\[
\cdot \longrightarrow J \longrightarrow \underline{\mathcal{O}}_{\mathcal{X}}
\longrightarrow \underline{\mathcal{O}}_{\mathcal{X}_0} \longrightarrow \cdot
\]

\[
W^{(\nu)} = W[\Lambda_0, \dots, \Lambda_\nu] \Big/ \Big(\Lambda_0^{p} - p!\,\Lambda_1,\;
\Lambda_1^{p} - \frac{p^2!}{(p!)^{p}}\,\Lambda_2,\; \dots,\;
\Lambda_{\nu-1}^{p} - \frac{p^{\nu}!}{(p^{\nu-1}!)^{p}}\,\Lambda_\nu,\;
\Lambda_\nu^{p} \;;\; \Lambda_0 - \struck{p}\lambda_0\Big)
\]

\[
A^{(\nu)} = W[\Lambda_0, \dots, \Lambda_\nu] \big/ \big(\Lambda_0^{p} - p!\,\Lambda_1,\; \dots,\; \Lambda_\nu^{p}\big)
\]

\[
\lambda_r = \frac{p^{p^{r}}}{p^{r}!} = \gamma^{p^{r}}_{\text{habituel}}(p)
\]

\[
\Lambda_i = \lambda_i + \Theta_i \qquad 0 \leq i \leq \nu .
\]

\[
\Theta_0^{a_0} \cdots \Theta_\nu^{a_\nu}, \qquad 0 \leq a_i \leq p-1 \quad \text{pour } i = 0, \dots, \nu .
\]

\[
c(\nu) = \nu + p\,\big[\,p^{\nu} - (p^{\nu-1} + p^{\nu-2} + \dots + 1)\,\big]
\]

\[
\begin{cases}
p^{i}\,\Theta_i \in I \\
p^{c(\nu)} \in I
\end{cases}
\]

\[
(\Theta_i + \lambda_i)^{p} = \frac{p^{i+1}!}{(p^{i}!)^{p}}\,(\Theta_{i+1} + \lambda_{i+1}) ,
\]

\[
\struck{\ill{}} \sum_{\alpha=1}^{p-1} C_p^{\alpha}\,\Theta_i^{\alpha}\lambda_i^{p-\alpha}
= \Big( -\lambda_i^{p} + \frac{p^{i+1}!}{(p^{i}!)^{p}}\,\struck{\Theta_{i+1}}
+ \frac{p^{i+1}!}{(p^{i}!)^{p}}\,\Theta_{i+1} \Big)
\]

\[
p^{i}\,\frac{p^{i+1}!}{(p^{i}!)^{p}}\,\Theta_{i+1} = 0
\]

\[
p^{i+1}\,\Theta_{i+1} = 0 .
\]

\[
(\lambda_\nu + \Theta_\nu)^{p} = 0 \quad \text{i.e.} \quad
\lambda_\nu^{p} + \sum_{i=1}^{p} C_i^{p}\,\Theta_\nu^{i}\lambda_\nu^{p-i} = 0
\]

\[
p^{\nu}\lambda_\nu^{p} \in I
\]

\[
v_p(p^{\nu}\lambda_\nu^{p}) = \nu + p\,v_p(\lambda_\nu)
= \nu + p\Big(p^{\nu} - \frac{p^{\nu}-1}{p-1}\Big) = c(\nu)
\]

\[
\sum w_{a_1, \dots, a_\nu}\, \dot{\Theta}_1^{a_1} \cdots \dot{\Theta}_\nu^{a_\nu}
\qquad 0 \leq a_i \leq p-1 \quad (1 \leq i \leq \nu)
\]

\[
\begin{cases}
w_{a_1, \dots, a_\nu} \in E_i \quad \text{si $a_i$ est la première coordonnée non nulle de } (a_1, \dots, a_\nu) \\
w_0 \in E_{c(\nu)}
\end{cases}
\]

\[
\sum w_{a_{r+1}, \dots, a_\nu}\, \Theta_{r+1}^{a_{r+1}} \cdots \Theta_\nu^{a_\nu}
\]

\[
\begin{cases}
w_{a_{r+1}, \dots, a_\nu} \in E_i\struck{\ill{}} \quad \text{si $a_{r+i}$ est le premier des $a_\alpha$ qui est } \neq 0 \\
w_0 \in E_{c(\nu) - r}
\end{cases}
\]

