In this section we are going to restate in a convenient form some results from [GS] and [Me].

Let $(A,m_{n}),n\geq 1$ be a non-unital $A_{\infty}$-algebra, $\Pi:A\to A$ be an idempotent which commutes with the differential $d=m_{1}$. In other words, $\Pi$ is a linear map of degree zero such that $d\Pi=\Pi d,\Pi^{2}=\Pi$. Assume that we are given an homotopy $H:A\to A[-1]$, $1-\Pi=dH+Hd$. Let us denote the image of $\Pi$ by $B$. Then we have an embedding $i:B\to A$ and a projection $p:A\to B$, such that $\Pi=i\circ p$.

Let us introduce a sequence of linear operations $m_{n}^{B}:B^{\otimes n}\to B[2-n]$ in the following way:

a) $m_{1}^{B}:=d^{B}=p\circ m_{1}\circ i$;

b) $m_{2}^{B}=p\circ m_{2}\circ(i\otimes i)$;

c) $m_{n}^{B}=\sum_{T}\pm m_{n,T},n\geq 3$.

Here the summation is taken over all oriented planar trees $T$ with $n+1$ tails vertices (including the root vertex), such that the (oriented) valency $|v|$ (the number of ingoing edges) of every internal vertex of $T$ is at least $2$. In order to describe the linear map $m_{n,T}:B^{\otimes n}\to B[2-n]$ we need to make some preparations. Let us consider another tree $\bar{T}$ which is obtained from $T$ by the insertion of a new vertex into every internal edge. As a result, there will be two types of internal vertices in ${\bar{T}}$: the “old” vertices, which coincide with the internal vertices of $T$, and the “new” ones, which can be thought geometrically as the midpoints of the internal edges of $T$.

To every tail vertex of ${\bar{T}}$ we assign the embedding $i$. To every “old” vertex $v$ we assign $m_{k}$ with $k=|v|$. To every “new” vertex we assign the homotopy operator $H$. To the root we assign the projector $p$. Then moving along the tree down to the root one reads off the map $m_{n,T}$ as the composition of maps assigned to vertices of $\bar{T}$. Here is an example of $T$ and $\bar{T}$:

Proof. Clear. $\blacksquare$

Sketch of the proof. The proof is quite straightforward, so we just briefly show main steps of computations.

First, one observes that $p$ and $i$ are homomorphisms of complexes. In order to prove the theorem we will replace for a given $n\geq 2$ each summand $m_{n,T}$ by a different one, and then compute the result in two different ways. Let us consider a collection of trees $\{\bar{T}_{e}\}_{e\in E(\bar{T})}$ such that $\bar{T}_{e}$ is obtained from $\bar{T}$ in the following way:

a) we split the edge $e$ into two edges by inserting a new vertex $w_{e}$ inside $e$;

b) the remaining part of $\bar{T}$ is unchanged.

We assign $d=m_{1}$ to the vertex $w_{e}$ edge, and keep all other assignments untouched. In this way we obtain a map $m_{n,\bar{T}_{e}}:B^{\otimes n}\to B[3-n]$.

Let us consider the following sum (with appropriate signs):

$$\hat{m}_{n}^{B}=\sum_{T}\sum_{e\in E(\bar{T})}\pm m_{n,\bar{T}_{e}}.$$

We can compute it in two different ways: using the relation $1-\Pi=dH+Hd$, and using the formulas for $d(m_{j}),\,j\geq 2$ given by the $A_{\infty}$-structure on $A$. The case of the relation $1-\Pi=dH+Hd=:d(H)$ gives

$$\hat{m}_{n}^{B}=d(m_{n}^{B})-m_{n}^{B,\Pi}+m_{n}^{B,1}$$

where $m_{n}^{B,\Pi}$ is defined analogously to $m_{n}^{B}$, with the only difference that we assign to a new vertex operator $\Pi$ instead of $H$ for some edge $e\in E_{i}(T)$. Similarly, the summand $m_{n}^{B,1}$ is defined if we assign to a new vertex operator $1=id_{A}$ instead of $H$. Formulas for $d(m_{j})$ are quadratic expressions in $m_{l},\,l<j$. This gives us another identity

$$\hat{m}_{n}^{B}=m_{n}^{B,1}$$

Thus we have $d(m_{n}^{B})=m_{n}^{B,\Pi}$, and it is exactly the $A_{\infty}$-constraint for the collection $(m_{n}^{B})_{n\geq 1}$. $\blacksquare$

Moreover, using similar technique, one can prove the following result.

For the convenience fo the reader we give an explicit formula for a canonical choice of $g$. The operator $g_{1}:B\to A$ is defined as the inclusion $i$. For $n\geq 2$ we define $g_{n}$ as the sum of terms $g_{n,T}$ over all planar trees $T$ with $n+1$ tails. Each term $g_{n,T}$ is similar to the term $m_{n,T}$ defined above, the only difference is that we insert operator $H$ instead of $p$ into the root vertex.

One can also construct an explicit $A_{\infty}$-quasi-isomorphism $A\to B$.