Calabi-Yau algebras and superpotentials

Abstract

We prove that complete $d$-Calabi-Yau algebras in the sense of Ginzburg are derived from superpotentials.

Appendices

Appendix A: The bar cobar formalism

1.1 Weak equivalences

We survey the bar cobar formalism for subsequent dualization to the pseudo-compact case. We use [16, 18, 24, 26, 28] as modern references. We use some notations that were already introduced in Sect. 5.

If $C\in \mathrm{Cog }(l)$, $A\in \mathrm{Alg }(l)$, then

$$\begin{aligned} \mathrm{Hom }_{l^e}(\bar{C},\bar{A}) \end{aligned}$$

is a DG-vector space and the convolution product $*$ makes it into a DG-algebra. A twisting cochain is an element $\tau \in \mathrm{Hom }_{l^e}(\bar{C},\bar{A})_1$ satisfying the Maurer-Cartan equation

$$\begin{aligned} d\tau +\tau *\tau =0 \end{aligned}$$

Let $\mathrm{Tw }(C,A)$ denote the set of twisting cochains in $\mathrm{Hom }_{l^e}(\bar{C},\bar{A})$. It is easy to show that $\mathrm{Tw }(-,A)$ is representable when restricted to complete augmented $l$-DG coalgebras. The representing object is called the bar construction on $A$ and is denoted by $BA$. Likewise $\mathrm{Tw }(C,-)$ is representable. The representing object is called the cobar construction on $C$ and is denoted by $\Omega C$. Thus, we obtain natural isomorphisms

$$\begin{aligned} \mathrm{Alg }(\Omega C,A)\cong \mathrm{Tw }(C,A)\cong \mathrm{Cog }(C,BA) \end{aligned}$$

(12.1)

(the right one if $C$ is cocomplete).

A weak equivalence between objects in $\mathrm{Alg }(l)$ is defined to be a quasi-isomorphism. This naive definition does not work for coalgebras. A morphism $p:C\rightarrow C'$ in $\mathrm{Cogc }(l)$ is said to be a weak equivalence if $\Omega p:\Omega C\rightarrow \Omega C'$ is a quasi-isomorphism. This leads to the following result.

Theorem 12.1

([24], Thm 1.3.12). The functors $(\Omega ,B)$ preserve weak equivalences, and furthermore, they define inverse equivalences between the categories $\mathrm{Alg }(l)$ and $\mathrm{Cogc }(l)$, localized at weak equivalences.

In particular, the counit/unit maps for (12.1), respectively, given by,

$$\begin{aligned} \Omega BA\rightarrow A\end{aligned}$$

(12.2)

$$\begin{aligned} C\rightarrow B\Omega C \end{aligned}$$

(12.3)

are weak equivalences.

These weak equivalences are part of a model structure on $\mathrm{Cogc }(l)$, which we will not fully specify. Let us mention, however, that every object is cofibrant and the fibrant objects are the $l$-DG-coalgebras, which are cofree when forgetting the differential [24, §1.3].

A weak equivalence between augmented $l$-DG-coalgebras is a quasi-isomorphism but not necessarily the other way around (see [24, §1.3.5] for a counter example). This can be repaired in the following typical case.

Proposition 12.2

([24], Prop. 1.3.5.1) Assume the gradings on $C,C'\in \mathrm{Cogc }(l)$ are concentrated in degrees $\ge$0. Then a weak equivalence between $C$ and $C'$ is the same as a quasi-isomorphism.

For completeness, we recall the standard constructions of $BA$ and $\Omega C$. If $V$ is a graded $l$-bimodule, then the tensor algebra $T_lV=\bigoplus _{n\ge 0} V^{\otimes _l n}$ becomes in a natural way an augmented graded $l$-coalgebra if we put $\overline{T_l V}= \bigoplus _{n> 0} V^{\otimes _l n}$ and define the coproduct on $T_lV$ as

$$\begin{aligned} \Delta (v_1|\cdots | v_n)=\sum _{i=0,\ldots ,n}(v_1| \cdots | v_i)\otimes (v_{i+1}|\cdots | v_n) \end{aligned}$$

where as customary $(v_1|\cdots |v_n)$ denotes $v_1\otimes \cdots \otimes v_n$ considered as an element of $V^{\otimes _l n}\subset T_lV$ and $()=1$.

If $A$ is an augmented $l$-DG-algebra then $BA=T_l(\Sigma \bar{A})$ with the codifferential $d$ on $T_l(\Sigma \bar{A})$ being defined via its Taylor coefficients $d_n:(\Sigma \bar{A})^{\otimes n}\hookrightarrow T(\Sigma \bar{A}) \xrightarrow {d}T(\Sigma \bar{A})\xrightarrow {\text {projection}} \Sigma A$

$$\begin{aligned} \begin{aligned} d_1(sa)&=-sda\\ d_2(sa{\mid } sb)&=(-1)^{|a|}s(ab)\\ d_n&=0\qquad n\ge 3 \end{aligned} \end{aligned}$$

(12.4)

for $a,b\in A$.

If $C$ is a DG-$l$-coalgebra, then $\Omega C=T_l(\Sigma ^{-1}\bar{C})$ and the differential is given by

$$\begin{aligned} d(s^{-1}c)=-s^{-1}dc+(-1)^{|c_{(1)}|}(s^{-1}c_{(1)}|s^{-1}c_{(2)}) \end{aligned}$$

(12.5)

for $c\in C$.

