\frac{1}{h}          \sum_{n=1}^{\infty}h^n S_n\big\}\Big]\\
          & = & \sum_{n=|t|}^{\infty}h^n\sum_{\substack{n_1+\dots+n_{|t|}=n\\n_i\geq 1}}C^i_t(S_{n_1},\dots,S_{n_{|t|}})\end{aligned}$$

which defines the $C^i_t$ which are multi-differential maps from $C^\infty(B_2)^{|t|}$ to $C^\infty(B_3)$.

We can now state the main Proposition of this section.

\[pertSGA\] The formal series $S_h= S_0 +\sum_{n\geq 1}h^n S_n$ satisfies the SGA equation iff for each $n>0$ we have

$$\sum_{\substack{t\in T\\ |t|\leq n}}\frac{1}{|t|!}\sum_{\substack{n_1+\dots+n_{|t|}=n\\n_i\geq 1}}C_t^1(S_{n_1},\dots,S_{n_{|t|}})
-C_t^2(S_{n_1},\dots,S_{n_{|t|}})=0.$$

Let us remark that for all $f\in C^\infty(B_2)$ we have that, $$C^1_\bullet(f)+C^1_\circ(f)-C^2_\bullet(f)-C^2_\circ(f) = dS$$ where $d:C^\infty (B_n)\rightarrow C^\infty(B_{n+1})$ is a differential (i.e., $d^2=0$) defined by the formula

$$d f(p_{1},\dots,p_{n+1}) =
f(p_{2},...,p_{n+1})-\sum_{i=1}^{n}(-1)^{i+1}f(p_{1},\dots,p_{i}+p_{i+1})$$$$+(-1)^ {n+1}f(p_{1},...,p_{n}).$$ This differential can be interpreted either as the Hochschild differential on symbols of multi-differential operators on $C^\infty(\mathbb R^d)$ or as the differential of the trivial symplectic groupoid cohomology over $\mathbb R^d$. This remark allows us to put the previous recursive equations into the form $$dS_n + H_n(S_{n-1},\dots,S_1)=0$$ which is exactly the analog of the recursive equation involved when considering star-products.

The remaining of this section is devoted to proving Proposition \[pertSGA\].

Proof of the Proposition {#defeqn}
------------------------

It follows from a series of little Lemmas. We are first interested in expanding $$\begin{aligned}
 \bar{p} &=&p_1+p_2+h\nabla_x S(p_{1},p_{2},\bar{x}),\\
 \bar{x} &=&x+h\nabla_{p_1}S(\bar{p},p_{3},x), 
 \end{aligned}$$ and $$\begin{aligned}
\tilde{p} &=&p_2+p_3+h\nabla_x S(p_{2},p_{3},\tilde{x}),\\
\tilde{x} &=&x+h\nabla_{p_2}S(p_{1},\tilde{p},x).
 \end{aligned}$$ as power series in $h$.

The method 