2.3. Let A and B be dg k-algebras. Recall that A is quasi-isomorphic to B, denoted $A \simeq B$ , if there exists a zig-zag of quasi-isomorphisms of dg k-algebras connecting A and B. A dg k-algebra K is called formal if it is quasi-isomorphic to $\mathrm {H}(K)$ .

4.1. An $\mathrm {A}_{\infty }$ -algebra is a graded Q-module A equipped with Q-linear maps

$$ \begin{align*} m_n \colon A^{\otimes n} \to A \quad\text{for } n \geqslant 1 \end{align*} $$

of degree $(n-2)$ satisfying the Stasheff identities

(4.1.1) $$ \begin{align} \sum_{\substack{r+s+t=n\\r,t \geqslant 0, s \geqslant 1}} (-1)^{r+st} m_{r+1+t} \left(\operatorname{\mathrm{id}}^{\otimes r} \otimes m_s \otimes \operatorname{\mathrm{id}}^{\otimes t}\right) = 0. \end{align} $$

Taking $n=1$ , this says that $m_1$ is a degree $-1$ square zero endomorphism of A, so we can (and will) make A a complex with $\partial =m_1$ . Taking $n=2$ yields a product satisfying the Leibniz rule $\partial m_2=m_2(\partial \otimes \operatorname {\mathrm {id}}+\operatorname {\mathrm {id}}\otimes \partial )$ . The next Stasheff identity, for $n=3$ , can be interpreted as saying that $m_2$ is associative up to a homotopy given by $m_3$ ; that is,

$$ \begin{align*} m_2( \operatorname{\mathrm{id}} \otimes m_2-m_2\otimes \operatorname{\mathrm{id}}) = \partial m_3+m_3(\partial \otimes \operatorname{\mathrm{id}}\otimes \operatorname{\mathrm{id}}+\operatorname{\mathrm{id}} \otimes \partial \otimes \operatorname{\mathrm{id}} + \operatorname{\mathrm{id}} \otimes \operatorname{\mathrm{id}}\otimes\partial ). \end{align*} $$

If for some n the Stasheff identity (4.1.1) holds for every integer less than n, then the obstruction

(4.1.2)

is a chain map $A^{\otimes n}\to A$ ; see [Reference Lefèvre-HasegawaLH03, Corollaire B.1.2].

A morphism of $\mathrm {A}_{\infty }$ -algebras $\varphi \colon A \to B$ consists of Q-linear maps

$$ \begin{align*} \varphi_n \colon A^{\otimes n} \to B \quad\text{for } n \geqslant 1 \end{align*} $$

of degree $(n-1)$ satisfying

(4.1.3)

where and , with .

If for some n the Stasheff identity (4.1.3) holds for every integer less than n, then we define

Then the Stasheff identity (4.1.3) holds if and only if

$$ \begin{align*} \operatorname{obs}_n^\varphi = m_1 \varphi_n + (-1)^n \varphi_n( m_1 \otimes \operatorname{\mathrm{id}}^{\otimes (n-1)} + \cdots + \operatorname{\mathrm{id}}^{\otimes (n-1)} \otimes m_1). \end{align*} $$

A morphism $\varphi $ of $\mathrm {A}_{\infty }$ -algebras is a quasi-isomorphism if the chain map $\varphi _1$ is a quasi-isomorphism of complexes. The morphism $\varphi $ is strict if $\varphi _n = 0$ for $n> 1$ . In this case, (4.1.3) simplifies to

(4.1.4) $$ \begin{align} \varphi_1 m_n^A = m_n^B (\varphi_1 \otimes \cdots \otimes \varphi_1). \end{align} $$

The composition of morphisms $\varphi \colon A \to B$ and $\psi \colon B \to C$ is defined by

4.2. An $\mathrm {A}_{\infty }$ -algebra A is strictly unital if there exists $1_A \in A_0$ such that

(4.2.1) $$ \begin{align} \begin{gathered} m_2(1_A \otimes a) = a = m_2(a \otimes 1_A) \quad\text{for all } a \in A \quad\text{and} \\ m_n(a_1 \otimes \cdots \otimes a_{i-1} \otimes 1_A \otimes a_{i+1} \otimes \cdots \otimes a_n) = 0 \quad\text{for all } 1 \leqslant i \leqslant n \end{gathered} \end{align} $$

for any $a_1, \ldots , a_n \in A$ and $n> 2$ . A morphism of strictly unital $\mathrm {A}_{\infty }$ -algebras $\varphi \colon A \to B$ is a morphism of $\mathrm {A}_{\infty }$ -algebras such that

(4.2.2) $$ \begin{align} \begin{gathered} \varphi_1(1_A) = 1_B \quad\text{and} \\ \varphi_n(a_1 \otimes \cdots \otimes a_{i-1} \otimes 1_A \otimes a_{i+1} \otimes \cdots \otimes a_n) = 0 \quad\text{for all } 1 \leqslant i \leqslant n \end{gathered} \end{align} $$

for any $a_j \in A$ and $n> 1$ .