1.2 Koszul duality

Let $A\in \mathrm{Alg }(l)$. We recall the standard model structure on $\mathrm{DGMod }(A)$.

• (1) The weak equivalences are the quasi-isomorphisms.

• (2) The fibrations are the surjective maps.

• (3) The cofibrations are the maps, which have the left lifting property with respect to the acyclic fibrations.

It is possible to describe cofibrations more explicitly as retracts of standard cofibrations, but we will not do it.

Now let $C\in \mathrm{Cogc }(l)$. The following model structure on $\mathrm{DGComod }(C)$ is defined in [28, §8.2].

• (1) The weak equivalences are the morphisms with a coacyclic cone.

• (2) The fibrations are surjective morphisms with kernel, which is injective when forgetting the differential.

• (3) The cofibrations are the injective morphisms.

An object is coacyclic if it is in the smallest subcategory of the homotopy category of $C$, which contains total complexes of short exact sequences and is closed under arbitrary coproducts. This model structure looks different from the one defined in [24, §2.2.2]. However, both model structures are Quillen equivalent to the one on $\mathrm{DGMod }(A)$ for $A=\Omega C$, defined above (see [24, Thm 2.2.2.2] and [28, §8.4]). So they have the same weak equivalences. Since they also have the same cofibrations, they are the same.

We now discuss this Quillen equivalence. Let $M\in \mathrm{DGComod }(C^\circ )$ and $N\in \mathrm{DGMod }(A)$. Then $M\otimes _l N$ becomes a left DG-module over $\mathrm{Hom }_{l^e}(\bar{C},\bar{A})$ if we let $\tau \in \mathrm{Hom }_{l^e}(\bar{C},\bar{A})$ act by

$$\begin{aligned} \delta _\tau =({\mathrm{id }}\otimes \mu )\circ ({\mathrm{id }}\otimes \tau \otimes {\mathrm{id }})\circ (\Delta \otimes {\mathrm{id }}) \in \mathrm{End }(M\otimes _l N) \end{aligned}$$

In particular, if $\tau \in \mathrm{Tw }(C,A)$, then $\delta _\tau$ satisfies the Maurer-Cartan equation in $\mathrm{End }(M\otimes _l N)$. We let $M\otimes _\tau N$ be equal to $M\otimes _l N$ but with $\delta _\tau$ added to the differential.

There exists also an analogue of this construction in case $M\in \mathrm{DGMod }(A^\circ )$ and $N\in \mathrm{DGComod }(C)$. We leave the easy to guess formulas to the reader.

Here are some useful identities

$$\begin{aligned} (M\otimes _\tau A)\otimes _A N&=M\otimes _\tau N\\ M\square _C (C\otimes _\tau N)&=M\otimes _\tau N \end{aligned}$$

There is an analogue of the twisting construction for $\mathrm{Hom }$. Let $M\in \mathrm{DGComod }(C)$ and $N\in \mathrm{DGMod }(A)$. Then $\mathrm{Hom }_l(M,N)$ becomes a left DG-module over $\mathrm{Hom }_{l^e}(\bar{C},\bar{A})$ if we let $\tau \in \mathrm{Hom }_{l^e}(\bar{C},\bar{A})$ act by

$$\begin{aligned} \delta _\tau (\phi )=\mu \circ (\tau \otimes \phi )\circ \Delta \end{aligned}$$

If $\tau \in \mathrm{Tw }(C,A)$ then we let $\mathrm{Hom }_\tau (M,N)$ be equal to $\mathrm{Hom }_l(M,N)$ but with $\delta _\tau$ added to the differential. Again this construction may also be performed with right (co)modules.

Now we have the following basic identities

$$\begin{aligned} \mathrm{Hom }_A(A\otimes _\tau M,N)&=\mathrm{Hom }_\tau (M,N)\\ \mathrm{Hom }_C(M,C\otimes _\tau N)&=\mathrm{Hom }_\tau (M,N) \end{aligned}$$

which yield a pair of adjoint functors [24, Theorem 2.2.2.2]

$$\begin{aligned} \begin{aligned} L:\mathrm{DGComod }(C)\rightarrow \mathrm{DGMod }(A):M\mapsto A\otimes _\tau M\\ R:\mathrm{DGMod }(A)\rightarrow \mathrm{DGComod }(C):N\mapsto C\otimes _\tau N \end{aligned} \end{aligned}$$

(12.6)

Below we let $\tau _u$ be the twisting cochain $\bar{C}\rightarrow \overline{\Omega C}$ given by the obvious map. This is the universal twisting cochain corresponding to the identity map $\Omega C\rightarrow \Omega C$ in (12.1). In [28, §8.4], it is shown that in case $A=\Omega C$ and $\tau =\tau _u$ the adjoint pair $(L,R)$ introduced above defines a Quillen equivalence. In particular, a map $M\rightarrow N$ in $\mathrm{DGComod }(C)$ is a weak equivalence if and only if $\Omega C\otimes _\tau M\rightarrow \Omega C\otimes _\tau N$ is a quasi-isomorphism.

The following result is proved in a similar way as Proposition 12.2.

Lemma 12.3

Assume that the grading on $C\in \mathrm{Cogc }(l)$ is concentrated in degrees $\ge$0 and $M,N\in \mathrm{DGComod }(C)$ are concentrated in degrees $\ge \! -n$ for certain $n$. Then a weak equivalence between $M,N$ is the same as a quasi-isomorphism.

