\[
  C_0=\{x\in C\mid R(x)=\emptyset\}=\text{ens.\ des positions terminales}
\]

\[
  C\smallsetminus C_0=\coprod_{j\in J}C(j)
\]

\[
  C_{-1}=\emptyset \qquad C_0=\{x\in C\mid \forall y\in R(x),\ y\in C_{-1}\}=\{x\in C\mid R(x)=\emptyset\}
\]

\[
  C_1=\{x\in C\mid \forall y\in R(x),\ y\in C_0\} \qquad \text{NB } C_1\supset C_0
\]

\[
  C_i=\{x\in C\mid \forall y\in R(x),\ y\in C_{i-1}\}
\]

\[
  C=C_{N-1}
\]

\[
  \Sigma\subset\bigl(C(j)\times C\bigr)\cap R \quad\text{tel que}
\]

\[
  x\in G(j)\Longleftrightarrow \exists\,y\in R(x),\ y\in G(j).
\]

\[
  \Sigma'(z)=\begin{cases}
    \Sigma(z) & \text{si } z\neq x\\
    \{y\} & \text{si } z=x
  \end{cases}
\]

\[
  x\in G(j)\Longleftrightarrow \forall y\in R(x),\ \text{on a } y\in G(j)
\]

\[
  z\in C(j), \qquad
  \Sigma(z)=\begin{cases}
    R(z) & \text{si } z\notin G(j)\\
    R(z)\cap G(j) & \text{si } z\in G(j)
  \end{cases}
\]

\[
  \Sigma(x_i)=R(x_i)\cap G(j) \quad\text{donc}\quad x_{i+1}\in\Sigma(x_i)
\]

\[
  \Sigma(z)=\begin{cases}
    R(z) & \text{si } z \text{ n'est pas conséquent de } x\\
    \Sigma_{y(z)}(z) & \text{si } \exists\, y(z)\in R(x),\ \text{avec } z\geqslant y(z).
  \end{cases}
\]

\[
  \Sigma(z)=\begin{cases}
    R(z) & \text{si } z\notin\Gamma\\
    R(z)\cap G(j) & \text{si } z\in\Gamma
  \end{cases}
\]

\[
  x_i\in G(j)\Longrightarrow x_{i+1}\in G(j) \qquad (0\leqslant i\leqslant N-1).
\]

\[
  C(J')=\bigcup_{j\in J'}C(j),
\]

\[
  \Sigma'(z)=\begin{cases}
    R(z) & \text{si } z\in G(j')\\
    R(z)\cap\overline{G}(j') & \text{sinon, i.e.\ } z\in\overline{G}(j')
  \end{cases}
\]

\[
  \overline{C}(j')\cap\overline{G}(j')=\bigl\{z\in\overline{C}(j')\ \big|\ \exists\,y\in R(z) \text{ avec } y\in\overline{G}(j') \text{ ou } (R(z)=\emptyset \text{ et } z\in\overline{G}_0(j'))\bigr\}
\]

\[
  C'=C \text{ ens.\ des positions}, \quad \text{et } R'=R
\]

\[
  C'(j')=C(j'),\qquad C'(j)=C(J')\qquad (J'=J-\{j'\}),
\]

\[
  G'_0(j')=G_0(j'),\qquad G'_0(j)=C_0\smallsetminus G'_0(j')=C_0\smallsetminus G_0(j')
\]

\[
  G_1(j)=G_0(j)\cup\left\{x\in C\ \middle|\ \begin{array}{l}
    \bigl(x\in C(j) \text{ et } R(x)\cap G_0(j)\neq\emptyset\bigr)\\
    \text{ou } \bigl(x\in(\overline{C}(j)\smallsetminus\overline{C}_0(j)) \text{ et } R(x)\subset G_0(j)\bigr)
  \end{array}\right\}
\]

\[
  G_i(j)=G_0(j)\cup\left\{x\in C\ \middle|\ \begin{array}{l}
    x\in C(j) \text{ et } R(x)\cap G_{i-1}(j)\neq\emptyset\\
    x\in(\overline{C}(j)\smallsetminus\overline{C}_0(j)) \text{ et } R(x)\subset G_{i-1}(j)
  \end{array}\right\}
\]

