots\otimes
a_{1}.$

The $A_{\infty}$-relations for $\mu$ are equivalent to the condition
$\mu_{A}^{2}=0$. One easily checks that $\mu_{A}$ is indeed a coderivation
with respect to the coproduct $\Delta$. This check explains the awkward flip
used in (3.1). If one uses “Polish notation” $\mu(a_{1},\ldots,a_{k})$ instead
of $\mu(a_{k},\ldots,a_{1})$, one can avoid this flip (see [31, 17]).

######  Definition 3.2.

Let $A$ be an $A_{\infty}$-algebra. Then the bar construction
$\operatorname{B}A$ is the dg coalgebra structure on $T(A[1])$ given by the
canonical coproduct $\Delta$ together with the coderivation $\mu_{A}$.

Modules over dg algebras comes with only an action map $A\otimes M\to A$ and a
differential $M\to M$. When $A$ is an $A_{\infty}$-algebra, one allows the
action maps to have higher components. For sake of section 3.2, we restrict
here to defining right $A$-modules. Left $A$-modules are defined analogously.

######  Definition 3.3.

Let $A$ be an $A_{\infty}$-algebra. Then a right $A$-module is a graded vector
space $M$ together with a degree $1$ map $\mu:M\otimes T(A[1])\to M$ of
satisfying the $A_{\infty}$-relations when combined with the product $\mu$ of
$A$ in a suitable way:

$\sum_{0\leq i\leq
k}(-1)^{\|a_{1}\|+\ldots+\|a_{i}\|}\mu(\mu(m,a_{k},\ldots,a_{i+1}),a_{i},\ldots,a_{1})\\\
+\sum_{0\leq j<i\leq
k}(-1)^{\|a_{1}\|+\ldots+\|a_{j}\|}\mu(m,a_{k},\ldots,a_{i+1},\mu(a_{i},\ldots,a_{j+1}),a_{j},\ldots,a_{1})=0.$

We shall only regard unital $A$-modules, in the sense that
$\mu(m,\mathrm{id})=m$ and $\mu^{\geq 3}(m,\ldots,\mathrm{id},\ldots)=0$.

An $A$-module can be captured elegantly as a coderivation, comparable to the
way that the $A_{\infty}$-product on $A$ can be captured via a coderivation:

######  Definition 3.4.

Let $A$ be an $A_{\infty}$-algebra and $M$ a graded vector space. Then we
regard the comodule map $\Delta_{M}:M\otimes T(A[1])\to
T(A[1])\otimes(M\otimes T(A[1])$ given by

$\Delta_{M}(m\otimes a_{k}\otimes\ldots\otimes a_{1})=\sum_{0\leq i\leq
k}(a_{i}\otimes\ldots\otimes a_{1})\otimes(m\otimes a_{k}\otimes\ldots\otimes
a_{i+1}).$

A map $f:M\otimes T(A[1])\to N\otimes T(A[1])$ is a coderivation if

$\Delta_{N}\circ f=(\mathrm{id}\otimes f)\circ\Delta_{M}.$

In the context of coderivations, we denote by $\mu_{A}$ also the map $M\otimes
T(A[1])\to M\otimes T(A[1])$ given by

$\mu_{A}(a_{k}\otimes\ldots\otimes a_{1}\otimes m)=\sum_{0\leq j<i\leq
k}(-1)^{\|a_{1}\|+\ldots+\|a_{j}\|}~{}m\otimes
a_{k}\otimes\ldots\otimes\mu(a_{i}\otimes\ldots\otimes
a_{j+1})\otimes\ldots\otimes a_{1}.$

######  Remark 3.3.

Whenever $f:M\otimes T(A[1])\to N\otimes T(A[1])$ is a coderivation, we can
consider its projection to $N$ which we denote by $f_{0}:M\otimes T(A[1])\to
N$. Conversely, if $f_{0}:M\otimes T(A[1])\to N$ is a graded linear map, we
can turn it into a coderivation $f:M\otimes T(A[1])\to N\otimes T(A[1])$. The
precise correspondence between $f$ and $f_{0}$ reads

$f(m\otimes a_{k}\otimes\ldots\otimes a_{1})=\sum_{0\leq i\leq
k}(-1)^{|f_{0}|(\|a_{1}\|+\ldots+\|a_{i}\|)}f_{0}(m\otimes
a_{k}\otimes\ldots\otimes a_{i+1})\otimes a_{i}\otimes\ldots\otimes a_{1}.$

In terms of Definition 3.4, a right $A$-module is simply a coderivation
$\mu_{M}:M\otimes T(A[1])\to M\otimes T(A[1])$ of degree $1$ such that
$(\mu_{M}+\mu_{A})^{2}=0$. The use of the letter $\mu_{A}$ is clearly an abuse
of notation, but we expect no confusion to arise.

Capturing an $A$-module in terms of a coderivation $\mu_{M}$ makes it
particularly straightforward to define a category of $A$-modules:

######  Definition 3.5.

Let $A$ be an $A_{\infty}$-algebra. Then
$\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A$ is the dg category of
finite-dimensional right $A$-modules, with structure specified as follows:

  * •

The hom space
$\operatorname{Hom}_{\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A}(M,N)$
is the space of coderivations $M\otimes T(A[1])\to N\otimes T(A[1])$.

  * •

The product $\mu^{2}_{\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A}$ is
ordinary composition.

  * •

The differential measures failure to be a module morphism:

$\mu^{1}_{\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A}(f)=(\mu_{A}+\mu_{N})\circ
f-(-1)^{|f|}f\circ(\mu_{A}+\mu_{M}).$

######  Remark 3.4.

It is readily checked by hand that
$\mu^{1}_{\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A}(f)$ is a
coderivation if $f$ is a coderivation.

Bimodules are another important tool in homological algebra. They can be
defined in a way analogous to left or right $A$-modules, see for instance [31,
Section 2]. We come back to bimodules in the dg case in section 3.4.

###  3.2 Koszul duality

In this section, we recapitulate Koszul duality as a preparation to the Cho-
Hong-Lau construction. Koszul duality is a construction connecting an
$A_{\infty}$-algebra $A$ with a dg algebra $A^{!}$, its Koszul dual.
Surprisingly, this construction also induces a correspondence between
$A_{\infty}$-modules over $A$ and twisted complexes over $A^{!}$:

$F:\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A\xrightarrow{~{}~{}~{}\phantom{\sim}~{}~{}~{}}\operatorname{Tw}A^{!}.$

Our aim is to recall as fast as possible that Koszul duality produces
functors. For further details we refer to [27] and [10, Section 12.5].

######  Definition 3.6.

An augmented $A_{\infty}\,$-algebra is an $A_{\infty}$-algebra $A$ with a
decomposition $A=\bar{A}\oplus\mathbb{C}\mathrm{id}$ such that
$\mu(a_{k},\ldots,a_{1})\in\bar{A}$ whenever $a_{1},\ldots,a_{k}\in\bar{A}$.

######  Remark 3.5.

If $A$ is an augmented $A_{\infty}$-algebra, many constructions for $A$ can be
carried out by working with the “augmented” tensor coalgebra $T(\bar{A}[1])$
instead of the full tensor coalgebra $T(A[1])$. For instance, to define an
$A$-module it suffices to provide the map $T(\bar{A}[1])\otimes M\to M$
instead of $T(A[1])\otimes M\to M$ since we only work with unital modules.

The canonical coproduct $\Delta:T(\bar{A}[1])\to T(\bar{A}[1])\otimes
T(\bar{A}[1])$ and for a graded vector space $M$ the comodule map
$\Delta_{M}:T(\bar{A}[1])\otimes M\to T(\bar{A}[1])\otimes
T(\bar{A}[1])\otimes M$ are defined as in the non-augmented case, this time
restricting to $T(\bar{A}[1])$ instead of $T(A[1])$.