1.3 $A_\infty$-algebras and minimal models

By definition, a (non-unital) $l$-$A_\infty$-algebra is an $l$-bimodule $A$ together with an $l$-coderivation $d$ of degree one and square zero on the coalgebra $T_l(\Sigma A)$ compatible with the augmentation. By this, we mean $d(1)=0$, $\epsilon \circ d=0$. We write $\tilde{B}A=(T_l(\Sigma A),d)$ and call $\tilde{B}A$ the bar construction of $A$. An $A_\infty$-morphism $A\rightarrow A'$ is a DG-coalgebra morphism $\tilde{B}A\rightarrow \tilde{B}A'$. We write $\mathrm{Alg }_\infty ^\bullet (l)$ for the category of $l$-$A_\infty$-algebras.

A coderivation on $T_l(\Sigma A)$ compatible with the augmentation is determined by “Taylor coefficients” ($n\ge 1$)

$$\begin{aligned} d_n:(\Sigma A)^{\otimes _l n}\hookrightarrow T_l(\Sigma A)\xrightarrow {d} T_l(\Sigma A) \xrightarrow {\text {projection}} \Sigma \bar{A} \end{aligned}$$

which are of degree one. Introducing suitable signs, the $d_n$ may be transformed into maps

$$\begin{aligned} m_n:A^{\otimes _l n}\rightarrow A \end{aligned}$$

of degree $2-n$ (see, e.g., [24, Lemme 1.2.2.1]). One has $m_1^2=0$, $m_1$ is a derivation for $m_2$, and $m_2$ is associative up to a homotopy given by $m_3$. We view $(A,m_1)$ as a complex and denote its homology by $H^*(A)$. In this way, $(H^*(A),m_2)$ becomes a graded $l$-algebra (without unit).

Likewise, an $A_\infty$-morphism $f:A\rightarrow A'$ is described by maps of degree $1-n$

$$\begin{aligned} f_n:A^{\otimes n}\rightarrow A' \end{aligned}$$

Here $f_1$ is a morphism of complexes $(A,m_1)\rightarrow (A',m'_1)$, which is compatible with the multiplications given by $m_2$, $m'_2$ up to a homotopy given by $f_2$. In particular, $H^*(f_1)$ defines a morphism of graded $l$-algebras.

A morphism $f:A\rightarrow A'$ in $\mathrm{Alg }_\infty ^\bullet (l)$ is said to be a quasi-isomorphism (or weak equivalence) if $f_1:(A,m_1)\rightarrow (A',m'_1)$ is a quasi-isomorphism.

The following is a basic result in the theory of $A_\infty$-algebras.

Proposition 12.4

([24], Cor. 1.4.14) Let $A\in \mathrm{Alg }_\infty ^\bullet (l)$ and let $(H^*(A),m_2)$ be its cohomology algebra. Then there exists a structure of an $l$-$A_\infty$-algebra on $H^*(A)$ of the form $(H^*(A),m_1=0,m_2,m_3,\ldots )$ together with a morphism in $\mathrm{Alg }_\infty ^\bullet (l)$: $f:H^*(A)\rightarrow A$, which lifts the identity $H^*(A)\rightarrow H^*(A)$.

An $A_\infty$-algebra with $m_1=0$ is said to be minimal. Following Kontsevich, one calls the $A_\infty$-algebra $(H^*(A),m_1=0,m_2,m_3,\ldots )$ a minimal model for $A$. It is unique up to non-unique isomorphism of $l$-$A_\infty$-algebras.

There is an obvious augmented version of the theory of $A_\infty$-algebras. An augmented $l$-$A_\infty$-algebra is an $l$-$A_\infty$-algebra $A$ equipped with a decomposition of $l$-bimodules $A=l\oplus \bar{A}$ such that $\bar{A}$ is a sub $l$-$A_\infty$-algebra of $A$ and $1\in l$ is a strict unit, i.e., $m_1(1)=0$, $m_2(1,a)=a$, $m_2(a,1)=a$ and $m_n(\ldots ,1,\ldots )=0$ for $n\ge 3$. Note that the $A_\infty$-structure on $A$ is completely determined by that of $\bar{A}$.

Likewise, an morphism of augmented $A_\infty$-algebras $f:A\rightarrow A'$ is a morphism of $l$-$A_\infty$-algebras that restricts to a morphism of $l$-$A_\infty$-algebras $\bar{A}\rightarrow \bar{A}'$ such that $f_1(1)=1$ and $f_n(\ldots ,1,\ldots )=0$ for $n\ge 2$. Again $f$ is completely determined by its restriction to $\bar{A}$. We denote the category of augmented $l$-$A_\infty$-algebras by $\mathrm{Alg }_\infty (l)$.

For $A\in \mathrm{Alg }_\infty (l)$, we put $BA=T_l(\Sigma \bar{A})$ and then the $A_\infty$-structure on $\bar{A}$ defines a codifferential on $BA$ compatible with the augmentation. Conversely, augmented $A_\infty$-algebras may be defined in terms of codifferentials on $T_l(\Sigma \bar{A})$ which are compatible with the augmentation.

If $A$ is an augmented $l$-$A_\infty$-algebra, then there is a (natural) $l$-$A_\infty$-morphism $A\rightarrow \Omega BA$ to the DG-algebra $\Omega BA$. This morphism is a quasi-isomorphism (see, e.g., [24, Lemma 2.3.4.3]). The DG-algebra $\Omega BA$ is called the DG-envelope of $A$.