\[
  G_i(j)\subset G_{i+1}(j)
\]

\[
  G_i(j)\subset G(j)
\]

\[
  G_\infty(j)=\bigcup G_i(j) \quad\text{donc}\quad G_\infty(j)\subset G(j)
\]

\[
  G_\infty(j)\cap C_i=G(j)\cap C_i
\]

\[
  \{0\}\ \big/\ \underset{x_1}{V_1}\ \big/\ V_1\dotplus\{x_2\},\ \underset{x_2}{V_2}\ \big/\ V_2+\{x_3\}
\]

\[
  \bigl\{V_2+x_3*V_1,\ V_2+x_3*V_1+\{x_2+x_3\}\bigr\},\quad V_3
\]

\[
  V_1+\{x_2\}\qquad V_2+\{x_3\}\qquad 2V_2+x_3*V_1+\{x_2+x_3\}
\]

\[
  3\,3\,3\,3\,1\,1\,1
\]

\[
  3\,2\,2\,2
\]

\[
  0\ \ a\ \ a' \qquad 0\ \ a\ \ a-a' \qquad\qquad 0\ \ a\ \ 2a \qquad 0\ \ a\ \ -a
\]

\[
  -2a\ \ -a\ \ 0\ \ a\ \ 2a \qquad\qquad 2a\neq 0
\]

\[
  \begin{array}{ccc} -2a & -a & 0\\ -a & 0 & a\\ 0 & a & 2a \end{array}
\]

\[
  (V_1\dotplus\{x\})*(V_1\dotplus\{x\})=2V_1\dotplus\{u\}+2\{x\}*V_1=2G+\{u\}\qquad (x^2=u)
\]

\[
  \{e\}\quad V_1\quad V_1\dotplus\{x\}\quad V_2
\]

\[
  1\,1\,1\,1 \qquad 2\,2 \qquad 2\,1\,1 \qquad 2\,1\,1 \qquad 4\,1\,1\,1 \qquad 2\,2\,1\,1 \qquad 1\,1\,1\,1\,1\,1 \qquad 2\,1\,1\,1
\]

\[
  V_1*(V_1+\{x\})=2V_1+x*V_1
\]

\[
  \underset{3}{E}+\underset{3}{x*E}\qquad (V_1+G)
\]

\[
  E_i=[-i,i]\quad 0\leqslant i\leqslant m \qquad\qquad \text{NB } E_m=G,\ E_1=\varepsilon
\]

\[
  F_j=[m-j,\ m+1+j]\quad 0\leqslant j\leqslant m-1 \qquad F_m=E_m=G
\]

\[
  F_i*F_j=\sum_0^m\beta_{ij}^k E_k \qquad \beta_{ij}^k\geqslant 0
\]

\[
  E_i*E_j=\sum\alpha_{ij}^k E_k \qquad \alpha_{ij}^k\geqslant 0
\]

\[
  E_i*F_j=\sum\gamma_{ij}^k F_k \qquad\qquad \varepsilon=\varepsilon^{-1}
\]

\[
  \mathcal{A}(g,\varepsilon)=\begin{cases} g*\mathcal{A}(0) & \text{si } \varepsilon=+1\\ g*\mathcal{B}(0) & \text{si } \varepsilon=-1\end{cases}
\]

\[
  \mathcal{A}(\tilde g)*\mathcal{A}(\tilde h)\subset\mathcal{A}(\tilde g\tilde h)
\]

\[
  |f,g|=\sup_{f'\sim f}\langle f',g\rangle=\sup_{g'\sim g}\langle f,g'\rangle=\sup_{\substack{f'\sim f\\ g'\sim g}}\langle f',g'\rangle
\]

\[
  |f',g|\leqslant|f,g|
\]

\[
  |f',A|\leqslant|f,A|
\]

\[
  \sum_0^n A(f')_i\leqslant\sum_0^n A(f)_i
\]

\[
  |f',g'|\leqslant|f,g|
\]