######  Remark 3.6.

If $M$ is a graded vector space, we denote its graded dual vector space by
${M}^{\vee}$. The dual of $T(\bar{A}[1])$ is equal to

${T(\bar{A}[1])}^{\vee}=\prod_{n=0}^{\infty}({\bar{A}[1]}^{\vee})^{\otimes
n}.$

Here we make the identification that reverses the order of tensor components:

$\displaystyle{V}^{\vee}_{1}\otimes\ldots\otimes{V}^{\vee}_{k}$
$\displaystyle\xrightarrow{~{}~{}~{}\sim~{}~{}~{}}{(V_{k}\otimes\ldots\otimes
V_{1})}^{\vee},$ (3.2)
$\displaystyle\varphi_{1}\otimes\ldots\otimes\varphi_{k}$
$\displaystyle\xmapsto{~{}~{}~{}\phantom{\sim}~{}~{}~{}}(-1)^{\sum_{1\leq
s<t\leq k}|\varphi_{s}||\varphi_{t}|}[(x_{k}\otimes\ldots\otimes
x_{1})\mapsto\varphi_{1}(x_{1})\ldots\varphi_{k}(x_{k})].$

The dual of a map $m:T(\bar{A}[1])\to T(\bar{A}[1])$ has the shape
${m}^{\vee}:{T(\bar{A}[1])}^{\vee}\to{T(\bar{A}[1])}^{\vee}$.

######  Definition 3.7.

Let $A$ be a finite-dimensional augmented $A_{\infty}$-algebra. Then its
Koszul dual $A^{!}$ is the dg algebra given by

$A^{!}={T(\bar{A}[1])}^{\vee}.$

Upon the identification of (3.2), the product on $A^{!}$ is defined as the
standard product on $\widehat{T({\bar{A}[1]}^{\vee})}$. The differential on
$A^{!}$ is given by

$dv=(-1)^{|v|+1}v\circ\mu_{A},\quad\forall
v\in{T(\bar{A}[1])}^{\vee}=\operatorname{Hom}_{\mathbb{C}}(T(\bar{A}[1]),\mathbb{C}).$

Here $\mu_{A}:T(\bar{A}[1])\to T(\bar{A}[1])$ denotes the product of $A$.

######  Example 3.1.

Pick a basis $x_{1},\ldots,x_{k}$ for $A$. Then we have the dual elements
${x}^{\vee}_{i}$ of degree $|{x}^{\vee}_{i}|=1-|x_{i}|$. Multiplication within
$A^{!}$ can be performed simply as
${x}^{\vee}_{i}\cdot{x}^{\vee}_{j}={x}^{\vee}_{i}{x}^{\vee}_{j}$. To interpret
this element as element of ${T(\bar{A}[1])}^{\vee}$, we have to apply the sign
flip from (3.2):

${x}^{\vee}_{i}{x}^{\vee}_{j}=(-1)^{|{x}^{\vee}_{i}||{x}^{\vee}_{j}|}{(x_{j}\otimes
x_{i})}^{\vee}.$

Here we have written ${(x_{j}\otimes x_{i})}^{\vee}$ for the element of
$A^{!}$ which sends the element $x_{j}\otimes x_{i}$ to $1$ and all other
basis tensors to zero. Now write the product of $A$ as

$\mu^{l}(x_{i_{l}},\ldots,x_{i_{1}})=\sum_{1\leq j\leq
k}c_{i_{l},\ldots,i_{1}}^{j}x_{j}.$

Then

$\displaystyle d_{A^{!}}({x}^{\vee}_{j})$
$\displaystyle=(-1)^{|{x}^{\vee}_{j}|+1}\sum_{\begin{subarray}{c}l\geq 1\\\
1\leq i_{l},\ldots,i_{1}\leq
k\end{subarray}}c_{i_{l},\ldots,i_{1}}^{j}{(x_{i_{l}}\otimes\ldots\otimes
x_{i_{1}})}^{\vee}$
$\displaystyle=(-1)^{|{x}^{\vee}_{j}|+1}\sum_{\begin{subarray}{c}l\geq 1\\\
1\leq i_{l},\ldots,i_{1}\leq
k\end{subarray}}c_{i_{l},\ldots,i_{1}}^{j}(-1)^{\sum_{1\leq s<t\leq
l}|{x}^{\vee}_{i_{s}}||{x}^{\vee}_{i_{t}}|}~{}{x}^{\vee}_{i_{1}}\ldots{x}^{\vee}_{i_{l}}.$

######  Remark 3.7.

The Koszul dual $A^{!}$ is indeed a dg algebra. Abstractly speaking, the dual
of the operator $\Delta$ is the ordinary product
${\Delta}^{\vee}:{T(\bar{A}[1])}^{\vee}\otimes{T(\bar{A}[1])}^{\vee}\to{T(\bar{A}[1])}^{\vee}$.
Dualizing $\mu_{A}^{2}=0$ gives $({\mu}^{\vee}_{A})^{2}=0$ and dualizing the
co-Leibniz rule for $\mu_{A}$ with respect to $\Delta$ gives the Leibniz rule
for $\mu_{A}$ with respect to ${\Delta}^{\vee}$. The signs can be checked by
hand.

######  Definition 3.8.

Let $A$ be an augmented $A_{\infty}$-algebra. Let $M$ and $N$ be graded vector
spaces and $f:M\otimes T(\bar{A}[1])\to N$ a linear map. Then the Koszul
transform of $f$ is the partial dual map ${f}^{\vee}:M\to N\otimes A^{!}$. It
is a graded linear map with characterizing property

$\forall m\in
M,~{}a\in\,T(\bar{A}[1]):\quad\langle{f}^{\vee}(m),a\rangle=f(m\otimes a).$

Here $\langle-,-\rangle$ liberally denotes the standard pairing between
${T(\bar{A}[1])}^{\vee}$ and $T(\bar{A}[1])$, in this case as map

$\langle-,-\rangle:(N\otimes{T(\bar{A}[1])}^{\vee})\otimes T(\bar{A}[1])\to
N.$

We copy Definition 2.2 and adapt it slightly to the dg case.

######  Definition 3.9.

Let $D$ be a dg algebra. Then the category $\operatorname{Add}D$ is the
category of formal shifted sums of copies of $D$ with hom spaces given as
spaces of matrices with entries $D$. The differential
$\mu^{1}_{\operatorname{Add}D}$ and product $\mu^{2}_{\operatorname{Add}D}$
are the linear and bilinear extension of differential and product of $D$, with
a sign change. On single matrix entries, the sign change is as follows:

$\displaystyle\mu^{1}_{\operatorname{Add}D}(a)$
$\displaystyle=(-1)^{k-l}d_{D}a,\quad\forall a:D[k]\to D[l],$
$\displaystyle\mu^{2}_{\operatorname{Add}D}(a,b)$
$\displaystyle=(-1)^{|a|_{D}(k-l)}ab,\quad\forall a:D[l]\to D[m],~{}b:D[k]\to
D[l].$

######  Definition 3.10.

Let $D$ be a dg algebra. Then a twisted complex over $D$ is an element
$X\in\operatorname{Add}D$ together with an element
$\delta\in\operatorname{End}_{\operatorname{Add}D}^{1}(X)$ such that $\delta$
is upper triangular and satisfies the Maurer-Cartan equation:

$\mu^{1}_{\operatorname{Add}D}(\delta)+\mu^{2}_{\operatorname{Add}D}(\delta,\delta)=0.$

The differential and product on $\operatorname{Tw}D$ follow the sign rule

$\displaystyle\mu^{1}_{\operatorname{Tw}D}(f)$
$\displaystyle=\mu^{1}_{\operatorname{Add}D}(f)+\mu^{2}_{\operatorname{Add}D}(\delta,f)+(-1)^{|f|+1}\mu^{2}_{\operatorname{Add}D}(f,\delta),$
$\displaystyle\mu^{2}_{\operatorname{Tw}D}(f,g)$
$\displaystyle=\mu^{2}_{\operatorname{Add}D}(f,g).$

The product $\mu^{2}_{\operatorname{Add}D}$ is comparable to a matrix product
and similarly $\mu^{1}_{\operatorname{Add}D}$ acts as entry-wise differential.
As preparation for the Koszul duality functor, we prove a few properties
regarding the Koszul transform. We show that the Koszul transform of $f\circ
g$ agrees with the matrix product $\mu^{2}_{\operatorname{Add}D}$ of
${f}^{\vee}$ and ${g}^{\vee}$ and that taking the Koszul transform of
$f\circ\mu_{A}$ amounts to taking the entry-wise differential
$\mu^{1}_{\operatorname{Add}D}$ of ${f}^{\vee}$. The precise statement is as
follows:

######  Lemma 3.1.

Let $A$ be a finite-dimensional augmented $A_{\infty}$-algebra. Let
$f:M\otimes T(\bar{A}[1])\to N\otimes T(\bar{A}[1])$ and $g:L\otimes
T(\bar{A}[1])\to M\otimes T(\bar{A}[1])$ be coderivations. Then