Lemma 12.5

If $A\rightarrow A'$ is an $A_\infty$-quasi-isomorphism, then $BA\rightarrow BA'$ is a weak equivalence.

Proof

We have to show that $\Omega BA\rightarrow \Omega BA'$ is a quasi-isomorphism. This follows from the fact that we have we have a commutative diagram

(12.7)

$\square$

Lemma 12.6

Assume that $C\in \mathrm{Cogc }(l)$ is weakly equivalent to $(T_l V,d)$. Then there is an augmented $l$-$A_\infty$-quasi-isomorphism $l\oplus \Sigma ^{-1} V \rightarrow \Omega C$.

Proof

Note that giving the codifferential $d$ on $T_l V$ is precisely the same as defining an augmented $l$-$A_\infty$-structure on $l+\Sigma ^{-1} V$. As $(T_l V,d)$ is fibrant (see above) the weak equivalence $C\rightarrow T_lV$ is represented by an actual map of augmented $l$-DG-coalgebras. As $(T_l V,d)=B( l+\Sigma ^{-1} V)$, we have the following quasi-isomorphisms

$$\begin{aligned} \Omega C\xrightarrow {DG} \Omega T_l V=\Omega B(l+\Sigma ^{-1} V)\xleftarrow {A_\infty } l+\Sigma ^{-1} V \end{aligned}$$

The first map is in particular an augmented $l$-$A_\infty$-quasi-isomorphism so it can be inverted (e.g., [24, Cor. 1.3.1.3]). This yields what we want. $\square$

1.4 The bar cobar formalism in the pseudo-compact case

In this paper, we use the bar cobar formalism in the context of pseudo-compact algebras and modules. To this end, we simply dualize everything we have explained above, using ${\mathbb D}$. Let $A,C$ be, respectively, objects in $\mathrm{PCAlg }(l)$ and $\mathrm{PCCog }(l)$. We put

$$\begin{aligned} \begin{aligned} BA&={\mathbb D}\Omega {\mathbb D}A\\ \Omega C&={\mathbb D}B{\mathbb D}C \end{aligned} \end{aligned}$$

(12.8)

We may interpret these definitions more concretely. For $V\in \mathrm{PCGr }(l)$ put

$$\begin{aligned} T_l V=\prod _{n\ge 0} V^{\otimes _l n} \end{aligned}$$

One checks that $T_lV$ is naturally a graded augmented pseudo-compact $l$-algebra and coalgebra. Then $BA=T_l(\Sigma \bar{A})$, $\Omega C=T_l(\Sigma ^{-1}\bar{C})$ with the differentials given by the formulas (12.4) (12.5).

We equip $\mathrm{PCAlgc }(l)$ with the dual model structure on $\mathrm{Cogc }(l)$. In particular morphism $p:A\rightarrow A'$ in $\mathrm{PCAlgc }(l)$ is a weak equivalence if $Bp:BA \rightarrow BA'$ is a quasi-isomorphism. An object is cofibrant if it is of the form $(T_l V,d)$ with $V\in \mathrm{PC }(l^e)$ and $d$ compatible with the augmentation.

By similar dualizing, we say that a weak equivalence between objects in $\mathrm{PCCog }(C)$ is the same as a quasi-isomorphism.

We equip the categories $\mathrm{PCDGComod }(C)$ and $\mathrm{PCDGMod }(A)$ with the duals of the model structures on $\mathrm{DGMod }({\mathbb D}C^\circ )$ and $\mathrm{DGComod }({\mathbb D}A^\circ )$.

We dualize the functors $L,R$ in the obvious way: $R={\mathbb D}L{\mathbb D}$, $L={\mathbb D}R {\mathbb D}$. They are given by the same formulas as (12.6) but now we use them with $C=BA$ and the universal (continuous) twisting cochain $\tau _u:\overline{BA}\rightarrow \bar{A}$.

A weak equivalence between objects in $\mathrm{PCDGComod }(C)$ is the same as a quasi-isomorphism. On the other hand, a morphism $M\rightarrow N$ is $\mathrm{PCDGMod }(A)$ is a weak equivalence if and only if $BA\otimes _{\tau _u} M\rightarrow BA\otimes _{\tau _u} N$ is a quasi-isomorphism. The derived categories of $A$ and $C$ are obtained from $\mathrm{PCDGMod }(A)$ and $\mathrm{PCDGComod }(C)$ by inverting weak equivalences.

1.5 Minimal models for pseudo-compact algebras

If $d$ is a differential on $T_l W$ with $W\in \mathrm{PC }(l^e)$ then we will denote its components $W\rightarrow W^{\otimes _l n}$ by $d_n$.

We first note that since ${\mathbb D}T_l W\cong T_l ({\mathbb D}W)$, specifying a differential on $T_l W$ is exactly the same as specifying an augmented $l$-$A_\infty$-structure on $l+\Sigma ^{-1} {\mathbb D}W$ (and this is an honest $A_\infty$-structure, not a pseudo-compact one).

For $A\in \mathrm{PCAlgc }(l)$, we define the Koszul dual of $A$ as (see also [18])

$$\begin{aligned} A^!=\Omega {\mathbb D}A \end{aligned}$$

(12.9)

Thus, $A^!$ is an honest augmented $l$-DG-algebra (not a pseudo-compact DG-algebra).