\[
  \forall\alpha,\beta\in\mathbb{R}^{+*},\quad E_\alpha(f)\ \text{et}\ E_\beta(\varepsilon_s\check g)\ \text{en relation d'inclusion}
\]

\[
  \sup_{\substack{f'\prec f\\ g'\prec g\\ h'\prec h}}(f'*g'*h')(0)\leqslant\sup_{h'\sim h}f*g*h'(0)\quad\Bigl(=\sup_{h'\sim h}\|f*g*h'\|_\infty\Bigr)
\]

\[
  \sup_{\substack{A'\sim A\\ f'\sim f\\ g'\sim g}}f'*g'*A'(0)\leqslant\sup_{A'\sim A}f*g*A'(0)\quad\Bigl(=\sup_{A'\sim A}\|f*g*A'\|_\infty\Bigr)
\]

\[
  f'_1*\cdots*f'_n\prec f_1*\cdots*f_n
\]

\[
  \|f'_1*\cdots*f'_n\|_\infty\leqslant\|f_1*\cdots*f_n\|_\infty
\]

\[
A' * B' \prec A * B
\]

\[
\| A' * B' * C' \|_\infty \leq \| A * B * C \|_\infty
\]

\[
\widetilde{S} = \{ f \in \mathbb{R}(G)^{+} \mid \text{a), \ill{} b)} \}
\]

\[
\begin{cases}
f = \sum_I \alpha_i A_i, & \alpha_i > 0,\quad A_i \in S_0 \quad (A_i) \text{ tot.\ ord.} \\
\exists\, s \in G, & A_i = \varepsilon_s * \check{A}_i \quad \forall i
\end{cases}
\]

\[
\begin{cases}
g = \sum_J \beta_j B_j, & \beta_j > 0,\quad B_j \in S_0 \quad (B_j) \text{ tot.\ ord.} \\
t \in G, & B_j = \varepsilon_t * \check{B}_j
\end{cases}
\]

\[
f * g = \sum_{(i,j) \in I \times J} \alpha_i \beta_j \, A_i * B_j
\]

\[
E_*(A_i * B_j) \subset S_0
\]

\[
\bigcap_{C \in \bigcup E_\alpha(A_i * B_j)} C \neq \emptyset
\]

\[
0 \to {}_2 G \to G \xrightarrow{\;2\;} G
\]

\[
C_0 = \{ x \in C \mid R(x) = \emptyset \} \subset C \qquad (\text{conf.\ \textit{terminales}})
\]

\[
\left.
\begin{array}{l}
C(j) = \{ x \in C \setminus C_0 \mid \alpha(x) = j \} \\
\overline{C}(j) = (C \setminus C_0) \setminus C(j)
\end{array}
\right\}
\]

\[
x_{i+1} \in R(x_i)
\]

\[
\Sigma(x) \subset R(x), \qquad \Sigma(x) \neq \emptyset .
\]

\[
G(j)
\]

\[
G(j) \cap C_0 = G_0(j)
\]

\[
\begin{array}{ll}
\text{a)} & G(j) \cap C(j) = \{ x_0 \in C(j) \mid R(x_0) \cap G(j) \neq \emptyset \} \\
\text{b)} & G(j) \cap \overline{C}(j) = \{ x_0 \in \overline{C}(j) \mid R(x_0) \subset G(j) \}
\end{array}
\]

\[
G(j) \cap C_0 = G_0(j) \quad ].
\]

\[
\begin{array}{l}
\Sigma_0(x_0) = \{ x_1 \} \\
\Sigma_0(x) = \Sigma_1(x) \quad \text{si } x \neq x_0
\end{array}
\]

\[
E(x) = \{ x_1 \in R(x_0) \mid \exists\ \Sigma_{x_1}\text{-partie de } x_1 \text{ à } x \}
\]

\[
C' = \{ x \in C \mid E(x) \neq \emptyset \}
\]

\[
\varepsilon(x) = \text{plus petit élément de } E(x)
\]

\[
\Sigma_0(x) =
\begin{cases}
R(x) & \text{si } x \in \overline{C}(j) \cup \complement C' \\
\Sigma_{\varepsilon(x)}(x) & \text{si } x \in C(j) \cap C'
\end{cases}
\]