$\displaystyle\mu^{2}_{\operatorname{Add}D}({f}^{\vee}_{0},{g}^{\vee}_{0})$
$\displaystyle={(f\circ g)_{0}}^{\vee}$ (3.3)
$\displaystyle\mu^{1}_{\operatorname{Add}D}({f}^{\vee}_{0})$
$\displaystyle=(-1)^{|f|+1}{(f\circ\mu_{A})_{0}}^{\vee}.$ (3.4)

###### Proof.

The simplest way to evaluate both statements is by choosing bases for $A$,
$L$, $M$ and $N$. Let $X,Y\in T(\bar{A}[1])$ be basis elements homogeneous
with respect to both tensor degree and $\|\cdot\|_{\bar{A}[1]}$. Let
$\varepsilon_{L}\in L$ and $\varepsilon_{M}\in M$ and $\varepsilon_{N}\in\,N$
be further basis elements. Let $f$ and $g$ be given coderivations. They are
determined solely by their first component $f_{0}$ and $g_{0}$. Since the
statement is linear in $f$ and $g$, we may assume that
$g_{0}(\varepsilon_{L}\otimes X)=\varepsilon_{M}$ and
$f_{0}(\varepsilon_{M}\otimes Y)=\varepsilon_{N}$ and that $f_{0}$ and $g_{0}$
vanish on all other basis elements for $L\otimes T(\bar{A}[1])$ and $M\otimes
T(\bar{A}[1])$. The Koszul transforms of $f$, $g$ read

$\displaystyle{f}^{\vee}(\varepsilon_{M})$
$\displaystyle=\varepsilon_{N}\otimes{Y}^{\vee},$
$\displaystyle{g}^{\vee}(\varepsilon_{L})$
$\displaystyle=\varepsilon_{M}\otimes{X}^{\vee}.$

Let us now show (3.3). We note that $(f\circ g)_{0}$ vanishes on all basis
elements, except

$(f\circ g)_{0}(\varepsilon_{L}\otimes X\otimes
Y)=(-1)^{|g|\|Y\|}f(g(\varepsilon_{L}\otimes X)\otimes
Y)=(-1)^{|g||{Y}^{\vee}|}\varepsilon_{N}.$

Therefore ${(f\circ g)}^{\vee}$ and ${f}^{\vee}\cdot{g}^{\vee}$ vanish on all
basis elements of $L$, except

$\displaystyle{(f\circ g)}^{\vee}(\varepsilon_{L})$
$\displaystyle=(-1)^{|g||{Y}^{\vee}|}\varepsilon_{N}\otimes{(X\otimes
Y)}^{\vee}$
$\displaystyle=(-1)^{|g||{Y}^{\vee}|+|{X}^{\vee}||{Y}^{\vee}|}\varepsilon_{N}\otimes{Y}^{\vee}{X}^{\vee}$
$\displaystyle=(-1)^{(|\varepsilon_{L}|-|\varepsilon_{M}|)|{Y}^{\vee}|}\varepsilon_{N}\otimes{Y}^{\vee}{X}^{\vee}$
$\displaystyle=(-1)^{(|\varepsilon_{L}|-|\varepsilon_{M}|)|{Y}^{\vee}|}({f}^{\vee}\cdot{g}^{\vee})(\varepsilon_{L})$
$\displaystyle=\mu^{2}_{\operatorname{Add}D}({f}^{\vee},{g}^{\vee})(\varepsilon_{L}).$

In the third row we have used that
$|g|+|\varepsilon_{L}|=|{X}^{\vee}|+|{X}^{\vee}|$. This proves (3.3).

We now prove (3.4). It is our aim to compute the composition
$(f\circ\mu_{A})_{0}=f_{0}\circ\mu_{A}$. Since $f_{0}$ vanishes on all basis
elements except $\varepsilon_{M}\otimes Y$ and (3.4) is linear in $\mu_{A}$
itself, we may simply assume that $\mu_{A}(\varepsilon_{M}\otimes
Z)=\varepsilon_{M}\otimes Y$ for some basis element $Z\in\,T(\bar{A}[1])$ and
$\mu_{A}$ vanishes on all other basis elements. Then $(f\circ\mu_{A})_{0}$
vanishes on all basis elements of $L\otimes T(\bar{A}[1])$, except

$(f\circ\mu_{A})_{0}(\varepsilon_{M}\otimes Z)=f_{0}(\varepsilon_{M}\otimes
Y)=\varepsilon_{N}.$

We see that ${(f\circ\mu_{A})_{0}}^{\vee}$ vanishes on all basis elements of
$M$, except

$\displaystyle{(f\circ\mu_{A})_{0}}^{\vee}(\varepsilon_{M})$
$\displaystyle=\varepsilon_{N}\otimes{Z}^{\vee}$
$\displaystyle=(-1)^{|{Y}^{\vee}|+1}\varepsilon_{N}\otimes d{Y}^{\vee}$
$\displaystyle=(-1)^{|f|+1}\mu^{1}_{\operatorname{Add}D}({f}^{\vee})(\varepsilon_{M}).$

In the last row, we have used that
$|\varepsilon_{M}|+|f|=|\varepsilon_{N}|+|{Y}^{\vee}|$. This finishes the
proof. ∎

######  Remark 3.8.

In Lemma 3.1, it is essential that $f$ and $g$ be coderivations. For instance,
$\mu_{A}:M\otimes T(\bar{A}[1])\to M\otimes T(\bar{A}[1])$ is not a
coderivation on its own and in fact its zeroth component $\mu_{M,0}$ vanishes,
while a composition $f\circ\mu_{A}$ may again have nonvanishing zeroth
component and therefore nonvanishing Koszul transform.

In Corollary 3.1, we recall the Koszul duality functor between modules over
$A$ and twisted complexes over $A^{!}$. The idea is to apply “Koszul
transform” the action map of every module. The construction is functorial,
therefore gives rise to a dg functor.

######  Corollary 3.1.

Let $A$ be a finite-dimensional augmented $A_{\infty}$-algebra. Then the
following defines a dg functor:

$\displaystyle F:\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A$
$\displaystyle\xrightarrow{~{}~{}~{}\phantom{\sim}~{}~{}~{}}\operatorname{Tw}A^{!},$
$\displaystyle(M,\mu_{M})$
$\displaystyle\xmapsto{~{}~{}~{}\phantom{\sim}~{}~{}~{}}(M\otimes
A^{!},{\mu}^{\vee}_{M,0}),$ $\displaystyle f$
$\displaystyle\xrightarrow{~{}~{}~{}\phantom{\sim}~{}~{}~{}}{f}^{\vee}_{0}.$

We call $F$ the Koszul duality functor of $A$.

###### Proof.

There are three items to check. First, we show that ${\mu}^{\vee}_{M,0}$
satisfies the Maurer-Cartan equation. Then, we check that $F$ preserves
differential and product. We start with the Maurer-Cartan equation:

$\displaystyle\mu^{1}_{\operatorname{Add}A^{!}}({\mu}^{\vee}_{M,0})+\mu^{2}_{\operatorname{Add}A^{!}}({\mu}^{\vee}_{M,0},{\mu}^{\vee}_{M,0})$
$\displaystyle={(\mu_{M,0}\circ\mu_{A})}^{\vee}+{(\mu_{M,0}\circ\mu_{M})}^{\vee}$
$\displaystyle={(\mu_{M}\circ\mu_{A}+\mu_{M}\circ\mu_{M}+\mu_{A}\circ\mu_{M})_{0}}^{\vee}=0.$

In the second row, we have used that $(\mu_{A}\circ\mu_{M})_{0}=0$. Next, for
every coderivation $f:M\otimes T(\bar{A}[1])\to N\otimes T(\bar{A}[1])$ we
have