Proposition 12.7

(Koszul duality, cfr [18]) There is an equivalence of triangulated categories

$$\begin{aligned} D(A)\rightarrow D((A^!)^\circ )^\circ \end{aligned}$$

which sends $\Sigma ^n l$ to $\Sigma ^{-n}A^!$.

Proof

We have

$$\begin{aligned} D(A)\cong D(BA)=D({\mathbb D}\Omega {\mathbb D}A)\cong D((\Omega {\mathbb D}A)^\circ )^\circ \end{aligned}$$

The functor realizing the indicated equivalence is given by

$$\begin{aligned} M\mapsto BA\otimes _{\tau _u} M\cong {\mathbb D}\Omega {\mathbb D}A\otimes _{\tau _u} M\cong {\mathbb D}({\mathbb D}M\otimes _{\tau _u} A^!) \mapsto {\mathbb D}M\otimes _{\tau _u} A^! \end{aligned}$$

We see that $l$ is indeed sent to $A^!$. $\square$

Corollary 12.8

We have as algebras

$$\begin{aligned} \mathrm{Ext }_A^*(l,l)\cong H^*(A^!)^\circ \end{aligned}$$

Proof

We have

$$\begin{aligned} \mathrm{Ext }_A^n(l,l)&=\mathrm{Hom }_{D(A)}(l,\Sigma ^n l)\\&=\mathrm{Hom }_{D(A^{!\circ })^\circ }(A^!,\Sigma ^{-n}A^!)\\&=\mathrm{Hom }_{D(A^{!\circ })}(\Sigma ^{-n} A^!,A^!)\\&=A^!_{n} \end{aligned}$$

One verifies that this identification inverts the order of the multiplication, whence the result. $\square$

Remark 12.9

One may show that $A^!$ actually computes $\mathrm{RHom }_A(l,l)^\circ$.

Proposition 12.10

Let $A\in \mathrm{PCAlgc }(l)$. Then $A$ there is a weak equivalence $\Omega {\mathbb D}A^!\rightarrow A$. Furthermore, the same holds with $A^!$ replaced by any augmented $l$-$A_\infty$-algebra quasi-isomorphic to it. Conversely, if $A$ is weakly equivalent to $(T_l W,d)$ then there is an $A_\infty$-quasi-isomorphism $l+\Sigma ^{-1} {\mathbb D}W\cong A^!$, where the $A_\infty$-algebra structure on $l+\Sigma ^{-1}{\mathbb D}W$ is as introduced above.

Proof

We have

$$\begin{aligned} \Omega {\mathbb D}A^!\cong \Omega {\mathbb D}\Omega {\mathbb D}A=\Omega B A \end{aligned}$$

and $\Omega B A$ is weakly equivalent to $A$ by applying ${\mathbb D}$ to (12.3). This implies that $A$ is weakly equivalent to $\Omega {\mathbb D}A^!$. The fact that $A^!$ may be replaced by any other algebra quasi-isomorphic to it follows from the fact that $\Omega {\mathbb D}A^!={\mathbb D}BA^!$ combined with Lemma 12.5.

Finally, by applying ${\mathbb D}$ to the conclusion of Lemma 12.6 with $C={\mathbb D}A$ and $V={\mathbb D}W$, we obtain $A^!\cong l+\Sigma ^{-1} {\mathbb D}W$. $\square$

Corollary 12.11

Let $A\in \mathrm{PCAlgc }(l)$. Then there exists a weak equivalence $(T_l W,d)\rightarrow A$ such that $d_1=0$.

Proof

We let $l+\Sigma ^{-1} W$ be a minimal augmented $l$-$A_\infty$-model for $A^!$. Then from Proposition 12.10 obtain that $A$ is weakly equivalent to $(T_l W,d)$ where $d_1=0$. $\square$

Following traditional terminology, we call a weak equivalence as in Corollary 12.11 a minimal model for $A$.

Corollary 12.12

Whenever we have a minimal model $T_l W\rightarrow A$, then $W\cong \Sigma ^{-1}({\mathbb D}\mathrm{Ext }^*_A(l,l))_{\le 0}$ and the $m_2$ multiplication on $l+\Sigma ^{-1} {\mathbb D}W\cong \mathrm{Ext }^*_A(l,l)$ for the induced $A_\infty$-structure corresponds to the opposite of the Yoneda multiplication on $\mathrm{Ext }^*_A(l,l)$.

Proof

By Proposition 12.10, we have an $A_\infty$-quasi-isomorphism $l+\Sigma ^{-1} {\mathbb D}W\cong A^!$ and hence an isomorphism as algebras

$$\begin{aligned} l+\Sigma ^{-1} {\mathbb D}W\cong H^*(l+\Sigma ^{-1}{\mathbb D}W)\cong H^*(A^!) \end{aligned}$$

It now suffices to apply Corollary 12.8. $\square$

Appendix B: Hochschild homology of pseudo-compact algebras

Let $A\in \mathrm{PCAlgc }(l)$. It is easy to see that the tensor product $-\otimes _A-$ satisfies the hypotheses of [30, Prop. 4.1] in both arguments, and hence, it may be left derived in both arguments. It is also easy to see that deriving the first argument gives the same result as deriving the second argument. Therefore, we make no distinction between the two and write the result as $-\overset{L}{\otimes }_A-$.