\[
\begin{cases}
\Lambda_i^{p} = \dfrac{p^{i+1}!}{(p^{i}!)^{p}}\,\Lambda_{i+1} & \text{si } 0 \leq i \leq \nu - 1 \\[6pt]
\Lambda_\nu^{p} = 0
\end{cases}
\]

\[
\Lambda_0^{a_0} \cdots \Lambda_\nu^{a_\nu} = 0 \quad \text{si} \quad \sum a_i p^{i} \geq p^{\nu+1} \qquad (a_i \geq 0)
\]

\[
\Lambda_0^{a_0}\Lambda_1^{a_1} \cdots \Lambda_\nu^{a_\nu}
= c\,\Lambda_0^{b_0} \cdots \Lambda_{\nu-1}^{b_{\nu-1}}\Lambda_\nu^{b_\nu}, \qquad c \in W,
\]

\[
\sum_0^{\nu} a_i p^{i} = \sum_0^{\nu} b_i p^{i} .
\]

\[
\Lambda_i^{a_i} = (\Lambda_i^{p})^{q_i}\,\Lambda_i^{a'_i}
= (c_i\Lambda_{i+1})^{q_i}\,\Lambda_i^{a'_i}
= c\,\Lambda_{i+1}^{q_i}\,\Lambda_i^{a'_i}
\]

\[
\Lambda_0^{a_0} \cdots \Lambda_\nu^{a_\nu}
= c\,\Lambda_0^{a_0} \cdots \Lambda_i^{a'_i}\,\Lambda_{i+1}^{a_{i+1} + q_i}\,\Lambda_{i+2}^{a_{i+2}} \cdots ,
\]

\[
\sum a_i p^{i} = \sum a'_i p^{i} ,
\]

\[
a'_i p^{i} + (a_{i+1} + q_i)\,p^{i+1} = \underbrace{(a'_i + q_i p)}_{a_i}\,p^{i} + a_{i+1}\,p^{i+1}
\]

\[
v_p(n!) = a_\nu\,v_p(p^{\nu}!) + \dots + a_{\mu+1}\,v_p(p^{\mu+1}!)
\]

\[
- v_p(p^{\mu+1}!) + (\mu+1) + (p-1)\big[\,v_p(p^{\mu}!) + \dots + v_p(p)\,\big]
\]

\[
v_p(1) = 0
\]

\[
v_p(p) = 1 + (p-1)\cdot 0 = 1
\]

\[
v_p(p^{2}) = 2 + (p-1)\cdot 1 = p+1
\]

\[
v_p(p^{3}) = 3 + \struck{(p-1)}(p+2)\struck{+1} = p^{2} + p + 1
\]

\[
v_p(p^{4}) = 4 + (p-1)\big[p^{2} + 2p + 3\big] = 4 + p^{3} + 2p^{2} + 3p - p^{2} - 2p - 3
= \struck{1+}\,p^{3} + p^{2} + p + 1
\]

\[
v_p(p^{\nu}) = p^{\nu-1} + p^{\nu-2} + \dots + 1 = \frac{p^{\nu} - 1}{p - 1}
\]

\[
v_p(p^{\nu+1}) = \nu + 1 + (p-1)\big[p^{\nu-1} + 2p^{\nu-2} + 3p^{\nu-3} + \dots + \nu p^{0}\big]
\]

\[
= \nu + 1 + p^{\nu} + 2p^{\nu-1} + 3p^{\nu-2} + \dots + \nu p
- p^{\nu-1} - 2p^{\nu-2} - \dots - (\nu-1)p - \nu
\]

\[
= p^{\nu} + p^{\nu-1} + \dots + p + 1
\]

\[
\sum a_\nu\,(p^{\nu-1} + p^{\nu-2} + \dots + 1)
\]

\[
a_1 + a_2(p+1) + a_3(p^{2} + p + 1) + a_4(p^{3} + p^{2} + p + 1) + \dots
\]

\[
v_p(n!)(p-1) = a_1(p-1) + a_2(p^{2}-1) + \dots + a_\nu(p^{\nu} - 1)
= n - a_0 - a_1 - \dots - a_\nu
\]

\[
\boxed{\;v_p(n!) = \frac{n - \sum a_i}{p - 1}\;}
\]

\[
v_p\Big(\frac{\pi^{n}}{n!}\Big) \geq 0 \quad \text{pour tout } n \ \ill{}
\]