$\displaystyle F(\mu^{1}(f))$ $\displaystyle={((\mu_{N}+\mu_{A})\circ
f-(-1)^{|f|}f\circ(\mu_{M}+\mu_{A}))_{0}}^{\vee}$
$\displaystyle={(\mu_{N}\circ
f)_{0}}^{\vee}-(-1)^{|f|}{(f\circ\mu_{M})_{0}}^{\vee}-(-1)^{|f|}{(f\circ\mu_{A})_{0}}^{\vee}$
$\displaystyle=\mu^{2}_{\operatorname{Add}A^{!}}({\mu}^{\vee}_{N,0},{f}^{\vee}_{0})-(-1)^{|f|}\mu^{2}_{\operatorname{Add}A^{!}}({f}^{\vee}_{0},{\mu}^{\vee}_{M,0})+\mu^{1}_{\operatorname{Add}A^{!}}({f}^{\vee}_{0})$
$\displaystyle=\mu^{1}_{\operatorname{Tw}A^{!}}(F(f)).$

If additionally $g:L\otimes T(\bar{A}[1])\to M\otimes T(\bar{A}[1])$ is a
coderivation, then

$F(\mu^{2}(f,g))=F(f\circ g)={(f\circ
g)_{0}}^{\vee}=\mu^{2}_{\operatorname{Add}A^{!}}({f}^{\vee},{g}^{\vee})=\mu^{2}_{\operatorname{Tw}A^{!}}(F(f),F(g)).$

This shows that $F$ is a dg functor and finishes the proof. ∎

######  Remark 3.9.

Strictly speaking, the object $F(M)=(M\otimes A^{!},{\mu}^{\vee}_{M,0})$ only
becomes a twisted complex upon choice of a graded basis for $M$. Furthermore,
${\mu}^{\vee}_{M,0}$ need not be an upper triangular matrix. However, if
$\bar{A}$ is concentrated in positive degrees, then sorting the basis elements
of $M$ in order of descending degree makes ${\mu}^{\vee}_{M,0}$ upper
triangular.

###  3.3 Classical Koszual duals

In this section, we comment on the relations of the modern with the classical
Koszul dual construction. Classical Koszul duality is namely a phenomenon
known for ordinary algebras and we recall here its typical properties: First,
the double Koszul dual $(A^{!})^{!}$ is $A$ again. Second, the Koszul dual
algebra is formal. Third, the Koszul algebra is the Ext algebra of its simple
module $\mathbb{C}=A/\bar{A}$. In the present section, we recall these
statements in the classical context and recall how they translate to modern
Koszul duality. A valuable source is [27].

Koszul duality has classically been a correspondence between Koszul algebras,
a class of ordinary algebras with quadratic relations:

Ordinary algebra$\frac{V\otimes V}{R}=\frac{\mathbb{C}\langle X,Y\rangle}{(XY-
YX)}$Ordinary
algebra$\frac{{V}^{\vee}\otimes{V}^{\vee}}{R^{\perp}}=\frac{\mathbb{C}\langle
X,Y\rangle}{(X^{2},Y^{2},XY+YX)}$Koszul

The relations on either side are the “orthogonal complement” of the relations
on the other side along the pairing $(V\otimes
V)\otimes({V}^{\vee}\otimes{V}^{\vee})\to\mathbb{C}$. In particular, classical
Koszul duality is an involution from the beginning.

Koszul duality for $A_{\infty}$-algebras is not a one-way street either. If
$A$ is an augmented finite-dimensional $A_{\infty}$-algebra, we regard the
double dual $(A^{!})^{!}$. It is possible that this dg algebra is quasi-
isomorphic to $A$ itself. However, the double Koszul dual construction applies
vector space duals twice so that finiteness conditions are required to match
$A$ exactly with $(A^{!})^{!}$.

A sufficient finiteness criterion can be formulated if one assumes that the
$A_{\infty}$-algebra $A$ has an additional grading, also referred to as Adams
grading. The direct sum decomposition $A=\mathbb{C}\mathrm{id}\oplus\bar{A}$
is supposed to be compatible and the products $\mu^{k}$ are assumed to be
homogeneous with respect to the Adams grading. One then says that $A$ is Adams
connected if the homogeneous part of $\bar{A}$ with respect to any Adams
degree $j\in\mathbb{Z}$ is finite-dimensional and vanishes either for all
$j\leq 0$ or all $j\geq 0$ [27]. This connectedness assumption is a sufficient
finiteness criterion and ensures that the double Koszul $(A^{!})^{!}$ is
quasi-isomorphic to $A$ again:

######  Theorem 3.1 ([27]).

Let $A$ be an augmented $A_{\infty}$-algebra. If $A$ is Adams connected, then
$(A^{!})^{!}$ and $A$ are quasi-isomorphic as $A_{\infty}$-algebras.

The classical Koszul dual is automatically an ordinary graded algebra, without
the need to pass to cohomology. From a modern perspective, Koszul duals are dg
algebras and only passing to cohomology $\operatorname{H}A^{!}$ and forgetting
the $A_{\infty}$-structure gives an ordinary graded algebra. Koszul algebras
are certain types of algebras distinguished by the property that their Koszul
dual $A^{!}$ tends to be a formal dg algebra. This way, one can forget the
higher structure on $\operatorname{H}A^{!}$ and recover classical Koszul
duality. For sake of completeness, we recall the definition here:

######  Definition 3.11 ([4]).

A graded associative algebra $A$ is Koszul if it is positively graded
$A=\bigoplus_{i\geq 0}A^{i}$, we have $A^{0}=\mathbb{C}\mathrm{id}$ and
$\mathbb{C}=A/\bigoplus_{i>0}A^{i}$ as an $A$-module has a resolution of
graded $A$-modules

$\ldots\to P^{2}\to P^{1}\to P^{0}\to A\to 0$

in which every $P^{i}$ is generated by its degree $i$ component:
$P^{i}=AP_{i}^{i}$.

In [27] a precise criterion was given for formality:

######  Theorem 3.2 ([27, Corollary 2.7]).

Let $A$ be an $(a,b)$-generated Koszul algebra in the sense of [27, Definition
2.6]. Then we have an $A_{\infty}$-quasi-isomorphism
$A^{!}\cong\operatorname{H}^{0}A^{!}$.

######  Remark 3.10.

Classical Koszul duality is full of correspondences between types of algebras.
Folklore statements include that Koszul algebras correspond to Koszul
algebras, Gorenstein corresponds to Gorenstein, Artin-Schelter regular
corresponds to Frobenius [27, Corollary D, E].

######  Remark 3.11.

The classical analog of $\operatorname{Tw}\operatorname{H}^{0}A^{!}$ is
$\operatorname{Perf}\operatorname{H}^{0}A^{!}$. This is the reason why Koszul
duality is classically formulated as triangulated equivalence between
categories of the form $\operatorname{D}\operatorname{Mod}A$ and
$\operatorname{Perf}\operatorname{H}^{0}A^{!}$, see for instance [27, Theorem
B]. These classical Koszul duality functors are typically complicated for the
reason that they do not take the possible higher structures on
$\operatorname{H}A^{!}$ into account. In the classical world, this is “solved”
by restricting to Koszul algebras. Thanks to modern Koszul duality, it is
possible to recover classical Koszul duality from the functor
$\operatorname{Mod}^{\mathrm{fd}}A\to\operatorname{Tw}A^{!}$. The idea is to
replace $A^{!}$ by its zeroth cohomology, which is an ordinary algebra.
Abstractly, we aim for a functor
$\operatorname{Tw}A^{!}\to\operatorname{Tw}\operatorname{H}^{0}A^{!}$ in order
to precompose it with the Koszul duality functor
$\operatorname{Mod}^{\mathrm{fd}}A\to\operatorname{Tw}A^{!}$. We follow this
route in section 3.8.