Now we work over $A^e$, which is considered as an object in $\mathrm{Mod }(l^e)$. Our aim is to show the following result

Proposition 13.1

If $A\in \mathrm{PCAlgc }(l)$, then $A\overset{L}{\otimes }_{A^e} A$ is computed by the standard Hochschild complex $(\mathrm{C }(A),b)= ((A\otimes _l T_l(\Sigma A))_l,d_A+d_{\text {Hoch}})$ where $d_{\text {Hoch}}$ is the usual Hochschild differential.

Proof

In Lemma 13.2, below we show that $A\otimes _{\tau _u} BA\otimes _{\tau _u} A$ is a cofibrant replacement for $A$ in $\mathrm{PCDGMod }(A^e)$.

Thus, we have the following formula

$$\begin{aligned} A\overset{L}{\otimes }_{A^e} A=A\otimes _{A^e} (A\otimes _{\tau _u} BA \otimes _{\tau _u} A) \end{aligned}$$

We have

$$\begin{aligned} A\otimes _{A^e} (A\otimes _{\tau _u} BA\otimes _{\tau _u} A)\cong ((A\otimes _l T_l(\Sigma \bar{A}))^l,d_A+d_{\text {Hoch}}) \end{aligned}$$

(13.1)

where $d_{\text {Hoch}}$ is the usual Hochschild differential.

The right-hand side of (13.1) is the reduced Hochschild complex. It is quasi-isomorphic to the standard Hochschild complex which has the form

$$\begin{aligned} ((A\otimes _l T_l(\Sigma A))^l,d_A+d_{\text {Hoch}}) \end{aligned}$$

(see [25, Prop. 1.6.5]). $\square$

Lemma 13.2

$A\otimes _{\tau _u} BA\otimes _{\tau _u} A$ is a cofibrant replacement for $A$ in $\mathrm{PCDGMod }(A^e)$

Proof

We have a Quillen equivalence

$$\begin{aligned}&L^e:\mathrm{PCDGComod }((BA)^e)\rightarrow \mathrm{PCDGMod }(A^e):N\mapsto A\otimes _{\tau _u} N\otimes _{\tau _u} A\nonumber \\&R^e:\mathrm{PCDGMod }(A^e)\rightarrow \mathrm{PCDGComod }(BA^e):M\mapsto BA\otimes _{\tau _u} M\otimes _{\tau _u} BA\qquad \quad \quad \end{aligned}$$

(13.2)

As $A\otimes _{\tau _u} BA\otimes _{\tau _u} A$ is a projective bimodule when forgetting the differential it is cofibrant.

Hence, we have to show that

$$\begin{aligned} \mu _{13}:A\otimes _{\tau _u} BA\otimes _{\tau _u} A\rightarrow A:a\otimes b\otimes c\mapsto a\epsilon (b)c \end{aligned}$$

is a weak equivalence in $\mathrm{PCDGMod }(A^e)$. By the Quillen equivalence between $\mathrm{PCDGMod }(A^e)$ and $\mathrm{PCDGComod }(BA^e)$, we may as well show that the adjoint map

$$\begin{aligned} \Delta _{13}:BA \rightarrow BA\otimes _{\tau _u} A \otimes _{\tau _u} BA:c\mapsto c_{(1)}\otimes 1 \otimes c_{(2)} \end{aligned}$$

(13.3)

is a weak equivalence, or equivalently a quasi-isomorphism of $BA$-bi-comodules. If we view (13.3) as a map of left comodules, then it is precisely the unit map

$$\begin{aligned} BA\rightarrow RL(BA) \end{aligned}$$

which is a weak equivalence (and hence, quasi-isomorphism) since $(L,R)$ forms a Quillen equivalence. $\square$

Appendix C: Symmetry for Hochschild homology

Assume that $A$ is an $l$-algebra and let $M$ be a finitely generated projective $A$-bimodule. Put $M^D=\mathrm{Hom }_{A^e}(M,A\otimes A)$. Then an element $\xi \in M\otimes _{A^e} M$ defines a bimodule map

$$\begin{aligned} \xi ^+:M^D\rightarrow M:\phi \mapsto \phi (\xi ')''\xi '' \phi (\xi ')' \end{aligned}$$

and conversely using the identification

$$\begin{aligned} \mathrm{Hom }_{A^e}(M^D,M)\cong M\otimes _{A^e} M \end{aligned}$$

any bimodule morphism $M^D\rightarrow M$ is of the form $\xi ^+$ for some $\xi \in M\otimes _{A^e}M$.

There is a ${\mathbb Z}/2{\mathbb Z}=\{1,\ss \}$-action on $M\otimes _{A^e} M$ such that $\ss (a\otimes b)=b\otimes a$. One checks that

$$\begin{aligned} \ss (\xi )^+=c\circ (\xi ^+)^D \end{aligned}$$

for the canonical isomorphism $c:M\mapsto M^{DD}:m\mapsto (\phi \mapsto \phi (m)''\otimes \phi (m)')$

Hence, if $\xi$ is symmetric and we view $c$ as an identification, then $(\xi ^+)^D=\xi ^+$.

What we have just explained extends to the case where $A$ is an $l$-DG-algebra and $M$ is a perfect object in $D(A^e)$ (where we now use the derived version of $(-)^D$ as introduced in Sect. 7). We will apply it in the case $M=A$. We will prove the following result.