4.3. An $\mathrm {A}_{\infty }$ -algebra is connective if it is concentrated in non-negative degrees. If $A = Q \oplus \bar {A}$ is a graded module and $1_A$ a free generator of the direct summand Q, then A is an $\mathrm {A}_{\infty }$ -algebra with a split unit.

A split unital $\mathrm {A}_{\infty }$ -algebra structure on a graded module concentrated in non-negative degrees is equivalent to the existence of Q-linear maps

$$ \begin{align*} \bar{m}_n \colon \bar{A}^{\otimes n} \to \bar{A} \quad\text{for } n \geqslant 1 \end{align*} $$

of degree $(n-2)$ and Q-linear maps

$$ \begin{align*} h_1 \colon \bar{A} \to Q \quad \text{and} \quad h_2 \colon \bar{A}^{\otimes 2} \to Q \end{align*} $$

of degrees $-1$ and $0$ , respectively, such that for $n \neq 2,3$ , the Stasheff identities (4.1.1) hold when replacing $m_i$ by $\bar {m}_i$ , and for $n=2,3$ ,

$$ \begin{align*} \sum_{\substack{r+s+t=n\\r,t \geqslant 0, s \geqslant 1}} (-1)^{r+st} \bar{m}_{r+1+t} (\operatorname{\mathrm{id}}^{\otimes r} \otimes \bar{m}_s \otimes \operatorname{\mathrm{id}}^{\otimes t}) + (h_{n-1} \otimes \operatorname{\mathrm{id}} - \operatorname{\mathrm{id}} \otimes h_{n-1}) = 0, \end{align*} $$

and additionally,

$$ \begin{align*} \begin{aligned} h_1 \bar{m}_1 &= 0 ,\quad h_1 \bar{m}_2 - h_2 (\bar{m}_1 \otimes \operatorname{\mathrm{id}} + \operatorname{\mathrm{id}} \otimes \bar{m}_1) = 0 \quad \text{and} \\& \qquad h_1 \bar{m}_3 + h_2 (\bar{m}_2 \otimes \operatorname{\mathrm{id}} - \operatorname{\mathrm{id}} \otimes \bar{m}_2) = 0\,; \end{aligned} \end{align*} $$

the former replaces the second and third Stasheff identity and the latter supplements the first three Stasheff identities. In particular, for $n> 2$ , the maps $\bar {m}_n$ are the appropriate restrictions of $m_n$ . For $n=2$ , we obtain $m_2$ by $\bar {m}_2 + h_2$ and additionally enforcing (4.2.1). For $n=1$ , we have $m_1 = \bar {m}_1 + h_1$ . This treatment is similar to [Reference BurkeBur15, Section 3] but is slightly more general since we allow $\bar {A}_0\neq 0$ and hence need $h_2$ as well as $h_1$ . If A were not connective, then we would also need maps $h_n$ for $n\geqslant 3$ . Taken together, the $h_n$ will correspond to the curvature term on the bar construction of A; see 4.7.

4.5. Fix a graded coalgebra $(C,\Delta )$ . Recall C is counital if there exists a counit map $\varepsilon \colon C \to Q$ such that

$$ \begin{align*} (\operatorname{\mathrm{id}} \otimes \varepsilon ) \Delta = \operatorname{\mathrm{id}} = (\varepsilon \otimes \operatorname{\mathrm{id}}) \Delta. \end{align*} $$

We say C is a curved dg coalgebra if it is equipped with a coderivation $\partial $ of degree $-1$ and a curvature term $h \colon C \to Q$ of degree $-2$ such that

$$ \begin{align*} \partial^2 = (h \otimes \operatorname{\mathrm{id}} - \operatorname{\mathrm{id}} \otimes h) \Delta \quad\text{and}\quad h \partial = 0. \end{align*} $$

A curved dg coalgebra C is connected if it is non-negatively graded, counital and $C_0 = Q$ . In this setting, we write $C = Q \oplus \bar {C}$ for $\bar {C} = \ker (\varepsilon )$ and set

for the restrictions to $\bar {C}$ . These maps satisfy the same relations as $\Delta $ , $\partial $ and h.

4.6. The tensor algebra on a graded Q-module V has underlying graded module , and the multiplication

The tensor algebra is bigraded by $\mathrm {T}^{a}_{(n)}(V)_i=(V^{\otimes n})_i$ .