Another classical statement of Koszul duality is that $A^{!}$ can be
interpreted as Ext algebra of the simple module $\mathbb{C}=A/\bar{A}$ of $A$.
As we recall in Lemma 3.2, this is also the case if $A$ is an
$A_{\infty}$-algebra. One can alternatively use this description of $A^{!}$ as
defining property. With regards to notation, let $A$ be an augmented
$A_{\infty}$-algebra. Then we denote by $\mathbb{C}$ the simple right
$A$-module $\mathbb{C}=A/\bar{A}$. Its action map $\mathbb{C}\otimes
T(\bar{A}[1])\to\mathbb{C}$ is simply zero. In these terms, $A^{!}$ is just
the dg algebra $\operatorname{Hom}_{rModfdA}(\mathbb{C},\mathbb{C})$:

######  Lemma 3.2.

Let $A$ be an augmented $A_{\infty}$-algebra. Then we have an isomorphism of
dg algebras

$A^{!}\cong\operatorname{Hom}_{\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A}(\mathbb{C},\mathbb{C}).$

###### Proof.

This follows easily from Corollary 3.1. Indeed, the Koszul duality functor $F$
sends the simple module $\mathbb{C}$ to the twisted complex
$(A^{!},0)\in\operatorname{Tw}A^{!}$ and therefore establishes a map between
the two dg endomorphism algebras. It is easy to see that $F$ is fully
faithful. Since the endomorphism algebra of $(A^{!},0)$ is just $A^{!}$, we
are done. ∎

######  Remark 3.12.

Similar to Lemma 3.2, let $(M,\mu_{M})$ be a finite-dimensional right
$A$-module. Then we have an isomorphism of right $A^{!}$-modules

$M\otimes
A^{!}\cong\operatorname{Hom}_{\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}}(\mathbb{C},M).$

The differential on $M\otimes A^{!}$ is induced from induced from
$\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}A$ and the action of $A^{!}$
is by (signed) right-multiplication. The action of $A^{!}$ on
$\operatorname{Hom}_{\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}}(\mathbb{C},M)$
is by composing with
$A^{!}\cong\operatorname{Hom}_{\operatorname{Mod}^{\mathrm{fd}}_{\mathrm{right}}}(\mathbb{C},\mathbb{C})$
on the right.

###  3.4 Calabi-Yau algebras

In this section, we recall Calabi-Yau algebras. This class of algebras is now
widely recognized as a noncommutative analog of Calabi-Yau manifolds. The
definition brings several technical difficulties and correspondingly a wide
range of adaptations and variants have been introduced in the literature. In
the present section, we follow the original definition of Ginzburg [19], in
particular we focus on the dg case.

We start by recalling opposite and enveloping algebras.

######  Definition 3.12.

Let $A$ be a dg algebra. Then the opposite algebra $A^{\operatorname{op}}$ of
$A$ is obtained by setting $A^{\operatorname{op}}=A$ as vector space and
defining

$d_{A^{\operatorname{op}}}(a)=-d_{A}(a),\quad
a\cdot_{A^{\operatorname{op}}}b=(-1)^{|a||b|}ba.$

The enveloping algebra $A^{\operatorname{e}}=A\otimes A^{\operatorname{op}}$
is the tensor product of $A$ and $A^{\operatorname{op}}$. It is an dg algebra
itself with product $(a\otimes b)(c\otimes d)=(-1)^{|b||c|}ac\otimes bd$ and
differential $d(a\otimes b)=da\otimes b+(-1)^{|a|}a\otimes db$.

######  Definition 3.13.

Let $A$ be a dg algebra. Then an $A$-bimodule is the same as an
$A^{\operatorname{e}}$-module. The space $A$ becomes naturally a bimodule over
$A$ by putting $(a\otimes b)x=(-1)^{|b||x|}axb$ for $a\otimes b\in
A^{\operatorname{op}}$ and $x\in\,A$. The space $A^{\operatorname{e}}$ is also
an $A$-bimodule, but in two different ways. The default action is the outer
action given by

$(a\otimes b).(x\otimes y)=(-1)^{|b||x|}ax\otimes yb,\quad a\otimes b\in
A^{\operatorname{e}},x\otimes y\in A^{\operatorname{e}}.$

The alternative is the inner action given by

$(a\otimes b).(x\otimes y)=(-1)^{|b||x|+|a||b|+|a||x|}xb\otimes ay,\quad
a\otimes b\in A^{\operatorname{e}},x\otimes y\in A^{\operatorname{e}}.$

The dg category of dg modules over a dg algebra can be defined similar to the
module category over an $A_{\infty}$-algebra. If $D$ is a dg algebra, we
denote this dg category by $\operatorname{Mod}D$. It is not the same as the
ordinary category with hom spaces the morphisms of dg modules, but rather the
differential measures failure to be a dg morphism. To define
$\operatorname{Mod}D$, one turns the dg algebra into an honest $A_{\infty}$ by
means of the sign flip $\mu^{1}(a)=(-1)^{|a|}$ and $\mu^{2}(a,b)=(-1)^{|b|}ab$
and then forms $\operatorname{Mod}D$ according to the recipe presented in
section 3.1. One may express the result more explicitly which we will not
attempt here.

In particular, we can form the category
$\operatorname{Mod}A^{\operatorname{e}}$. When
$M\in\operatorname{Mod}_{A^{\operatorname{e}}}$, we can regard the hom space

$M^{\mathrm{D}}=\operatorname{Hom}_{\operatorname{Mod}A^{\operatorname{e}}}(M,A^{\operatorname{e}}).$

For us, it is most important that $M^{\mathrm{D}}$ is again a dg
$A^{\operatorname{e}}$-module, often called the dual bimodule of $M$. Its
differential is the differential
$\mu^{1}_{\operatorname{Mod}A^{\operatorname{e}}}$ and the
$A^{\operatorname{e}}$-action on $M^{\mathrm{D}}$ is the inner action on the
codomain $A^{\operatorname{e}}$. It is elementary to check that
$M^{\mathrm{D}}$ is indeed a dg module. According to Kontsevich and Soibelman,
the bimodule $A^{\mathrm{D}}$ is to be viewed as “inverse dualizing bibundle”
$F\mapsto F\otimes K_{X}^{-1}[-\operatorname{dim}X]$ of the noncommutative
manifold defined by $A$ [22, Definition 8.1.6].

When $M$ is a dg module, the $n$-th left shift $M[n]$ also becomes an object
of $\operatorname{Mod}A^{\operatorname{e}}$. Reflecting the definition of
Calabi-Yau manifolds in the commutative world, the dg algebra $A$ is called
Calabi-Yau if the dual bimodule $A^{\mathrm{D}}$ is quasi-isomorphic to a
shift of $A$:

######  Definition 3.14.

Let $A$ be a dg algebra. Then $A$ is Calabi-Yau of dimension $n\geq 1$ (CY$n$)
if $A^{\mathrm{D}}[n]$ and $A$ are quasi-isomorphic in the category
$\operatorname{Mod}A^{\operatorname{e}}$.

######  Remark 3.13.

The original definition [19, Definition 3.2.3] requires that the quasi-
isomorphism is a self-dual morphism. Van den Bergh has shown in [33,
Proposition C.1] that this condition is typically automatic.

The definition of the hom space
$\operatorname{Hom}_{A^{\operatorname{e}}}(A,A^{\operatorname{e}})$ takes the
degrees of the chosen resolution of $A$ and the degrees of
$A^{\operatorname{e}}$ into account. In particular, requiring
$\operatorname{Hom}_{A^{\operatorname{e}}}(A,A^{\operatorname{e}})$ and $A[n]$
to be quasi-isomorphic cannot be easily translated into a property regarding
resolutions on $A$. If $A$ is however an ordinary algebra (concentrated in
degree zero), then the definition simplifies as follows:

######  Lemma 3.3 ([19, (3.2.5)]).

Let $A$ be an associative algebra which has a finite projective $A$-bimodule
resolution of finitely generated bimodules. Then $A$ is Calabi-Yau of
dimension $n\geq 1$ if and only if

$\operatorname{HH}^{k}(A,A^{\operatorname{e}})\cong\begin{cases}A&\text{if
}k=n,\\\ 0&\text{else}.\end{cases}$ (3.5)

Here $\operatorname{HH}^{k}(A,A^{\operatorname{e}})$ is equipped again with
the inner $A$-bimodule action and the isomorphism is meant as $A$-bimodules.

###### Proof.

By definition $\operatorname{HH}^{k}(A,A^{\operatorname{e}})$ is the
cohomology of
$\operatorname{Hom}_{A^{\operatorname{e}}}(A,A^{\operatorname{e}})$, together
with the additional $A$-bimodule action. We now prove both directions.

If $A$ is CY$n$ then
$\operatorname{Hom}_{A^{\operatorname{e}}}(A,A^{\operatorname{e}})$ and $A[n]$
are quasi-isomorphic as objects of $\operatorname{Mod}A^{\operatorname{e}}$.
There exist closed morphisms $f$ and $g$ between them and a morphism $h$ of
degree $-1$ such that $fg=\mathrm{id}+d(h)$. In particular $f$ and $g$ define
quasi-inverse morphisms of complexes and we conclude the cohomology of
$\operatorname{Hom}_{A^{\operatorname{e}}}(A,A^{\operatorname{e}})$ is $A$,
concentrated in degree $n$.