\[
v_p\Big(\frac{\pi^{n}}{n!}\Big) \to +\infty \quad \text{pour } n \to +\infty
\]

\[
v_p\Big(\frac{\pi^{n}}{n!}\Big) = \struck{(p-1)}\; n\,v_p(\pi) - \frac{n - \mathrm{chif}(n)}{p-1} \geq 0
\]

\[
\Updownarrow \qquad v_p(\pi) \geq \frac{1 - \frac{\mathrm{chif}(n)}{n}}{p-1}
\]

\[
n = a_0 + a_1 p + \dots + a_\nu p^{\nu}
\]

\[
\mathrm{chif}(n) = a_0 + a_1 + \dots + a_\nu
\]

\[
\mathrm{chif}(n) \leq (p-1)\nu\struck{\ill{}}
\]

\[
n \geq p^{\nu}
\]

\[
\frac{\mathrm{chif}(n)}{n} \leq \frac{(p-1)\nu}{p^{\nu}} \to 0
\]

\[
v_p\Big(\frac{\pi^{n}}{n!}\Big) \geq 0 \ \text{pour tout } n
\iff v_p\Big(\frac{\pi^{n}}{n!}\Big) > 0 \ \text{pour tout } n
\iff v_p(\pi) \geq \frac{1}{p-1} \quad \text{i.e.} \quad \pi^{p-1} \in pV
\]

\[
(p-1)\,v_p\Big(\frac{\pi^{n}}{n!}\Big) = n\big[(p-1)v_p(\pi) - 1\big] + \mathrm{chif}(n)
\]

\[
n\Big(\alpha + \frac{\mathrm{chif}(n)}{n}\Big) \sim \alpha n
\]

\[
v_p\Big(\frac{\pi^{n}}{n!}\Big) \to +\infty \iff v_p(\pi) > \frac{1}{p-1}
\quad \text{i.e.} \quad p = \text{unit}\cdot\pi^{r}, \ \text{avec } r < p-1
\]

\[
(1.1) \qquad c_E : F_Y^{*}(g_*(E)) \longrightarrow g_*(F_X^{*}(E))
\]

\[
(1.2) \qquad F_E : F_X^{*}(E) \longrightarrow E
\]

\[
g_*(F_E) : g_* F_X^{*} E \longrightarrow g_*(E)
\]

\[
(1.3) \qquad \struck{c_E}\; g_*(F_E) \circ c_E : F_Y^{*}(g_*(E)) \longrightarrow g_*(E)
\]

\[
(1.4) \qquad c^{G} : F_X^{*}(g^{*}(G)) \xrightarrow{\;\sim\;} g^{*}(F_Y^{*}(G))
\]

\[
(1.5) \qquad F_G : F_Y^{*}(G) \longrightarrow G
\]

\[
g^{*}(F_G) : g^{*}F_Y^{*}(G) \longrightarrow g^{*}(G) ,
\]

\[
(1.6) \qquad g^{*}(F_G) \circ c^{G} : F_X^{*}(g^{*}(G)) \longrightarrow g^{*}(G)
\]

\[
\alpha, \beta : X' \rightrightarrows X
\]

\[
E \in \mathrm{Ob}\,X, \qquad F_E : \alpha^{*}(E) \longrightarrow \beta^{*}(E)
\]

\[
\omega : X \longrightarrow \mathcal{Q}(\alpha, \beta)
\]

\[
F : \omega\beta \longrightarrow \omega\alpha
\]

\[
f : \varphi\beta \longrightarrow \varphi\alpha
\]

\[
g \longmapsto (g \circ \omega,\; g * F) \;:\;
\underline{\mathrm{Homtop}}(\mathcal{Q}(\alpha, \beta), Y) \longrightarrow
\underline{\mathrm{Homtop}}(X, \alpha, \beta ; Y)
\]

\[
\mathcal{Q}(u) : \mathcal{Q}(\alpha, \beta) \longrightarrow \mathcal{Q}(\lambda, \mu)
\]

\[
\omega_X^{*}(R^{i}v_*(\mathbb{E})) \longrightarrow R^{i}u_{0*}(\omega_X^{*}(\mathbb{E})) = R^{i}u_{0*}(E) ,
\]

\[
  \pi^{n} : x \mapsto x^{(n)} = \pi^{n}(x) \qquad n \in \mathbb{N}^{+}
\]