Proposition 14.1

$H_d(\ss )$ acts trivially on $\mathrm{HH }_d(A)=H_d(A\overset{L}{\otimes }_{A^e}A)$. Hence, if $A$ is homologically smooth then any $\eta :A^D\rightarrow {\Sigma ^{-d}} A$ is automatically self-dual.

Proof

For our purpose, we may and we will assume that $A$ is cofibrant. We will use the complex $Y(A) =\Sigma \Omega _l^1 A\otimes \Sigma \Omega _l^1 A\oplus \Sigma (\Omega _l^1 A)\oplus \Sigma (\Omega _l^1 A)_l \oplus (A\otimes A)_l$ to compute $A\overset{L}{\otimes }_{A^e} A$ (see (10.5)).

Taking homology for rows and columns in $Y(A)$, we get two maps

$$\begin{aligned} l,r:Y(A)\rightarrow X(A) \end{aligned}$$

where by a slight abuse of notation we have written $X(A)$ for $\mathrm{cone }((\Omega ^1_lA)_\natural \xrightarrow {\partial _1} A_l)$ (see Sect. 6.2). Note that by Proposition 6.2 $X(A)$ computes the Hochschild homology of $A$.

We claim that $l,r$ are homotopy equivalent. To prove this, we will describe $l$ and $r$ explicitly:

$$\begin{aligned} l(s\omega _1\otimes s\omega _2)&=0&(\text {on}\, \Sigma \Omega _l^1 A\otimes _{A^e}\Sigma \Omega _l^1 A)\\ l(s\omega _1)&=0&\text {(on the first copy of}\, (\Sigma \Omega _l^1 A)_l)\\ l(s\omega _2)&=s\omega _{2,\natural }&\text {(on the second copy of}\, \Sigma (\Omega _l^1 A)_l) \\ l(a\otimes b)&=\overline{ab}&(\text {on}\, (A\otimes _l A)_l) \end{aligned}$$

taking into account that in (10.4), $s\omega _1$ is represented by $s\omega _1\otimes (1\otimes 1)$, $s\omega _2$ is represented by $(1\otimes 1)\otimes s\omega _2$, $a\otimes b$ is represented by $(a\otimes b)\otimes (1\otimes 1)$ and taking homology for rows/columns corresponds to taking homology in the first/second factor. For the last line one needs to take into account the identification (10.6).

Likewise, we have

$$\begin{aligned} r(s\omega _1\otimes s\omega _2)&=0&(\text {on}\, \Sigma \Omega _l^1 A\otimes _{A^e}\Sigma \Omega _l^1 A)\\ r(s\omega _1)&=s\omega _{1,\natural }&\text {(on the first copy of}\, (\Sigma \Omega _l^1 A)_l)\\ r(s\omega _2)&=0&\text {(on the second copy of}\, \Sigma (\Omega _l^1 A)_l)\\ r(a\otimes b)&=(-1)^{|a||b|}\overline{ba}&(\text {on}\, (A\otimes _l A)_l) \end{aligned}$$

Thus, for the difference $m=l-r$

$$\begin{aligned} m(s\omega _1\otimes s\omega _2)&=0\\ m(s\omega _1)&=-s\omega _{1,\natural }\\ m(s\omega _2)&=s\omega _{2,\natural }\\ m(a\otimes b)&=\overline{[a,b]} \end{aligned}$$

Now we define a map of degree $-1$

$$\begin{aligned} h:Y(A)\rightarrow X(A) \end{aligned}$$

by

$$\begin{aligned} h(s\omega _1\otimes s\omega _2)&=0\\ h(s\omega _1)&=0\\ h(s\omega _2)&=0\\ h(a\otimes b)&=s(aDb)_{\natural } \end{aligned}$$

Now we compute $dh=[d,h]=\partial _1\circ h+h\circ (\partial _1\otimes 1+1\otimes \partial _1)$. To this end we have to know $\partial _1^{\text {hor}}$ and $\partial _1^{\text {ver}}$. We compute

$$\begin{aligned} \partial _1^{\text {hor}}(aDb)&=(\partial _1\otimes 1)(aDb\otimes (1\otimes 1))\\&=(ab\otimes 1-a\otimes b) \otimes (1\otimes 1)\\&=ab\otimes 1-a\otimes b \end{aligned}$$

$$\begin{aligned} \partial _1^{\text {ver}}(aDb)&=(1\otimes \partial _1)((1\otimes 1)\otimes aDb)\\&=(1\otimes 1)\otimes (ab\otimes 1-a\otimes b)\\&=1\otimes ab-(-1)^{|a||b|}b\otimes a \end{aligned}$$

taking into account the identification (10.6).

We now find

$$\begin{aligned} (dh)(s\omega _1\otimes s\omega _2)=0 \end{aligned}$$

$$\begin{aligned} (dh)(s(aDb))&=h(ab\otimes 1-a\otimes b)\\&=-s(aDb)_{\natural }&\text {(on the first copy of}\, \Sigma \Omega ^1_l A) \end{aligned}$$

$$\begin{aligned} (dh)(s(aDb))&=h((1\otimes ab-(-1)^{|a||b|}b\otimes a)\\&=s(Dab)_{\natural }-(-1)^{|a||b|}s(bDa)_{\natural }\\&=s(aDb)_{\natural }&\text {(on the second copy of}\, \Sigma \Omega ^1_l A) \end{aligned}$$

$$\begin{aligned} (dh)(a\otimes b)&=\partial _1(s(aDb)_{\natural })&\\&=\overline{[a,b]} \end{aligned}$$

Hence, $h$ is indeed a homotopy connecting $l$ and $r$.