The tensor coalgebra $\mathrm {T}^{c}(V)$ on a graded Q-module V has underlying graded module , and the comultiplication

The tensor coalgebra is bigraded by $\mathrm {T}^{c}_{(n)}(V)_i=(V^{\otimes n})_i$ . The data of an $\mathrm {A}_{\infty }$ -algebra can equivalently be encoded as a differential on a tensor coalgebra, as we see next.

4.7. Let A be a split unital connective $\mathrm {A}_{\infty }$ -algebra. Then the tensor coalgebra has an induced curved dg coalgebra structure. The curvature term has components

$$ \begin{align*} h_1 {\scriptstyle\mathsf{\Sigma}}^{-1} \colon \mathrm{T}^{c}_{(1)}{(\Sigma \bar{A})} \to Q \quad \text{and} \quad h_2 ({\scriptstyle\mathsf{\Sigma}}^{-1})^{\otimes 2} \colon \mathrm{T}^{c}_{(2)}(\Sigma \bar{A}) \to Q \end{align*} $$

and zero otherwise. The coderivation $\partial $ has components

$$ \begin{align*} (-1)^{\frac{k(k+1)}{2}} \sum_{\substack{i+j=n-k\\i,j \geqslant 0}} (\operatorname{\mathrm{id}}^{\otimes i} \otimes {\scriptstyle\mathsf{\Sigma}} \bar{m}_{k} ({\scriptstyle\mathsf{\Sigma}}^{-1})^{\otimes k} \otimes \operatorname{\mathrm{id}}^{\otimes j}) \colon \mathrm{T}^{c}_{(n)}{(\Sigma \bar{A})} \to \mathrm{T}^{c}_{(n-k+1)}{(\Sigma \bar{A})} \end{align*} $$

for $k \geqslant 1$ , and zero otherwise. The map $\partial $ is well defined since A is concentrated in non-negative homological degree. With this structure, is a connected curved dg coalgebra, and we call

the bar construction of A. For $a_1, \ldots , a_n \in \bar {A}$ , we write

For a split unital connective $\mathrm {A}_{\infty }$ -algebra, the canonical projection and inclusion induce a degree $-1$ map of graded modules

Let C be a connected curved dg coalgebra. Then the algebra has an induced dg algebra structure. The differential $m_1$ has components

$$ \begin{align*} \begin{aligned} -\bar{h} {\scriptstyle\mathsf{\Sigma}} \colon \mathrm{T}^{a}_{(1)}&{(\Sigma^{-1} \bar{C})} \to \mathrm{ T}^{a}_{(0)}{(\Sigma^{-1} \bar{C})} ,\quad -{\scriptstyle\mathsf{\Sigma}}^{-1} \bar{\partial} {\scriptstyle\mathsf{\Sigma}} \colon \mathrm{T}^{a}_{(1)}{(\Sigma^{-1} \bar{C})} \to \mathrm{T}^{a}_{(1)}{(\Sigma^{-1} \bar{C})} \\& \qquad\text{and}\quad ({\scriptstyle\mathsf{\Sigma}}^{-1})^{\otimes 2} \bar{\Delta} {\scriptstyle\mathsf{\Sigma}} \colon \mathrm{T}^{a}_{(1)}{(\Sigma^{-1} \bar{C})} \to \mathrm{ T}^{a}_{(2)}{(\Sigma^{-1} \bar{C})}, \end{aligned} \end{align*} $$

and zero otherwise. With this structure, is a split unital connective dg algebra, and we call

the cobar construction of C. For $c_1, \ldots , c_n \in \bar {C}$ , we write

For a connected curved dg coalgebra, the canonical inclusion and projection maps induces a degree $-1$ map of graded modules

$$ \begin{align*} C \to \Sigma^{-1} C \twoheadrightarrow \Sigma^{-1} \bar{C} \hookrightarrow \mathsf{\Omega}{(C)}. \end{align*} $$

4.9. A morphism $\varphi \colon (C,\Delta ,\varepsilon ,\partial ,h) \to (C',\Delta ',\varepsilon ',\partial ',h')$ of connected curved dg coalgebras consists of Q-linear maps

$$ \begin{align*} \varphi_0 \colon C \to Q \quad \text{and} \quad \varphi_1 \colon C \to C' \end{align*} $$

of degree $-1$ and 0, respectively, satisfying

$$ \begin{align*} \begin{aligned} &\varepsilon = \varepsilon ' \varphi_1 ,\quad h' \varphi_1 = h - \varphi_0 \partial + (\varphi_0 \otimes \varphi_0) \Delta , \\\partial' \varphi_1 = \varphi_1 \partial &+ (\varphi_0 \otimes \varphi_1 - \varphi_1 \otimes \varphi_0) \Delta \quad \text{and} \quad \Delta' \varphi_1 = (\varphi_1 \otimes \varphi_1) \Delta\,; \end{aligned} \end{align*} $$

see [Reference PositselskiPos11, Chapter 4]. This induces a map of dg algebras $\mathsf {\Omega }{(\varphi )} \colon \mathsf {\Omega }{(C)} \to \mathsf {\Omega }{(C')}$ , and we say $\varphi $ is a weak equivalence if $\mathsf {\Omega }{(\varphi )}$ is a quasi-isomorphism.