\[
  \pi^{n}(x+y) = \sum_{p+q=n} \pi^{p}(x)\,\pi^{q}(y) \qquad x, y \in J
\]

\[
  \text{(1.5.bis)}\qquad \pi^{p_{1}}(x)\,\pi^{p_{2}}(x) \cdots \pi^{p_{r}}(x)
  = \frac{(p_{1} + \cdots + p_{r})!}{p_{1}! \cdots p_{r}!}\; \pi^{p_{1} + \cdots + p_{r}}(x)
\]

\[
  \text{1.5.ter}\qquad n!\; \pi^{n}(x) = \struck{\ill{}}\; x^{n}
\]

\[
  \text{1.6.}\qquad \pi^{n}(\lambda x) = \lambda^{n}\, \pi^{n}(x) \qquad \lambda \in A,\ x \in \struck{A}.
\]

\[
  \Gamma^{*}(M) = \coprod_{n \geqslant 0} \Gamma^{n}(M),
\]

\[
  \pi^{n} : \Gamma^{+}(M) \longrightarrow \Gamma^{+}(M) \qquad x \mapsto x^{(n)}
\]

\[
  \pi^{n}_{(p)} : \Gamma^{p}(M) \longrightarrow \Gamma^{np}(M) \qquad \struck{x \mapsto x^{(n)}}
\]

\[
  x \mapsto \pi^{n}_{(p)}(x^{(p)}) : \underline{M} \longrightarrow \underline{\Gamma^{np}(M)}
\]

\[
  (x^{(p)})^{(n)} = x^{(pn)}\, \frac{(pn)!}{(p!)^{n}\, n!}
\]

\[
  M \longrightarrow \Gamma^{+}(M) ;
\]

\[
  (x.y)^{(n)} - x^{n} * y^{(n)}
\]

\[
  A \times J' / \alpha(\uncertain{J}) , \quad \text{où} \quad \alpha : \uncertain{J} \to A \times J'
\]

\[
  \alpha(x) = (x, -x')
\]

\[
  \Gamma\big(\operatorname{Sym}_{\Lambda}(M),\ \operatorname{Sym}^{+}_{\Lambda}(M)\big)
  \simeq \Gamma^{*}_{\Lambda}(M)
\]

\[
  1 + J \;\rightleftarrows\; J, \qquad
  \log(1+x) = \sum_{n \geqslant 0} (-1)^{n}\,(n-1)!\; x^{(n)}, \qquad
  \exp x = \sum_{n \geqslant 0} x^{(n)}
\]

\[
  1 + J \subset A^{*}
\]

\[
  H^{*}(X_{\mathrm{DR\,st}},\ \mathcal{O}) \Longleftarrow
  E_{2}^{p,q} = H^{p}\big(\nu \mapsto H^{q}(\widetilde{X}^{\nu+1}, \mathcal{O})\big)
\]

\[
  \Delta^{(\nu)}_{X/S}(i) = \varinjlim_{j} \Delta^{(\nu)}_{X/S}(i, j)
\]

\[
  \widetilde{X}^{\nu+1} = \varinjlim_{i, j} \Delta^{(\nu)}_{X/S}(i, j).
\]

\[
  H^{q}\big(\mathrm{DR}_{\mathrm{st}}/(U, U') ;\ \mathcal{O}\big) \simeq H^{q}(U', \mathcal{O}_{U'})
\]

\[
  \begin{cases}
    H^{q}(\widetilde{X}^{\nu+1}, \mathcal{O}) = 0 & \text{si } q > 0 \\
    H^{0}(\widetilde{X}^{\nu+1}, \mathcal{O}) = \varprojlim_{i,j}
      H^{0}\big(\Delta^{(\nu)}_{X/S}(i, j),\ \mathcal{O}_{\Delta^{\nu}_{X/S}(i,j)}\big)
  \end{cases}
\]

\[
  H^{n}(X_{\mathrm{DR\,strat}}, \mathcal{O}) \simeq
  H^{n}\big(\nu \mapsto H^{0}(\widetilde{X}^{\nu+1}_{/X})\big)
\]

\[
  H^{*}(X_{\mathrm{DR\,strat}}, \mathcal{O}) \simeq H^{*}(X_{\mathrm{Zar}}, \mathcal{E}^{*})
\]