\[
E(y) \ni \varepsilon(x)
\]

\[
\varepsilon(y) \leq \varepsilon(x).
\]

\[
(x_0, x_1, x_2, \dots)
\]

\[
\Sigma'(x) =
\begin{cases}
R(x) & \text{si } x \in C(j) \cup G(j) \\
R(x) \cap \complement G(j) & \text{si } x \in \overline{C}(j) \cap \complement G(j)
\end{cases}
\]

\[
G_0(j) \cap G_0(j') = \emptyset .
\]

\[
N = \{ x \in C \mid x \text{ admissible contre } j, \text{ et contre } j' \}.
\]

\[
\begin{cases}
G^{*} = \{ x \in C \setminus C_0 \mid x \in G(\alpha(x)) \} \\
P^{*} = \{ x \in C \setminus C_0 \mid x \in G(\bar\alpha(x)) \}
\end{cases}
\]

\[
N^{*} = (C \setminus C_0) \setminus (G^{*} \cup P^{*}) = N \cap (C \setminus C_0)
\]

\[
\begin{cases}
G(j) \cap C(j) = G^{*} \cap C(j) \\
G(j) \cap \overline{C}(j) = G(j) \cap C(j') = P^{*} \cap C(j') \\
G(j) \cap C_0 = G_0(j)
\end{cases}
\]

\[
\hat\alpha : C \to J
\]

\[
\hat\alpha(y) \neq \hat\alpha(x)
\]

\[
\begin{array}{l}
G = \{ x \in C \mid x \in G(\hat\alpha(x)) \} \\
P = \{ x \in C \mid x \in G(\bar{\hat\alpha}(x)) \}
\end{array}
\]

\[
\begin{cases}
G \cap (C \setminus C_0) = G^{*} \\
P \cap (C \setminus C_0) = P^{*} \\
C \setminus (G \cup P) = N
\end{cases}
\]

\[
\hat C(j) = \{ x \in C \mid \hat\alpha(x) = j \}
\]

\[
\begin{cases}
G(j) \cap \hat C(j) = \hat C(j) \cap G \\
G(j) \cap \hat C(\bar\jmath) = \hat C(\bar\jmath) \cap P
\end{cases}
\]

\[
\begin{array}{lll}
x \in G & \Longleftrightarrow \exists\, y \in R(x) \text{ tel que } y \in P & \text{i.e. } R(x) \cap P \neq \emptyset \\
x \in P & \Longleftrightarrow \forall\, y \in R(x), \text{ on a } y \in G & \text{i.e. } R(x) \subset G
\end{array}
\]

\[
\begin{array}{l}
G \cap (C \setminus C_0) = \{ x \in C \setminus C_0 \mid R(x) \cap P \neq \emptyset \} \\
P \cap (C \setminus C_0) = \{ x \in C \setminus C_0 \mid R(x) \subset G \}
\end{array}
\]

\[
\widetilde G \subset G, \qquad \widetilde P \subset P
\]

\[
N(\mathbb{A}) \xrightarrow{\;\exp\;} 1 + N(\mathbb{A}) \qquad \exp x = \sum_{n \geq 0} \frac{x^n}{n!}
\]

\[
\log(1+y) = \sum_{n \geq 1} (-1)^{n+1} \frac{y^n}{n} \qquad (y \in N(\mathbb{A}))
\]

\[
\exp(x+y) = \exp x \, \exp y \qquad \text{NB } \exp 0 = 1
\]

\[
\log(uv) = \log u + \log v \qquad \text{NB } \exp 1 = 0
\]

\[
x \mapsto \exp x : L \xrightarrow{\;\sim\;} 1 + L
\]

\[
J\mathbb{A} \longrightarrow 1 + J\mathbb{A}
\]

\[
J\mathbb{A} = \{ x \in \mathbb{A} \mid \alpha(x) = 0 \}, \qquad
1 + J\mathbb{A} = \{ x \in \mathbb{A} \mid \alpha(x) = 1 \}
\]

\[
L = \{ u \in \mathcal{U} = \operatorname{End}_K(M) \mid u \otimes_K K_0 = 0 \} = \{ u \mid u(M) \subset JM \}
\]