Conversely, assume (3.5) holds. The conclusion is a general statement
regarding complexes. Let $P^{\bullet}$ be a complex with homology concentrated
in a single degree, namely $\varphi:A\xrightarrow{\sim}\operatorname{H}^{k}P$.
Then there is a quasi-isomorphism $\psi:A\to P^{\bullet}$ of complexes given
by $a\mapsto\varphi(a)\in\operatorname{H}^{k}P\subseteq P^{k}$. This map is
indeed a map of chain complexes since the differential on $A$ and on the image
$\psi(A)$ vanishes. The map is a quasi-isomorphism since on cohomology this
map is just $\varphi$. This finishes the proof. ∎

######  Remark 3.14.

In contrast to dg algebras, two $A_{\infty}$-algebras can not be tensored
easily to form a tensor $A_{\infty}$-algebra. Instead, the construction is
difficult and has attracted various literature [26, 28]. This includes an
advanced construction of the tensor product $A\otimes A^{\operatorname{op}}$
as $A_{\infty}$-algebra.

This way, $A\otimes A^{\operatorname{op}}$ itself becomes an $A$-bimodule.
This construction should not be confused with the naive bimodule action of $A$
mentioned in [29, Theorem 1.8]. In this naive action, $A$ acts on the two
tensor components of the graded vector space $A\otimes A^{\operatorname{op}}$
separately. In the present section, we avoid all difficulties by restricting
to the case of $A$ being a dg algebra whenever we need the algebra
$A^{\operatorname{e}}$ or $A\otimes A^{\operatorname{op}}$ as $A$-bimodule.

In the definition of a category of modules $\operatorname{Mod}A$, one is not
limited to choosing the specific resolution $T(A[1])\otimes M$ of $M$. Instead
one may choose an arbitrary “projective replacement”. In case $A$ is an
$A_{\infty}$-algebra, this is a topic of research, but if $A$ is a dg algebra,
then the correct notion of “projective replacement” is to be K-projective:

######  Definition 3.15 ([27, Section 4.2]).

Let $A$ be a dg algebra. Then a dg module $P$ is K-projective if the dg hom
space $\operatorname{Hom}_{\operatorname{Mod}A}(P,Q)$ is acyclic whenever $Q$
is an acyclic dg module. A K-projective replacement of
$M\in\operatorname{Mod}^{\mathrm{fd}}A$ is a K-projective dg module $P$
together with a quasi-isomorphism $P\to M$.

######  Remark 3.15.

Keller shows in [21, Section 3.2] that K-projective replacements can be found
by resolving a module into a complex of so-called dg-projective modules. A
module $P$ is called dg-projective if it is a cofibrant object with respect to
the projective model structure on the category of dg categories. Explicitly,
$P$ is dg-projective if for every surjective quasi-isomorphism
$L\twoheadrightarrow M$, every morphism $P\to M$ factors through $L$.

######  Remark 3.16.

The hom spaces in $\operatorname{Mod}^{\mathrm{fd}}A$ enjoy various names in
the literature, thanks to the fact that their cohomology can also be built as
derived hom functor. Generally, the cohomology hom spaces
$\operatorname{H}^{\bullet}\operatorname{Hom}_{\operatorname{Mod}^{\mathrm{fd}}A}(M,N)$
may be denoted $\operatorname{RHom}_{A}^{\bullet}(M,N)$ or
$\operatorname{Ext}_{A}^{\bullet}(M,N)$. When $M=N$, the dg hom space
$\operatorname{Hom}_{\operatorname{Mod}^{\mathrm{fd}}A}(M,M)$ is actually a dg
algebra itself and its minimal model becomes an $A_{\infty}$-algebra. This
$A_{\infty}$-algebra is commonly denoted
$\operatorname{RHom}_{A}^{\bullet}(M,M)$ as well, despite the fact that
classical derived hom functors do not retain homotopy information.

######  Remark 3.17.

If $A$ is a dg algebra, then the Hochschild cohomology
$\operatorname{HH}^{\bullet}(A,M)$ with coefficients in an $A$-bimodule $M$ is
by definition equal to
$\operatorname{H}^{\bullet}\operatorname{Hom}_{\operatorname{Mod}A^{\operatorname{e}}}(A,M)$.

###  3.5 Van den Bergh and Serre duality

In this section, we recall Van den Bergh and Serre duality for ordinary (non-
dg) Calabi-Yau algebras. As it turns out, the Calabi-Yau property for ordinary
algebras is closely related to having a self-dual bimodule resolution. This
fact has been observed and exploited in [9].

######  Definition 3.16 ([9]).

Let $A$ be an algebra and $0\to P^{n}\to\ldots\to P^{0}\to A$ be a projective
$A$-bimodule resolution of $A$ by finitely generated bimodules. Then
$P^{\bullet}$ is a self-dual resolution if
$\operatorname{Hom}_{A^{\operatorname{e}}}(P^{\bullet},A^{\operatorname{e}})\cong
P^{n-\bullet}$ as complexes of $A$-bimodules:

${0}$${\operatorname{Hom}_{A^{\operatorname{e}}}(P^{0},A^{\operatorname{e}})}$${\ldots}$${\operatorname{Hom}_{A^{\operatorname{e}}}(P^{n},A^{\operatorname{e}})}$${0}$${0}$${P^{n}}$${\ldots}$${P^{0}}$${0}$

$\scriptstyle\sim$

$\scriptstyle\sim$

######  Lemma 3.4.

If $A$ has a selfdual projective bimodule resolution of length $n\geq 1$ then
$A$ is CY$n$.

###### Proof.

Let $P^{\bullet}$ be a selfdual bimodule resolution. By definition, the
Hochschild cohomology $\operatorname{HH}^{k}(A,A^{\operatorname{e}})$ is the
homology in degree $k$ of the complex
$\operatorname{Hom}_{A^{\operatorname{e}}}(P^{\bullet},A^{\operatorname{e}})$.
By selfduality, this homology is just the homology in degree $n-k$ of
$P^{\bullet}$, which is $A$ if $k=n$ and zero otherwise. This shows that $A$
is CY$n$. ∎

A bimodule resolution of $A$ gives rise to resolutions for all left and right
$A$-modules. The idea is to tensor the resolution on the right or left side
with $M$, respectively:

######  Lemma 3.5 ([7, Lemma 2.4]).

Let $A$ be an algebra and $P^{\bullet}\to A$ a projective bimodule resolution.
Let $M$ be a left $A$-module. Then $P^{\bullet}\otimes_{A}M\to M$ is a
resolution for $M$.

Van den Bergh duality is a Poincaré-style theorem for Hochschild homology and
cohomology of CY$n$ algebras. This duality was first observed in [8] and we
recall it as follows:

######  Theorem 3.3 ([8]).

Let $A$ be a CY$n$ algebra. If $A$ has a finite projective resolution of
finitely generated bimodules, then

$\operatorname{HH}^{k}(A,M)\cong\operatorname{HH}_{n-k}(A,M).$

###### Proof.

We recall here the proof in the easy case where $A$ has a selfdual resolution
$P^{\bullet}$. We compute

$\displaystyle\operatorname{HH}^{k}(A,M)$
$\displaystyle=\operatorname{H}^{k}\operatorname{Hom}_{A^{\operatorname{e}}}(P^{\bullet},M)$
$\displaystyle=\operatorname{H}^{k}\big{(}\operatorname{Hom}_{A^{\operatorname{e}}}(P^{\bullet},A^{\operatorname{e}})\otimes_{A^{\operatorname{e}}}M\big{)}$
$\displaystyle=\operatorname{H}^{k}(P^{n-\bullet}\otimes_{A^{\operatorname{e}}}M)=\operatorname{HH}_{n-k}(A,M).$

∎

As we recapitulate in Lemma 3.6, a CY$n$ algebra has the property that the
$n$-th shift is a Serre functor for its derived category. In [9], CY$n$
algebras were precisely defined by this characteristic property. It is unclear
to which extent the definitions are equivalent.