4.10. Let A be an $\mathrm {A}_{\infty }$ -algebra. An $\mathrm {A}_{\infty }$ -module over A is a graded module M equipped with maps

$$ \begin{align*} m_n^M \colon A^{\otimes (n-1)} \otimes M \to M \quad\text{for } n \geqslant 1 \end{align*} $$

of degree $(n-2)$ , satisfying

$$ \begin{align*} \sum_{\substack{r+s+t=n\\r \geqslant 0, s,t \geqslant 1}} (-1)^{r+st} m^M_{r+1+t} \left(\operatorname{\mathrm{id}}^{\otimes r} \otimes m_s \otimes \operatorname{\mathrm{id}}^{\otimes t}\right) + \sum_{\substack{r+s=n\\r \geqslant 0, s \geqslant 1}} (-1)^r m^M_{r+1} \left(\operatorname{\mathrm{id}}^{\otimes r} \otimes m_s^M\right) = 0. \end{align*} $$

If A is strictly unital, we say an $\mathrm {A}_{\infty }$ -module M over A is strictly unital if

$$ \begin{align*} \begin{aligned} &m_2(1_A \otimes m) = m \quad\text{for all } m \in M \quad\text{and} \\m_n(a_1 \otimes \cdots \otimes a_{i-1} &\otimes 1_A \otimes a_{i+1} \otimes \cdots \otimes a_{n-1} \otimes m) = 0 \quad \text{for all } 1 \leqslant i \leqslant n-1 \end{aligned} \end{align*} $$

for any $a_1, \ldots , a_{n-1} \in A$ and $m \in M$ with $n \neq 2$ .

If A is connective and has a split unit, then a strictly unital $\mathrm {A}_{\infty }$ -module structure over A on M is equivalent to the existence of maps

$$ \begin{align*} \bar{m}^M_n \colon \bar{A}^{\otimes (n-1)} \otimes M \to M \quad\text{for } n \geqslant 1 \end{align*} $$

of degree $(n-2)$ such that for $n \neq 2,3$ , the Stasheff identities hold when replacing m by $\bar {m}$ , and for $n=2,3$ , there is an extra curvature term $h_{n-1} \otimes \operatorname {\mathrm {id}}$ similar to 4.3.

4.11. Let A be a split unital connective $\mathrm {A}_{\infty }$ -algebra. The data of a strictly unital $\mathrm {A}_{\infty }$ -module structure over A is equivalent to that of a strictly unital dg module structure over $\mathsf {\Omega }{(\mathsf {B}{(A)})}$ . Explicitly, if $\{\bar {m}^M_n\}$ is a strictly unital $\mathrm {A}_{\infty }$ -module structure on a graded module M, then the dg module structure on M is given by the same differential $m^M_1$ , and the multiplication $\mathsf {\Omega }{(\mathsf {B}{(A)})} \otimes M \to M$ induced by

$$ \begin{align*} -(-1)^{\frac{n(n-1)}{2}} m^M_{n+1} (({\scriptstyle\mathsf{\Sigma}}^{-1})^{\otimes n} {\scriptstyle\mathsf{\Sigma}} \otimes \operatorname{\mathrm{id}}_M) \colon \Sigma^{-1} \bar{\mathsf{B}}_{(n)}{(A)} \otimes M \to M. \end{align*} $$

Moreover, this construction is natural in A and M, and any quasi-isomorphism of $\mathrm {A}_{\infty }$ -modules over A yields a quasi-isomorphism of dg modules over $\mathsf {\Omega }{(\mathsf {B}{(A)})}$ .

Proof. Since $\varepsilon $ is surjective, we may choose a splitting $A = \bar {A} \oplus Q$ such that $\varepsilon $ maps the free generator of Q to the unit of R. We inductively construct higher multiplication maps $m_n$ on A satisfying the nth Stasheff identity. To begin with, we set where $\partial $ is the differential of A.

