\[
p_B : B \longrightarrow B \otimes_A B
\]

\[
\delta_B x = p_B x - (x \otimes 1 + 1 \otimes x) \in J \otimes J
\]

\[
x \in B, \quad p_B x = \sum y_i z_i \;\Longrightarrow\;
\sum \check{y}_i z_i = \sum y_i \check{z}_i = \varepsilon(x) . 1
\]

\[
\varepsilon(\check{x}) = \varepsilon(x)
\]

\[
\Delta_{\mathcal{U}} : \mathcal{U} \longrightarrow \mathcal{U} \otimes_A \mathcal{U},
\qquad
\eta_{\mathcal{U}} : \mathcal{U} \longrightarrow A
\]

\[
x \in \mathcal{U}, \quad \Delta_{\mathcal{U}} x = \sum y_i \otimes z_i
\;\Longrightarrow\;
\sum y_i \check{z}_i = \sum \check{y}_i z_i = \eta_{\mathcal{U}}(x) . 1_{\mathcal{U}} .
\]

\[
q_C : C \longrightarrow B \otimes_A C
\]

\[
\left\lbrace
\begin{array}{l}
q_{\mathcal{U}}(y) . 1_C = \eta_{\mathcal{U}}(y) . 1_{C'} \\[4pt]
x, y \in C,\ u \in \mathcal{U},\ \Delta_{\mathcal{U}} u = \sum v_i \otimes w_i
\;\Longrightarrow\;
p_{\mathcal{U}}(u)(xy) = \sum \bigl(p_{\mathcal{U}}(v_i) x\bigr)\bigl(p_{\mathcal{U}}(w_i) y\bigr)
\end{array}
\right.
\]

\[
\left\lbrace
\begin{array}{l}
u . 1_C = \eta(u) . 1_C \\[4pt]
x, y \in C,\ u \in \mathcal{U},\ \Delta_{\mathcal{U}} u = \sum v_i \otimes w_i
\;\Longrightarrow\; u(xy) = \sum (v_i x)(w_i y)
\end{array}
\right.
\]

\[
B \otimes C \xleftarrow{\ \mathrm{id}_B \otimes \Delta_C\ } B \otimes C \otimes C
\xleftarrow{\ q_C \otimes \mathrm{id}_C\ } C \otimes C
\]

\[
h : \mathcal{D} \longrightarrow C
\]

\[
h(x) = h'(x) . 1_C \qquad ??
\]

\[
x \in C' \iff q_C x = 1_B \otimes x
\]

\[
x \otimes 1_C = 1_C \otimes x \quad\text{dans } C \otimes C
\]

\[
q_C : B/JB \longrightarrow (B \otimes B)/J.(B \otimes B)
\]

\[
\text{(**)} \qquad \varphi : C \otimes_A C \longrightarrow \mathrm{Hom}_A(\mathcal{U}, C)
\]

\[
\psi : C \otimes \mathcal{U} \longrightarrow \mathrm{Hom}(C, C)
\]

\[
\boxed{\ \psi(x \otimes u) . y = x\, u(y)\ }
\]

\[
\varphi : C \otimes C \longrightarrow \mathrm{Hom}_A(\mathcal{U}, C)
\]

\[
\boxed{\ \varphi(x \otimes y) . u = u(x) . y\ }
\]

\[
u . x = \eta(u) . x
\]

\[
G \times E \xrightarrow{\ (q_E, p_2)\ } E \times E
\]

\[
E \times E \xrightarrow{\ g_E\ } G ,
\]

\[
\text{(*)} \qquad
\left\lbrace
\begin{array}{l}
p_B : B \longrightarrow B \otimes B \\
q_C : C \longrightarrow B \otimes C
\end{array}
\right.
\]

\[
\varphi : C \otimes C \xrightarrow{\ (q_C, p_2)\ } B \otimes C
\]

\[
g_C : B \longrightarrow C \otimes C
\]

\[
\psi : \mathrm{Hom}_A(B, C) \longrightarrow \mathrm{Hom}_A(C, C)
\]

\[
\psi' : \mathrm{Hom}(C, C) \longrightarrow \mathrm{Hom}(B, C)
\]

\[
\psi' : \mathrm{Hom}_A(C, C) \longrightarrow \mathrm{Hom}(B, C)
\]

\[
\mathrm{Hom}(C, C) \otimes B \longrightarrow C
\]

\[
\psi'_1 : B \longrightarrow \mathrm{Hom}(\mathrm{Hom}(C, C), C)
\]

\[
\rho : \mathrm{Hom}(\mathrm{Hom}(C, C), C) \longrightarrow C \otimes C
\]

\[
(C' \otimes C)' \otimes C = C \otimes C' \otimes C
\]

\[
\varepsilon : C' \longrightarrow A
\]

\[
g_C = \rho \circ \psi'_1 : B \longrightarrow C \otimes C
\]

\[
\mathcal{U} = A[X] / X^p A[X]
\]

\[
\boxed{\ \Delta X = X \otimes 1 + 1 \otimes X\ }
\]

\[
D^i \quad (0 \leq i \leq p-1) \quad \text{forment une \emph{base} du $C$-module } \mathrm{Hom}_A(C, C)
\]

\[
C \simeq A[u]/(u^p - a), \qquad \text{où } a \in A .
\]

\[
Du = 1
\]

\[
Du^n = n u^{n-1} .
\]

\[
\begin{pmatrix}
1 & u & u^2 & u^3 & \cdots & u^{p-1} \\
0 & 1 & 2u & 3u^2 & \cdots & (p-1) u^{p-2} \\
0 & 0 & 2.1 & 3.2.u & \cdots & (p-1)(p-2) u^{p-3} \\
0 & 0 & 0 & 3.2.1 & \cdots & (p-1)(p-2)(p-3) u^{p-4} \\
\vdots & & & & \ddots & \vdots \\
0 & 0 & 0 & 0 & \cdots & (p-1) \cdots 1
\end{pmatrix}
\]

\[
0!\, 1!\, 2! \cdots (p-1)!
\]

\[
0 \longrightarrow \mathcal{B}_X \longrightarrow G_{a,X}
\xrightarrow{\ \Phi\ } G_{a,X} \longrightarrow 0
\qquad \Phi = \text{Frobenius}
\]

\[
\Delta T = T \otimes 1 + 1 \otimes T .
\]

\[
\underline{O}[T] \xleftarrow{\ \varphi\ } \underline{O}[T]
\]

\[
\Phi(T) = T^p
\]

\[
G_{a,X} \xrightarrow{\ \mathrm{proj}\ } X \longrightarrow Y .
\]

\[
0 \to H^0(X, \mathcal{B}_X) \to A \xrightarrow{\ F\ } A \longrightarrow
H^1(X, \mathcal{B}_X) \to H^1(X, \underline{O}_X) \xrightarrow{\ F\ }
H^1(X, \underline{O}_X)
\]

\[
A[X^{n/p}] \simeq A[T]/(T^p - X^n),
\]

\[
A[T]/(T^3 - X^2) = A[X^{2/3}],
\]