######  Lemma 3.6 ([7, Proposition 2.3]).

Let $A$ be an algebra with a finite projective resolution of finitely
generated bimodules. If $A$ is CY$n$, then for all finite-dimensional
$A$-modules $M,N$ there are natural isomorphisms of graded vector spaces

${\operatorname{Ext}_{A}^{\bullet}(M,N)}^{\vee}\cong\operatorname{Ext}_{A}^{n-\bullet}(N,M).$

###### Proof.

We recall here the proof from [7]. The first step of the proof is to realize
that thanks to Lemma 3.5, $\operatorname{Ext}^{k}_{A}(M,N)$ equals the
Hochschild cohomology
$\operatorname{HH}^{k}(A,\operatorname{Hom}_{\mathbb{C}}(M,N))$:

$\operatorname{Ext}^{k}_{A}(M,N)=\operatorname{H}^{k}\operatorname{Hom}_{A}(P^{\bullet}\otimes
M,N)=\operatorname{H}^{k}\operatorname{Hom}_{A^{\operatorname{e}}}(P^{\bullet},\operatorname{Hom}_{\mathbb{C}}(M,N))=\operatorname{HH}^{k}(A,\operatorname{Hom}_{\mathbb{C}}(M,N)).$

Here $\operatorname{Hom}_{\mathbb{C}}(M,N)$ is the $A$-bimodule with the left
factor acting on $N$ and the right factor acting on $M$: $(a\varphi
b)(m)=a\varphi(bm)$.

The second step is to show
$\operatorname{HH}^{k}(A,{M}^{\vee})\cong{\operatorname{HH}_{k}(A,M)}^{\vee}$
for finite-dimensional $A$-bimodules $M$. Here the vector space dual
${M}^{\vee}$ is also an $A$-bimodule by letting the left factor act from the
right on $M$ and the right factor act from the left on $M$, explicitly
$(a\varphi b)(m)=\varphi(bma)$. We compute

$\operatorname{HH}^{k}(A,{M}^{\vee})=\operatorname{H}^{k}\operatorname{Hom}_{A^{\operatorname{e}}}(P^{\bullet},\operatorname{Hom}_{\mathbb{C}}(M,\mathbb{C}))=\operatorname{Hom}_{\mathbb{C}}(P^{\bullet}\otimes_{A^{\operatorname{e}}}M,\mathbb{C})={\operatorname{HH}_{k}(A,M)}^{\vee}.$

Combining the first two steps with Van den Berg duality, we conclude

$\displaystyle\operatorname{Ext}_{A}^{k}(M,N)$
$\displaystyle\cong\operatorname{HH}^{k}(A,\operatorname{Hom}_{\mathbb{C}}(M,N))$
$\displaystyle=\operatorname{HH}^{k}(A,{\operatorname{Hom}_{\mathbb{C}}(N,M)}^{\vee})$
$\displaystyle\cong{\operatorname{HH}_{k}(A,\operatorname{Hom}_{\mathbb{C}}(N,M))}^{\vee}$
$\displaystyle\cong{\operatorname{HH}^{n-k}(A,\operatorname{Hom}_{\mathbb{C}}(N,M))}^{\vee}$
$\displaystyle\cong{\operatorname{Ext}^{n-k}_{A}(N,M)}^{\vee}.$

This finishes the proof. ∎

######  Remark 3.18.

The duality between the two Ext spaces can also be interpreted as a pairing

$\langle-,-\rangle:\operatorname{Ext}_{A}^{\bullet}(M,N)\times\operatorname{Ext}_{A}^{n-\bullet}(N,M)\to\mathbb{C}.$

This pairing can also be expressed by means of the traces
$\operatorname{Tr}_{M}$, $\operatorname{Tr}_{N}$ on
$\operatorname{Ext}_{A}^{\bullet}(M,M)$ and
$\operatorname{Ext}_{A}^{\bullet}(N,N)$ as

$\langle x,y\rangle=\operatorname{Tr}_{N}(x\circ
y)=(-1)^{|x||y|}\operatorname{Tr}_{M}(y\circ x),\quad
x\in\operatorname{Ext}_{A}^{\bullet}(M,N),\quad
y\in\operatorname{Ext}_{A}^{n-\bullet}(N,M).$

Here $\circ$ denotes the composition of extensions, equivalently the product
of $\operatorname{H}\operatorname{Mod}A$. The trace
$\operatorname{Tr}_{M}:\operatorname{Ext}_{A}^{\bullet}(M,M)\to\mathbb{C}$ is
induced from the trace on the complex
$\operatorname{Hom}_{A}(P^{\bullet}\otimes M,P^{\bullet}\otimes M)$ which
computes $\operatorname{Ext}^{\bullet}(M,M)$. The correct signs were computed
in [9, Appendix].

###  3.6 Jacobi algebras

In this section, we recall characterizations of ordinary (non-dg) CY3
algebras. The core idea is that most CY3 algebras are the Jacobi algebra of a
quiver with superpotential. Conversely, most quivers with superpotential gives
rise to a CY3 algebra. In the present section, we recall the precise criteria
from [9]. In particular, we recall superpotentials and their associated Jacobi
algebras as well as a candidate bimodule resolution for Jacobi algebras. This
section serves as a direct preparation for section 6.

We start by fixing terminology for cyclic elements of quiver algebras as
follows:

######  Definition 3.17.

Let $Q$ be a quiver. A path in $Q$ is a cycle if it starts and ends at the
same vertex. If $p$ is a cycle in $Q$, we denote by
$p_{\operatorname{cyc}}\in\mathbb{C}Q$ the sum of its cyclic permutations. We
extend this assignment linearly to $\mathbb{C}Q$ and denote it by $p\mapsto
p_{\operatorname{cyc}}$ as well. An element $W\in\mathbb{C}Q$ is cyclic if it
lies in the image of this map. Explicitly, $W$ is cyclic if it is a linear
combination of cycles whose coefficients are invariant under cyclic
permutation:

$W=\sum_{\text{cycles }a_{k}\ldots a_{1}}\lambda_{a_{k}\ldots
a_{1}}a_{k}\ldots a_{1},\quad\text{with}\quad\forall
i=1,\ldots,k:~{}~{}\lambda_{a_{k}\ldots a_{1}}=\lambda_{a_{i-1}\ldots
a_{i+1}a_{i}}.$

A superpotential on a quiver $Q$ is defined as a cyclic element
$W\in\mathbb{C}Q$ which is a linear combination of paths of length at least
two:

######  Definition 3.18.

A superpotential is a cyclic element $W\in\mathbb{C}Q_{\geq 2}$. Its relations
are the elements

$\partial_{a}W=\sum_{\begin{subarray}{c}\text{paths }a_{k}\ldots a_{1}\\\
\text{with }a_{k}=a\end{subarray}}\lambda_{a_{k}\ldots a_{1}}a_{k-1}\ldots
a_{1},\quad a\in Q_{1}.$

Its Jacobi algebra is given by

$\operatorname{Jac}(Q,W)=\frac{\mathbb{C}Q}{(\partial_{a}W)}.$

Here $(\partial_{a}W)$ denotes the two-sided ideal generated by the partial
derivatives $\partial_{a}W$ for $a\in Q_{1}$.

######  Remark 3.19.

A typical assumption in the literature is that the paths are of length at
least three. In fact, length two term gives rise to a single arrow being
contained in one relation $\partial_{a}W$. The effect is that this arrow is
killed in the Jacobi algebra.

The original paper of Ginzburg [9] formulated the expectation that all CY3
algebras “appearing in nature” are Jacobi algebras of quivers with
superpotential. This expectation was largely verified in [9], with the core
result that a quiver algebra with graded relations which is CY3 is necessarily
of the form $\operatorname{Jac}(Q,W)$:

######  Theorem 3.4 ([9, Theorem 3.1]).

Let $Q$ be a quiver and let $A=\mathbb{C}Q/I$ be the quotient by a finitely
generated graded ideal $I\subseteq\mathbb{C}Q_{\geq 2}$. If $A$ is CY3, then
there exists a superpotential $W$ such that
$\mathbb{C}Q/I\cong\operatorname{Jac}(Q,W)$.

The idea of the proof is to explore the structure of $I$ in terms of
resolutions for the simple modules of $A$. These first bits of these
resolutions are standard and do not depend on the ideal $I$. In contrast, the
last bits depend on $I$ but can be guessed by consideration on the dimension
of the Ext spaces based on the assumption that $A$ is CY3.