\[
1 + L = \{ u \in \mathcal{U} = \operatorname{End}_K(M) \mid u \otimes_K K_0 = \mathrm{id}_{M_0} \}.
\]

\[
\exp : L \xrightarrow{\;\sim\;} 1 + L
\]

\[
\overline{\mathbb{G}}_{a}(K') = N(K') \subset K' \qquad (K' \ K\text{-alg.\ comm.} \dots)
\]

\[
K' \mapsto (\mathcal{U}_{K'})^{*} \qquad \mathrm{Alg}_{/K} \to \mathrm{Gr}
\]

\[
\begin{array}{l}
\overline{\mathbb{G}}_{a,K} \to \mathrm{Un}(\mathcal{U}) \\
\mathbb{G}_{a,K} \to \mathrm{Un}(\mathcal{U})
\end{array}
\]

\[
\bar e_x : \overline{\mathbb{G}}_{a,K} \to \mathrm{Un}(\mathcal{U})
\]

\[
\bar e_x(\lambda) = \exp(\lambda x)
\]

\[
\bar e_x \in \mathrm{Hom}_{\mathrm{gr}}(\overline{\mathbb{G}}_{a,K}, \mathrm{Un}(\mathcal{U}))
\]

\[
(*) \qquad x \mapsto \bar e_x : \mathcal{U} \to \mathrm{Hom}_{\mathrm{gr}}(\overline{\mathbb{G}}_{a,K}, \mathrm{Un}(\mathcal{U}))
\]

\[
\begin{cases}
e_x : \mathbb{G}_{a,K} \to \mathrm{Un}(\mathcal{U}) \\
e_x(\lambda) = \exp(\lambda x)
\end{cases}
\]

\[
e_x \in \mathrm{Hom}_{\mathrm{gr}}(\mathbb{G}_{a,K}, \mathrm{Un}(\mathcal{U}))
\]

\[
(**) \qquad x \mapsto e_x : N(\mathcal{U}) \to \mathrm{Hom}_{\mathrm{gr}}(\mathbb{G}_{a,K}, \mathrm{Un}(\mathcal{U}))
\]

\[
e_x(\lambda) = \sum \frac{x^n}{n!} \lambda^n
\]

\[
e_x = e_y \Longleftrightarrow \frac{x^n}{n!} = \frac{y^n}{n!} \ \ \forall n \in \mathbb{N} \Longleftrightarrow x = y
\]

\[
\bar\varphi(\lambda) = \sum_{n \geq 0} a_n \lambda^n \qquad a_n \in \mathcal{U}
\]

\[
\begin{array}{ccc}
\sum a_n (\lambda + \mu)^n & = & \sum a_i \lambda^i \sum a_j \mu^j \\
\| & & \| \\
\sum_{i,j} a_{i+j} \frac{(i+j)!}{i!\,j!} \lambda^i \mu^j & & \sum a_i a_j \lambda^i \mu^j
\end{array}
\]

\[
\frac{(i+j)!}{i!\,j!}\, a_n = a_i a_j
\]

\[
n a_n = a_1 a_{n-1} \quad \text{i.e.} \quad a_n = \tfrac{1}{n} a_1 (a_{n-1})
\]

\[
a_n = \frac{1}{n!} x^n \qquad x = a_1
\]

\[
\varphi(\lambda) = \sum a_n \lambda^n \qquad a_n \in \mathcal{U}, \text{ les } a_n \text{ nuls sauf un nb fini}
\]

\[
\begin{array}{l}
\mathbb{G}_a \xrightarrow{\;\varphi\;} \mathbf{Aut}_K(M) \\
\overline{\mathbb{G}}_a \xrightarrow{\;\bar\varphi\;} \mathbf{Aut}_K(M)
\end{array}
\]

\[
\begin{array}{ll}
(*) & u \mapsto \bar e_u : \mathrm{End}_K(M) \to \mathrm{Hom}_{\mathrm{gr}}(\overline{\mathbb{G}}_{a,K}, \mathbf{Aut}_K(M)) \\
(**) & u \mapsto e_u : N(\mathrm{End}_K(M)) \to \mathrm{Hom}_{\mathrm{gr}}(\mathbb{G}_{a,K}, \mathbf{Aut}_K(M))
\end{array}
\]