For $n=2$ , we consider the commutative diagram

The morphism of complexes $m_2^A \colon A^{\otimes 2} \to A$ exists because $\varepsilon $ is a surjective quasi-isomorphism, the left vertical arrow is injective in each degree and the cokernel in each degree is projective; see, for example, [Reference Dwyer, Spaliński and JamesDS95, Section 7]. The morphism $m_2^A$ satisfies the desired properties by construction.

For $n> 2$ , the obstruction $\operatorname {obs}_n^A$ from (4.1.2) is a chain map. We have a short exact sequence of complexes

$$ \begin{align*} 0 \to \sum_{i+j=n-1} A^{\otimes i} \otimes Q \otimes A^{\otimes j} \xrightarrow{\eta_n} A^{\otimes n} \to \bar{A}^{\otimes n} \to 0. \end{align*} $$

By direct computation, we obtain $\operatorname {obs}_n^A \eta _n = 0$ . So the obstruction $\operatorname {obs}_n^A$ factors through $\bar {A}^{\otimes n}$ . We consider the diagram

Since $\bar {A}^{\otimes n}$ is, as graded modules, a direct summand of $A^{\otimes n}$ , and the higher multiplications $m_i^A$ for $i < n$ commute with $\varepsilon $ , the right triangle commutes up to the homotopy $m_n^R \varepsilon ^{\otimes n}$ . Then there exists a chain map $\alpha $ such that the left triangle commutes up to a homotopy $\sigma $ . Since $\varepsilon $ is surjective, we may assume $m_n^R \varepsilon ^{\otimes n} = \varepsilon \sigma $ by [Reference Avramov, Elías, Giral, Miró-Roig and ZarzuelaAvr98, Proposition 1.3.1]. That is,

satisfies the nth Stasheff identity (4.1.4).

5.2. In the setup of Proposition 5.1, we can also transfer $\mathrm {A}_{\infty }$ -module structures: If M is a strictly unital $\mathrm {A}_{\infty }$ -module over R and with semifree resolution $\gamma \colon G \to M$ over Q, in the sense discussed later in 6.2, then there exists a strictly unital $\mathrm {A}_{\infty }$ -module structure on G over A and $\gamma $ is a strict morphism of $\mathrm {A}_{\infty }$ -modules; compare with [Reference BurkeBur18]. When the homology of M is bounded below (for example, if M is an honest module), one can take G to be a bounded below complex of free Q-modules.

Proof. The unit $Q\to S$ factors through $\varepsilon $ and $\varphi _1$ , so by [Reference Dwyer, Spaliński and JamesDS95, Section 7], there is a chain map $\psi _1 \colon A \to R$ such that $\varphi _1 = \varepsilon \psi _1$ and $\psi _1(1_A) = 1_R$ .

Let $n \geqslant 2$ and assume that for $i < n$ , the chain maps $\psi _i \colon A^{\otimes i} \to R$ exist, the ith Stasheff identities (4.1.3) and (4.2.2) hold, and $\varepsilon \psi _i = \varphi _i$ . A computation shows that $\operatorname {obs}_n^\psi $ and $\operatorname {obs}_n^\varphi $ vanish when any of its inputs is $1_A$ . Hence, we can view $\operatorname {obs}_n^\psi $ and $\operatorname {obs}_n^\varphi $ as maps on $\bar {A}^{\otimes n}$ . Taking homology classes in $\operatorname {Hom}{(\bar {A}^{\otimes n},S)}$ , we have

$$ \begin{align*} \varepsilon [\operatorname{obs}_n^\psi] = [\operatorname{obs}_n^\varphi]= 0, \end{align*} $$

since $\varepsilon \operatorname {obs}_n^\psi = \operatorname {obs}_n^\varphi $ and $\varphi $ is a morphism of strictly unital $\mathrm {A}_{\infty }$ -algebras; cf. 4.1. Since $\varepsilon $ is a surjective quasi-isomorphism, and using the assumptions on A, the induced map $\operatorname {Hom}({\bar {A}^{\otimes n}},R)\to \operatorname {Hom}{(\bar {A}^{\otimes n},S)}$ is a quasi-isomorphism, and hence, $\operatorname {obs}_n^\psi $ is a boundary in $\operatorname {Hom}{(\bar {A}^{\otimes n},R)}$ . That is, there is $\bar {\psi }_n \colon \bar {A}^{\otimes n}\to R$ such that

$$\begin{align*}\operatorname{obs}_n^\psi=m_1^R\bar{\psi}_n+(-1)^n\bar{\psi}_n(\bar{m}_1^A \otimes \operatorname{\mathrm{id}}^{\otimes n} + \cdots + \operatorname{\mathrm{id}}^{\otimes n} \otimes \bar{m}_1^A). \end{align*}$$

Setting , we now have $\psi _1,\ldots , \psi _n$ satisfying the required identities (4.1.3) and (4.2.2), completing the induction.

