\[
(F,G) \longmapsto F \circ G
\]

\[
W(\Omega_0) \times W(\Omega_0) \longrightarrow W(\Omega_0)
\]

\[
(\lambda F)(x) = \lambda F(x)
\]

\[
(F+G)(x) = F(x) + G(x)
\]

\[
(FG)(x) = F(x)G(x)
\]

\[
W(\Omega_0) \longrightarrow \operatorname{End} W(A)
\]

\[
F(x+y) = \Delta F(x,y)
\qquad \Omega_0 \otimes \Omega_0
\qquad A \otimes A
\qquad
\begin{matrix} k_0 \\ | \\ \mathbb{Z} \end{matrix}
\]

\[
F_n(T) = \frac{T(T-1)\cdots(T-n+1)}{n!} \in \mathbb{Q}[T]
\]

\[
(*) \qquad F_m(T)F_n(T) = \sum_{0 \leq p \leq m+n} c^{p}_{m,n} F_p(T)
\]

\[
F_m(F_n(T)) = \sum_{0 \leq p \leq mn} d^{p}_{m,n} F_p(T)
\]

\[
\bigoplus_n \mathbb{Z}\cdot F_n \subset \mathbb{Q}[T]
\]

\[
F_m(P(T) + Q(T)) = \sum \gamma^{m}_{ij} F_i(P(T)) F_j(Q(T))
\]

\[
\delta : \Omega \longrightarrow \Omega \otimes_{\mathbb{Z}} \Omega
\]

\[
F_m(P(T)Q(T)) = \sum \delta^{m}_{ij} (P(T))
\]

\[
F_m(x+y) \qquad F_m(xy)
\]

\[
\sum a_i T^i \qquad a_i \in \mathbb{Q}
\]

\[
\Omega \subset K_0[T], \qquad
\Omega = \{\, F \in K_0[T] \mid F(\lambda) \in k_0 \ \ \forall \lambda \in k_0 \,\}
\]

\[
\begin{cases}
(F+G)(x) = F(x) + G(x) \\
(\lambda F)(x) = \lambda(F(x)) \\
(FG)(x) = F(x)G(x) \\
(F \circ G)(x) = F(G(x))
\end{cases}
\]

\[
F(x+y) \overset{?}{=} \sum_i G_i(x) H_i(y)
\]

\[
F(xy) \overset{?}{=} \sum_\alpha J_\alpha(x) K_\alpha(y)
\]

\[
F(X+Y) \overset{\text{déf}}{=} \Delta_a F(X,Y)
\]

\[
F(XY) \overset{\text{déf}}{=} \Delta^{\times}_{m} F(X,Y)
\]

\[
F \circ (P+Q) = \sum_i (G_i \circ P)(H_i \circ Q)
\]

\[
F \circ (PQ) = \sum_\alpha (J_\alpha \circ P)(K_\alpha \circ Q)
\]

\[
F \circ \lambda T = \sum_\alpha J_\alpha(\lambda) K_\alpha
\qquad \Bigl( = \sum_\alpha K_\alpha(\lambda) J_\alpha \Bigr)
\]

\[
x+y, \qquad xy
\]

\[
(F,G) \longmapsto F \circ G
\]

\[
\begin{cases}
+ \ \text{ass., commutative, avec } 0 \\
xy \ \text{associative, commutative, biadditive} \\
x \circ y \ \text{associative, unitaire}
\end{cases}
\]

\[
\begin{cases}
(F+F') \circ G = F \circ G + F' \circ G \\
(FF') \circ G = (F \circ G)(F' \circ G)
\end{cases}
\]

\[
F \circ (G' + G'') = \sum_i (F'_i \circ G')(F''_i \circ G'')
\]

\[
F \circ (G'G'') = \sum_i (P'_\alpha \circ G')(Q''_\alpha \circ G'')
\]

\[
(F \circ 0) \circ G = F \circ \bigl(\underbrace{0 \circ G}_{0}\bigr) = F \circ 0 ,
\]

\[
(F,\lambda) \longmapsto F(\lambda) \overset{\text{déf}}{=} F \circ \lambda
\]

\[
\Omega^{0} = \{\, F \in \Omega \mid F \circ 0 \overset{\text{déf}}{=} F(0) = 0 \,\}
\qquad \text{On a} \quad \Omega = \Omega_0 \oplus \Omega^{0}
\]

\[
1 \circ F = 1 \qquad \forall F \in \Omega .
\]

\[
\Omega_A = \operatorname{Appl}(A,A) .
\]

\[
\underline{\Omega}_{A/k} = \operatorname{End}(W_k(A))
\]

\[
\Omega(\struck{\ill{}}) = \operatorname{End}_{\text{faisceaux d'ens.}}(A) ,
\]

\[
\boxed{(\Omega_A)_0 \simeq \Gamma A \, ?}
\]

\[
F \longmapsto (\lambda \mapsto F \circ \lambda) : \Omega \longrightarrow
\operatorname{Appl}(\Omega_0,\Omega_0) = \Omega_{\Omega_0}
\]

\[
\Omega \subset K_0[T], \qquad
\Omega = \{\, P \in K_0[T] \mid P(\lambda) \in k_0 \ \ \forall \lambda \in k_0 \,\}
\]