\[
\begin{array}{ll}
\bar e_u(\lambda) = \exp(\lambda u) = \sum_{n \geq 0} \frac{u^n_{K'}}{n!} \lambda^n & \lambda \in \overline{\mathbb{G}}_a(K') = N(K') \\
e_u(\lambda) = \text{— — —} & (\lambda \in \mathbb{G}_a(K') = K')
\end{array}
\]

\[
\bar\varphi(\lambda) = \sum_{n \geq 0} u_n \lambda^n \qquad u_n \in \mathrm{End}_K(M),
\]

\[
\varphi(\lambda) = \sum_{n \geq 0} u_n \lambda^n \qquad u_n \in \mathrm{End}_K(M) \text{ nuls sauf un nb fini.}
\]

\[
\sum_{n \geq 0} \lambda^n \frac{v^n}{n!} f = \sum_{n \geq 0} \lambda^n f \frac{u^n}{n!}
\]

\[
\sum_{n \geq 1} \frac{1}{n!} (v^n f - f u^n) \lambda^n = 0
\]

\[
v^n f - f u^n = 0 \qquad \forall n \geq 1
\]

\[
\underbrace{(\exp v) \circ f - f \circ \exp u}_{\displaystyle \sum_{n \geq 1} \frac{1}{n!} \underbrace{(v^n f - f u^n)}_{\delta_n \ (\text{déf})}} = 0
\]

\[
\delta_n = v^{n-1} \delta + v^{n-2} \delta u + \dots + v \delta u^{n-2} + \delta u^{n-1}
\]

\[
\delta + \frac{1}{2}(v \delta + \delta u) + \frac{1}{3!}(v^2 \delta + v \delta u + \delta u^2) + \dots = 0 \qquad (\text{somme finie})
\]

\[
\delta = \sum_{i+j \geq 1} c_{ij}\, v^i \delta u^j \qquad \Bigl(c_{ij} \in K,\ c_{ij} = \frac{1}{(i+j)!}\Bigr)
\]

\[
\mathcal{U} = \mathrm{End}_K(M)
\]

\[
\Theta, \Theta' : \mathcal{U} \to \mathcal{U} \qquad K\text{-endom.}
\]

\[
\Theta(x) = vx, \qquad \Theta'(x) = xu
\]

\[
\delta = \sum_{i+j \geq 1} c_{ij}\, \Theta^i \Theta'^j \delta = \rho \delta
\]

\[
\rho = \sum_{i+j \geq 1} c_{ij}\, \Theta^i \Theta'^j \in \mathrm{End}_K(\mathcal{U})
\]

\[
\delta = \rho^N \delta \qquad \forall N \in \mathbb{N}
\]

\[
\delta = 0 \qquad \text{cqfd.}
\]

\[
\bar e_u \otimes \bar e_v = \bar e_{u \otimes \mathrm{id}_N + \mathrm{id}_M \otimes v}
\]

\[
e_u \otimes e_v = e_{u \otimes \mathrm{id}_N + \mathrm{id}_M \otimes v}
\]

\[
\delta(f) = vf - fu
\]

\[
\mathcal{M}(K') \text{ est un } K'\text{-module}
\]

\[
\mathcal{M}(K') \to \mathcal{M}(K'') \quad (K' \to K'')\text{-semi-linéaire}
\]

\[
\bar e_u : \overline{\mathbb{G}}_{a,K} \to \mathbf{Aut}_K(\mathcal{M})
\]

\[
\bar e_u(\lambda) = \exp(\lambda u)
\]

\[
\mathrm{End}_K(\mathcal{M}) \to \mathrm{Hom\,gr}(\overline{\mathbb{G}}_{a,K}, \mathbf{Aut}_K(\mathcal{M}))
\]

\[
N(\mathrm{End}_K(\mathcal{M})) \to \mathrm{Hom\,gr}(\mathbb{G}_{a,K}, \mathbf{Aut}_K(\mathcal{M}))
\]