Proof. Suppose that the minimal Q-free resolution A of R has an $\mathrm {A}_{\infty }$ -structure $\{m_n\}$ with the stated property. If $A'$ is a Q-free dg algebra resolution of R, then A and $A'$ are quasi-isomorphic as $\mathrm {A}_{\infty }$ -algebras over Q by Proposition 5.3. Therefore, $A\otimes _Qk$ and $R\otimes ^{\mathsf {L}}_Qk=A'\otimes _Qk$ are quasi-isomorphic as $\mathrm {A}_{\infty }$ -algebras over k.

If $m_n\otimes _Qk =0$ for $n\geqslant 3$ , then $A\otimes _Qk$ is a graded algebra, canonically isomorphic to $\operatorname {Tor}^{Q}(R,k)$ . Two dg k-algebras are quasi-isomorphic as dg algebras if and only if they are quasi-isomorphic as $\mathrm {A}_{\infty }$ -algebras [Reference KadeishviliKad82], and we can conclude that $R\otimes ^{\mathsf {L}}_Qk$ is formal.

Suppose conversely that $R\otimes ^{\mathsf {L}}_Q k$ is formal. By Proposition 5.1, the minimal Q-free resolution A of R admits an $\mathrm {A}_{\infty }$ -structure $\{m_n^{\prime }\}$ . Using the same reasoning as above, since $R\otimes ^{\mathsf {L}}_Q k$ is formal $A\otimes _Qk$ and $\operatorname {Tor}^{Q}(R,k)$ are quasi-isomorphic as $\mathrm {A}_{\infty }$ -algebras over k. By the uniqueness of minimal models (that is, $\mathrm {A}_{\infty }$ -algebras over a field having zero differential; see [Reference KadeishviliKad82]), there is an isomorphism of $\mathrm {A}_{\infty }$ -algebras

$$\begin{align*}\psi\colon (\operatorname{Tor}^{Q}(R,k),0, \mu, 0, \ldots) \xrightarrow{\cong} (A \otimes_Q k, 0, m_2^{\prime} \otimes k, m_3^{\prime} \otimes k, \ldots), \end{align*}$$

where $\mu $ is the ordinary product on $\operatorname {Tor}^{Q}(R,k)$ . We may make the identification $A \otimes _Q k=\operatorname {Tor}^{Q}(R,k)$ and choose lifts $\Psi _i\colon A^{\otimes i} \to A$ with $\Psi _i\otimes _Qk=\psi _i$ . By Nakayama’s lemma, $\Psi _1$ is an isomorphism and we can inductively define operations $m_n\colon A^{\otimes n} \to A$ by the formula

By construction, the map $\Psi \colon (A,\{m_n^{\prime }\}) \to (A,\{m_n\})$ now satisfies the Stasheff morphism identities (4.1.3), and it follows that $(A,\{m_n\})$ is an $\mathrm {A}_{\infty }$ -algebra, isomorphic to $(A,\{m_n^{\prime }\})$ . Finally, from $\Psi \otimes _Q k=\psi $ , it follows that $m_2\otimes _Qk=\mu $ and $m_n\otimes _Qk =0$ for $n\geqslant 3$ , as stated in the proposition.

Proof. We prove this by induction on n. It is clear for $n=1$ since $\varphi _1$ is a chain map. For $n\geqslant 2$ , since $\bar {m}_s^{T}=0$ for all s, we can rearrange the Stasheff morphism identities (4.1.3):

$$\begin{align*}\bar{\varphi}_1\bar{m}_n=-\sum_{\substack{r+s+t=n\\r,t \geqslant 0, s \geqslant 1}} (-1)^{r+st} \bar{\varphi}_{r+1+t} \left(\operatorname{\mathrm{id}}^{\otimes r} \otimes \bar{m}_s \otimes \operatorname{\mathrm{id}}^{\otimes t}\right). \end{align*}$$

We can assume by induction that $(\bar {m}_s(\bar {A}^{\otimes s}))_{<N}=0$ for $s<n$ . Since each $\varphi _r$ increases degree by $r-1$ , this implies that the right-hand side above is zero in degrees $i<N$ . Since $(\bar {\varphi }_1)_{<N}$ is injective, it follows that $(\bar {m}_n(\bar {A}^{\otimes n}))_{<N}=0$ .