Now we have the following commutative diagram of complexes.

The top line comes from the fact that $Y(A)$ is obtained from tensoring the bimodule resolution of $A$ with itself over $A^e$.

Taking homology, we obtain

Since $r$ and $l$ are homotopy we have $H_d(l)=H_d(r)$ and we are done. $\square$

Appendix D: Koszul duality for Hochschild homology

The definition of the Hochschild mixed complex may be dualized to coalgebras. If $C$ is a counital $l$-DG-coalgebra, then the Hochschild mixed complex of $C$ is $(\mathrm{C }(C),b,B)$ where $(\mathrm{C }(C),b)$ is the sum total complex of a double complex of the form

$$\begin{aligned} 0\rightarrow C^l \xrightarrow {\partial } (C\otimes _l C)^l \xrightarrow {\partial } (C\otimes _l C\otimes _l C)^l\rightarrow \cdots \end{aligned}$$

where $l$ denotes the centralizer and $\partial$ is the dual of the Hochschild differential. $B$ is the dual of the Connes differential. There exist a similar normalized mixed complex denoted by $(\bar{\mathrm{C }}(C),b,B)$. The mixed complexes $\bar{\mathrm{C }}(C)$ and $\mathrm{C }(C)$ are quasi-isomorphic (see [25, Prop. 1.6.5]).

The following result is well known.

Proposition 15.1

Let $C\in \mathrm{Cogc }(l)$. There is a quasi-isomorphism of mixed complexes.

$$\begin{aligned} \mathrm{C }(\Omega C)\rightarrow \mathrm{C }(C) \end{aligned}$$

(15.1)

Proof

Put $A=\Omega C$. Since $C$ is cocomplete $A$ is cofibrant. Hence, we may apply Proposition 6.2 to obtain a quasi-isomorphism

$$\begin{aligned} \mathrm{C }(A)\rightarrow M X(A) \end{aligned}$$

By the dual version of [29, Theorem 4], we have an isomorphism of complexes

$$\begin{aligned} (\Omega _l^1 A)_\natural \cong \Sigma ^{-1}\mathrm{C }(\bar{C}) \end{aligned}$$

By definition $MX(A)=\mathrm{cone }( (\Omega _l^1 A)_\natural \xrightarrow {\partial _1} A_l)$. Since $C$ is augmented we have $C=l\oplus \bar{C}$. We get isomorphisms as graded vector spaces

$$\begin{aligned} \mathrm{cone }( (\Omega _l^1 A)_\natural \xrightarrow {\partial _1} A_l)&=\mathrm{C }(\bar{C})\oplus A_l \\&=\mathrm{C }(\bar{C})\oplus T_l(\Sigma ^{-1}\bar{C})\\&=\bar{\mathrm{C }}(C) \end{aligned}$$

One checks that this isomorphism is compatible with $(b,B)$ and hence yields an isomorphism of mixed complexes

$$\begin{aligned} M X(A)\cong (\bar{\mathrm{C }}(C),b,B) \end{aligned}$$

Combining this with the standard quasi-isomorphism of mixed complexes (see [25, Prop. 1.6.5])

$$\begin{aligned} (\bar{\mathrm{C }}(C),b,B)\rightarrow (\mathrm{C }(C),b,B) \end{aligned}$$

yields indeed a quasi-isomorphism as in (15.1). $\square$

Corollary 15.2

Let $A\in \mathrm{PCAlgc }(l)$. Then we have a quasi-isomorphism of mixed complexes

$$\begin{aligned} C(A^!)\rightarrow {\mathbb D}\mathrm{C }(A) \end{aligned}$$

(15.2)

This quasi-isomorphism is natural in $A$ (taking into account that $A\mapsto A^!$ is a contravariant functor).

Proof

This follows the fact that ${\mathbb D}\mathrm{C }(A)=\mathrm{C }({\mathbb D}A)$ together with Proposition 15.1. The naturality of (15.2) follows from the naturality of (15.1). $\square$

Corollary 15.3

A weak equivalence $A\rightarrow A'$ in $\mathrm{PCAlgc }(l)$ induces a quasi-isomorphism $\mathrm{C }(A)\rightarrow \mathrm{C }(A')$ of mixed complexes.

Proof

By definition of $(-)^!$, we obtain that $A^{\prime !}\rightarrow A^{!}$ is a quasi-isomorphism in $\mathrm{Alg }(l)$. Hence, this induces a quasi-isomorphism $\mathrm{C }(A^{\prime !})\rightarrow \mathrm{C }(A^{!})$. By Corollary 15.2, we get a commutative diagram of mixed complexes

So the rightmost map is indeed a quasi-isomorphism. $\square$

Corollary 15.4

Assume that $A\rightarrow A'$ is a weak equivalence in $\mathrm{PCAlgc }(l)$ between homologically smooth algebras. Then $A$ is exact $d$ Calabi-Yau if and only if this is the case for $A'$.

Proof

Taking into account Corollary 15.3, we have to prove that $\eta \in \mathrm{HH }_d(A)$ is non-degenerate if and only if its image in $\mathrm{HH }_d(A')$ is non-degenerate. This is a formal verification, which we leave to the reader. $\square$