\[
\begin{cases}
\mathbb{E}_0 = \mathrm{id} \\
\mathbb{E}_{\Theta + \Theta'} = \mathbb{E}_\Theta + \mathbb{E}_{\Theta'} & \text{si } [\Theta, \Theta'] = 0
\end{cases}
\]

\[
  \mathrm{St}_{V'} : \Omega(V) \to \Omega(W) \qquad \text{fonctorielle en } (V, V')
  \]

\[
  \Omega(W, V) \wedge \Omega(V) \xrightarrow{\sim} \Omega(W)
  \]

\[
\Omega(V, W) \wedge \Omega(W) \xrightarrow{\sim} \Omega(W) \qquad \text{pour } \dim V = n
\]

\[
\underbrace{\Omega(\mathbb{R}^{n-1}, \mathbb{R}^n)}_{\mathbb{1}} \wedge \Omega(\mathbb{R}^n) \to \Omega(\mathbb{R}^{n-1}) \qquad \text{i.e.} \qquad \omega_n \xrightarrow{\sim} \omega_{n-1}
\]

\[
u_n = \begin{pmatrix} u_{n-1} & \vdots \\ 0 \ \cdots \ 0 & \lambda \end{pmatrix}
\]

\[
\varepsilon_n(u_n) = \mathrm{sg}(\lambda)\, \varepsilon_{n-1}(u_{n-1})
\]

\[
\varepsilon_1 = \mathrm{sg} : \mathbb{R}^* \to \{\pm 1\}
\]

\[
\varepsilon_n(u_n) = \mathrm{sg} \circ \det(u_n) \qquad u \in GL(n, \mathbb{R})
\]

\[
U \underbrace{\subset}_{\omega'} W \underbrace{\subset}_{\omega} V \qquad \Omega(V/W) \wedge \Omega(V) \to \Omega(W)
\]

\[
\mathrm{St}_\omega : \Omega(V) \to \Omega(W)
\]

\[
\mathrm{St}_{\omega'} : \Omega(W) \to \Omega(U)
\]

\[
\mathrm{St}_{\omega'} \mathrm{St}_\omega : \Omega(V) \to \Omega(U)
\]

\[
I(\omega, \omega') \in \Omega
\]

\[
\Omega(V) \wedge \Omega(V') \to \Omega(V \times V')
\]

\[
\omega' \times \omega = (-1)^{\dim V \cdot \dim V'}\, \omega \times \omega'
\]

\[
\begin{array}{ccccccccc}
\mathbb{R} & \subset & \mathbb{R}^2 & \subset & \mathbb{R}^3 & \text{—} & \mathbb{R}^{n-1} & \subset & \mathbb{R}^n \\
\cup & & \cup & & \cup & & \cup & & \cup \\
\mathbb{R}^+ & \subset & \mathbb{R} \times \mathbb{R}^+ & \subset & \mathbb{R}^2 \times \mathbb{R}^+ & \subset & \mathbb{R}^{n-2} \times \mathbb{R}^+ & \subset & \mathbb{R}^{n-1} \times \mathbb{R}^+
\end{array}
\]

\[
\begin{cases}
\mathrm{St}(\omega \times \omega') = \mathrm{St}(\omega) \times \omega' \\
\mathrm{St}(\omega \times \omega') = \omega \times \mathrm{St}\,\omega'
\end{cases}
\]

\[
\mathrm{St}(e_1 \ldots e_{p-1}\, e_p\, f_1 \ldots f_q) = (-1)^q\, e_1 \ldots e_{p-1}\, \hat e_p\, f_1 \ldots f_q
\]

\[
\mathcal{V} = \coprod_{n \in \mathbb{N}} \mathcal{V}_n(k)
\]

\[
  \Delta(V + V') \overset{\varphi_{V,V'}}{\simeq} \Delta(V) \otimes \Delta(V')
  \]

\[
  \alpha : W_1 + W_2 \xrightarrow{\sim} V \qquad
  \alpha' : W_1 + W_2' \xrightarrow{\sim} V \qquad
  \lambda : W_2 \simeq W_2'
  \]