Construction 6.1. Ignoring differentials for now, there is a well-known exact sequence of graded ${\mathsf {\Omega }}{(C)}$ -bimodules

(6.1.1) $$ \begin{align} 0 \to \mathsf{\Omega}{(C)} \otimes \Sigma^{-1} \bar{C} \otimes \mathsf{\Omega}{(C)} \xrightarrow{\iota} \mathsf{\Omega}{(C)} \otimes Q \otimes \mathsf{\Omega}{(C)} \xrightarrow{\mu} \mathsf{\Omega}{(C)} \to 0, \end{align} $$

where $\iota (x \otimes \langle c \rangle \otimes y)= \mu (x,\langle c \rangle ) \otimes 1 \otimes y - x \otimes 1 \otimes \mu (\langle c \rangle ,y)$ for $c \in \bar {C}$ and $x,y \in \mathsf {\Omega }{(C)}$ , and $\mu $ the multiplication map.

We give ${\mathsf {\Omega }}{(C)} \otimes \Sigma ^{-1} \bar {C} \otimes \mathsf {\Omega }{(C)}$ the unique differential $\partial ^\iota $ making $\iota $ a chain map, and we set

We write $\partial ^\tau $ for the differential on ${\mathsf {\Omega }}{(C)} \otimes ^{\tau } C \otimes ^{\tau } \mathsf {\Omega }{(C)}$ ; this is a dg ${\mathsf {\Omega }}{(C)}$ -bimodule whose underlying graded bimodule is ${\mathsf {\Omega }}{(C)} \otimes C \otimes \mathsf {\Omega }{(C)}$ , using the evident multiplication by ${\mathsf {\Omega }}{(C)}$ on either side.

For a right dg ${\mathsf {\Omega }}{(C)}$ -module F and a left dg ${\mathsf {\Omega }}{(C)}$ -module G, we define

Explicitly, its underlying graded module is $F \otimes C \otimes G$ and the differential is

$$ \begin{align*} \begin{aligned} \partial^\tau &= \partial^F \otimes \operatorname{\mathrm{id}}_C \otimes \operatorname{\mathrm{id}}_G + \operatorname{\mathrm{id}}_F \otimes \partial^C \otimes \operatorname{\mathrm{id}}_G + \operatorname{\mathrm{id}}_F \otimes \operatorname{\mathrm{id}}_C \otimes \partial^G \\ &\quad + \left(\mu (\operatorname{\mathrm{id}}_F \otimes {\scriptstyle\mathsf{\Sigma}}^{-1} p) \otimes \operatorname{\mathrm{id}}_C \otimes \operatorname{\mathrm{id}}_G - \operatorname{\mathrm{id}}_F \otimes \operatorname{\mathrm{id}}_C \otimes \mu({\scriptstyle\mathsf{\Sigma}}^{-1} p \otimes \operatorname{\mathrm{id}}_G)\right) (\operatorname{\mathrm{id}}_F \otimes \Delta \otimes \operatorname{\mathrm{id}}_G), \end{aligned} \end{align*} $$

where $p \colon C \to \bar {C}$ is the natural projection and we use $\mu $ for the right action of ${\mathsf {\Omega }}{(C)}$ on F and the left action of ${\mathsf {\Omega }}{(C)}$ on G.

6.2. Given a dg algebra A, recall that a dg A-module F is semifree if it admits an exhaustive filtration

$$\begin{align*}0= F(-1)\subseteq F(0)\subseteq F(1)\subseteq \ldots \subseteq F, \end{align*}$$

where each subquotient $F(i)/F(i-1)$ is a sum of shifts of A. As a matter of terminology, a semifree dg A-bimodule is a semifree dg module over $A \otimes A^{\mathrm {op}}$ .

Every dg A-module M admits a semifree resolution in the sense that there exists a surjective quasi-isomorphism $F\xrightarrow {\simeq } M$ , with F a semifree dg A-module. Such resolutions are unique up to homotopy; see [Reference Félix, Halperin and ThomasFHT01, Chapter 6] for this fact, as well as other details regarding semifree dg modules.

Proof. By construction, (6.1.1) is a short exact sequence of dg ${\mathsf {\Omega }}{(C)}$ -bimodules, and so it induces an exact triangle in the derived category of dg ${\mathsf {\Omega }}{(C)}$ -bimodules. We obtain the quasi-isomorphism by comparing this triangle to the triangle associated to the cone construction for ${\mathsf {\Omega }}{(C)} \otimes ^{\tau } C \otimes ^{\tau } \mathsf {\Omega }{(C)}$ .

The dg module ${\mathsf {\Omega }}{(C)} \otimes ^{\tau } C \otimes ^{\tau } \mathsf {\Omega }{(C)}$ is semifree as a dg ${\mathsf {\Omega }}{(C)}$ -bimodule since C is free as a module over Q and non-negatively graded.