Conversely, not every Jacobi algebra is CY3. There is however a precise
criterion due to [9] as well. The criterion is formulated in terms of a
“candidate” bimodule resolution for $A$. We start with the following notation:

######  Definition 3.19.

Let $W\in\mathbb{C}Q_{\geq 2}$ be a superpotential. Then the
$\mathbb{C}Q_{0}$-bimodule generated by $W$ in $\mathbb{C}Q$ is denoted

$W=\mathbb{C}Q_{0}WQ_{0}=\bigoplus_{v\in
Q_{0}}\mathbb{C}vWv\subseteq\mathbb{C}Q.$

The relations space is denoted

$R=\operatorname{span}\\{\partial_{a}W~{}|~{}a\in Q_{1}\\}.$

To introduce the candidate bimodule resolution, let $Q$ be a quiver,
$W\in\mathbb{C}Q_{\geq 3}$ a superpotential. Let us temporarily write
$A=\operatorname{Jac}(Q,W)$ for the Jacobi algebra. The candidate bimodule
resolution has the shape

$0\to
A\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{C}Q_{0}}}W\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{C}Q_{0}}}A\overset{g_{1}}{\to}A\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{C}Q_{0}}}R\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{C}Q_{0}}}A\overset{g_{2}}{\to}A\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{C}Q_{0}}}\mathbb{C}Q_{1}\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{C}Q_{0}}}A\overset{g_{3}}{\to}A\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{C}Q_{0}}}A\to
A\to 0.$ (3.6)

######  Remark 3.20.

The maps in the sequence (6.1) are described as follows:

  * •

For the map $g_{1}$, let $w=\sum_{i\in I}r_{i}a_{i}=\sum_{j\in J}b_{i}r_{j}\in
W$, where all $a_{i}$ an $b_{i}$ are arrows and the $r_{i},r_{j}\in R$ are
compatible elements. The map $g_{1}$ sends the element $1\otimes w\otimes 1$
to

$g_{1}(1\otimes w\otimes 1)=\sum_{i\in I}1\otimes r_{i}\otimes
a_{i}-\sum_{j\in J}b_{j}\otimes r_{j}\otimes 1.$

  * •

For the map $g_{2}$, let $r\in R$ and for $d\geq 0$ write $r=\sum_{i\in
I_{d}}p_{i}^{(d)}a_{i}^{(d)}q_{i}^{(d)}\in R$, where $p_{i}^{(d)}$ are paths
of length $d$ and $a_{i}^{(d)}$ are arrows. The map $g_{2}$ sends the element
$1\otimes r$ to

$g_{2}(1\otimes r\otimes 1)=\sum_{d\geq 0}\sum_{i\in I_{d}}p_{i}^{(d)}\otimes
a_{i}^{(d)}\otimes q_{i}^{(d)}.$

  * •

The map $g_{3}$ is given by $1\otimes a\otimes 1\mapsto a\otimes 1-1\otimes
a$.

  * •

The fourth map is simply the contraction map.

The sequence (6.1) is clearly a chain complex. Whether or not it is exact is
an indicator on whether $A$ is CY3 or not:

######  Theorem 3.5 ([9, Theorem 4.3]).

Let $Q$ be a quiver and $W\in\mathbb{C}Q_{\geq 3}$ a superpotential. Then the
algebra $\operatorname{Jac}(Q,W)$ is CY3 if and only if the sequence (3.6) is
a bimodule resolution for $A$.

###  3.7 Cyclic $A_{\infty}$-algebras

In this section, we recall the correspondence of cyclic $A_{\infty}$-algebras
and Calabi-Yau dg algebras via Koszul duality. In Lemma 3.6 and 3.2, we have
already seen that $A^{!}$ is the endomorphism algebra of the simple module
$\mathbb{C}$ and that modules over a CY$n$ algebra enjoy Serre duality of
degree $n$. This is a strong indication that $\operatorname{H}A^{!}$ has a
cyclic structure of degree $n$. As we recall in this section, this is the norm
and cyclic $A_{\infty}$-algebras correspond to Calabi-Yau algebras under
Koszul duality:

$A$Cyclic$A^{!}$Calabi-YauKoszul

In this section, we start by recalling the notion of cyclic
$A_{\infty}$-algebras. Then we explain that their Koszul duals are so-called
deformed dg-preprojective algebras, which are Calabi-Yau. This explains the
correspondence of cyclic $A_{\infty}$-algebras and Calabi-Yau dg algebras via
Koszul duality. Our primary reference is [33]. We start by recalling cyclic
$A_{\infty}$-algebras, a generalizations of Frobenius algebras:

######  Definition 3.20.

Let $A$ be an $A_{\infty}$-algebra. Then $A$ is cyclic of degree $n$ if it is
equipped with a bilinear nondegenerate pairing $\langle-,-\rangle:A\times
A\to\mathbb{C}$ of degree $-n$ such that $\langle
x,y\rangle=(-1)^{|x||y|}\langle y,x\rangle$ and

$\langle\mu^{k}(a_{k+1},\ldots,a_{2}),a_{1}\rangle=(-1)^{\|a_{k+1}\|(\|a_{k}\|+\ldots+\|a_{1}\|)}\langle\mu^{k}(a_{k},\ldots,a_{1}),a_{k+1}\rangle.$

######  Remark 3.21.

The pairing is a noncommutative and categorical manifestation of a symplectic
form. This is one of the reasons that cyclic $A_{\infty}$-algebras are
dominant in the theory of Fukaya categories. In our context, the notion of
cyclic Fukaya categories is reserved for the A-side of mirror symmetry, while
Calabi-Yau algebras are reserved for the B-side of mirror symmetry.

When discussing Koszul duality for cyclic $A_{\infty}$-algebras, we frequently
need to choose a basis compatible with the pairing $\langle-,-\rangle$. We
recall this piece of linear algebra as follows:

######  Lemma 3.7.

Let $V$ be a finite-dimensional graded vector space over $\mathbb{C}$ and
$\langle-,-\rangle:V\times V\to\mathbb{C}$ a nondegenerate bilinear form of
degree $-n$ with $\langle x,y\rangle=(-1)^{|x||y|}\langle y,x\rangle$. Then
there is a graded basis for $V$ in which $\langle-,-\rangle$ takes the form

$\displaystyle\langle-,-\rangle$
$\displaystyle=\left(\begin{array}[]{c|c|c}\begin{matrix}&I_{k}\\\ \pm
I_{k}&\end{matrix}&&\\\ \hline\cr&I_{l}&\\\
\hline\cr&&\begin{matrix}&-I_{m}\\\ I_{m}&\end{matrix}\end{array}\right),$
(3.10) $\displaystyle\pm I_{k}$
$\displaystyle=\operatorname{diag}((-1)^{(n+1)|x_{1}|},\ldots,(-1)^{(n+1)|x_{k}|}).$

Here $k=\operatorname{dim}V_{<n/2}$ and the first $k$ basis elements
$x_{1},\ldots,x_{k}$ can be freely chosen. The matrix $I_{l}$ appears only if
$n$ is even and $n/2$ is even. The symplectic part consisting of $I_{m}$ and
$-I_{m}$ only appears if $n$ is even and $n/2$ is odd. In these cases we have
$l=\operatorname{dim}V_{n/2}$ and $m=\operatorname{dim}V_{n/2}/2$,
respectively.

###### Proof.

Let $x_{1},\ldots,x_{k}$ be a given graded basis for $V_{<n/2}$. The
restricted pairing $\langle-,-\rangle:V_{<n/2}\times V_{>n/2}\to\mathbb{C}$
gives an isomorphism $\varphi:V_{>n/2}\xrightarrow{\sim}{V}^{\vee}_{<n/2}$ and
the basis $x_{1},\ldots,x_{k}$ gives rise to a dual basis
${x}^{\vee}_{1},\ldots,{x}^{\vee}_{i}\in{V}^{\vee}_{<n/2}$. Pick $x_{j}^{*}\in
V_{>n/2}$ such that $\varphi(x_{j}^{*})={x}^{\vee}_{j}$. Then

$\langle
x_{i},x_{j}^{*}\rangle=\varphi(x_{j}^{*})(x_{i})={x}^{\vee}_{j}(x_{i})=\delta_{ij},\quad\langle
x_{j}^{*},x_{i}\rangle=(-1)^{(n-|x_{j}|)|x_{i}|}\delta_{ij}=(-1)^{(n+1)|x_{i}|}\delta_{ij}.$

Picking $x_{1},\ldots,x_{k},x_{1}^{*},\ldots,x_{k}^{*}$ as first part of the
basis sets up the first diagonal block of the matrix. If $n$ is odd, then