6.4. Let $\varphi \colon Q \to R$ be a finite local homomorphism with $A\to R$ a free resolution of R over Q. Fix an R-complex M and a semifree resolution $\gamma \colon G \to M$ over Q. By Proposition 5.1, there exists a split unital $\mathrm {A}_{\infty }$ -algebra structure $\{m_n\}$ on A and a strictly unital $\mathrm {A}_{\infty }$ -module structure $\{m_n^G\}$ over A on G. Then by 4.11, this induces a dg module structure over $\mathsf {\Omega }{(\mathsf {B}{(A)})}$ on G.

Suppose further that C is a connected curved dg coalgebra with counit $\varepsilon \colon C\to Q$ , equipped with a weak equivalence of connected curved dg coalgebras ; cf. 4.9. Then R and G each have an induced dg module structure over ${\mathsf {\Omega }}{(C)}$ .

Proof. The map $G \to M$ is a quasi-isomorphism of $\mathrm {A}_{\infty }$ -modules over A, and by 4.11, it is a quasi-isomorphism of dg modules over $\mathsf {\Omega }{(\mathsf {B}{(A)})}$ , and thus over ${\mathsf {\Omega }}{(C)}$ . From Lemma 6.3, we obtain the quasi-isomorphism of (left) dg ${\mathsf {\Omega }}{(C)}$ -modules

$$ \begin{align*} \beta\colon \mathsf{\Omega}{(C)} \otimes^{\tau} C \otimes^{\tau} G = \mathsf{\Omega}{(C)} \otimes^{\tau} C \otimes^{\tau} \mathsf{\Omega}{(C)} \otimes_{\mathsf{\Omega}{(C)} } G \xrightarrow{\simeq} \mathsf{\Omega}{(C)} \otimes_{\mathsf{\Omega}{(C)}} G \cong G. \end{align*} $$

Since G is semifree over Q, it follows that ${\mathsf {\Omega }}{(C)} \otimes ^{\tau } C \otimes ^{\tau } G$ is semifree as a left dg module over ${\mathsf {\Omega }}{(C)}$ .

The claimed quasi-isomorphism fits into the commutative diagram below

where $\alpha $ is the map induced by the quasi-isomorphism of dg algebras

(6.5.1) $$ \begin{align} \mathsf{\Omega}{(C)}\xrightarrow{\simeq} \mathsf{\Omega}{(\mathsf{B}{(A)})} \xrightarrow{\simeq} A \xrightarrow{\simeq} R\,; \end{align} $$

where the fact that the second map is a quasi-isomorphism follows from the derived version of Nakayama’s lemma since, upon applying $-\otimes _Q k$ , the map becomes a quasi-isomorphism [Reference Lefèvre-HasegawaLH03, Section 2.2.1].

As the composition in (6.5.1) is a quasi-isomorphism and ${\mathsf {\Omega }}{(C)} \otimes ^{\tau } C \otimes ^{\tau } G $ is a semifree resolution of M over ${\mathsf {\Omega }}{(C)}$ , it follows that $\alpha $ is a quasi-isomorphism. We have already justified that $\beta $ and $\gamma $ are quasi-isomorphisms, accounting for the downward arrow on the right. As both legs of the triangle in the diagram above are quasi-isomorphisms, the horizontal map is a quasi-isomorphism, as claimed.

Proof. If the stated conditions hold, then $R\otimes ^{\mathsf {L}}_Q k$ is formal by Proposition 5.4, and is Koszul; therefore, $\varphi $ is Koszul by definition.

Assume, conversely, that $\varphi $ is Koszul. Since $\operatorname {Tor}^{Q}(R,k)$ is Koszul, it admits a compatible weight grading making it quadratic. That is, we have an isomorphism

$$ \begin{align*} \operatorname{Tor}_i^Q{(R,k)}_{(w)} \cong \big(\mathrm{T}^{a}_{(w)}{(\bar{V})}/(\bar{W})\big)_i \end{align*} $$

identifying the product on Tor with the product on the quotient of the tensor algebra, where $\bar {V} = \operatorname {Tor}^{Q}(R,k)_{(1)}$ and $\bar {W} \subseteq \bar {V} \otimes _k \bar {V}$ is a graded subspace.

Let V be a free graded Q-module such that $V\otimes k =\bar {V}$ , and choose a direct summand $W \subseteq V \otimes V$ such that $W\otimes k =\bar {W}$ . If we define , then A is a free, bigraded Q-module and

Therefore, we may equip A with a differential making it the minimal Q-resolution of R. We have constructed (1) and (3) satisfying condition (i). Since $R\otimes ^{\mathsf {L}}_Qk$ is formal, by Proposition 5.4, there is an $\mathrm {A}_{\infty }$ -structure on A as required for (2), inducing the algebra structure on $\operatorname {Tor}^{Q}(R,k)$ and satisfying the conditions (ii) and (iii).