\[
  B = A\Big[\struck{\ill{}}\ \{\Gamma^{n}(x)\}_{\substack{x \in I \\ n \geqslant 2}}\Big] \Big/ \text{relations}
\]

\begin{align*}
  \Gamma^{n}(x+y) &= \Gamma^{n}x + \Gamma^{n}y + \sum_{\substack{p, q \geqslant 1 \\ p+q = n}} \Gamma^{p}(x)\,\Gamma^{q}(y) \qquad (x, y \in I)
    \quad [\Gamma^{1}x = x \text{ par déf.}] \\
  \Gamma^{n}(xy) &= x^{n}\,\Gamma^{n}(y) \qquad x \in I,\ y \in A \\
  \struck{\Gamma^{m}(\Gamma^{n}(x))} &\struck{= \tfrac{(mn)!}{(n!)^{m}\, m!}\, \Gamma^{mn}(x)} \qquad x \in I \\
  \Gamma^{m}(x)\,\Gamma^{n}(x) &= \frac{(m+n)!}{m!\, n!}\; \Gamma^{m+n}(x)
\end{align*}

\[
  n = a_{0} + a_{1} p + a_{2} p^{2} + \cdots + a_{\nu} p^{\nu}, \qquad 0 \leqslant a_{i} \leqslant p-1,
\]

\[
  v_{p}(n) = \sum a_{i}\, v_{p}(p^{\nu}) \qquad \Big[\text{N.B. } v_{p}(p^{\nu}) = \frac{p^{\nu}-1}{p-1}\Big]
\]

\[
  \gamma^{n}(x) = c_{n}\; \gamma^{1}(x)^{a_{0}}\, \gamma^{p}(x)^{a_{1}} \cdots \big(\gamma^{p^{\nu}}(x)\big)^{a_{\nu}}
\]

\[
  c_{n} = \frac{\prod_{i} \big[(p^{i}\,!)/p^{\,v_{p}(p^{i}!)}\big]^{a_{i}}}{\big[n!/p^{\,v_{p}(n!)}\big]}
  = \text{partie première à } p \text{ de } \frac{\prod (p^{i}\,!)^{a_{i}}}{n!}
\]

\begin{align*}
  \Gamma^{p^{\nu}}(x+y) &= \Gamma^{p^{\nu}}x + \Gamma^{p^{\nu}}y
    + \sum_{\substack{\alpha, \beta \geqslant 1 \\ \alpha + \beta = p^{\nu}}} \struck{\ill{}}\ \Gamma^{\alpha}(x)\,\Gamma^{\beta}(y)
  \qquad \underbrace{\phantom{xxxxxxxxxxxxxx}}_{\text{expression en les } \Gamma^{p^{i}}(x),\ \Gamma^{p^{j}}(y),\ i, j < \nu} \\
  \Gamma^{p^{\nu}}(xy) &= x^{p^{\nu}}\, \Gamma^{p^{\nu}}(y) \\
  \Gamma^{p^{m}}(x)\,\Gamma^{p^{n}}(x) &= \frac{(p^{m} + p^{n})!}{p^{m}!\; p^{n}!}\; \Gamma^{p^{m}+p^{n}}(x) \qquad \ldots
\end{align*}

\[
  \begin{cases}
    \Gamma^{m}(\lambda)\,\Gamma^{n}(\lambda) - \dfrac{(m+n)!}{m!\, n!}\; \Gamma^{m+n}(\lambda) \\[6pt]
    \struck{\ill{}} \quad (x^{n} - y^{n})\,\Gamma^{n}(\lambda) \qquad \text{pour } x, y \in A \text{ tels que } (x-y)\lambda = 0 .
  \end{cases}
\]

\[
  A\big[(\Gamma^{p^{\nu}}(\lambda))_{\nu \geqslant 1}\big]
\]

\[
  \begin{cases}
    \Gamma^{m}(\lambda)\,\Gamma^{n}(\lambda) - \dfrac{(m+n)!}{m!\, n!}\; \Gamma^{m+n}(\lambda) \\[6pt]
    (x^{n} - y^{n})\,\Gamma^{n}(\lambda) \qquad \text{si } x, y \in A \text{ tels que } (x-y)\lambda = 0
  \end{cases}
\]

\[
  \boxed{\ \Gamma^{p^{\nu}}(\lambda)^{\uncertain{e}} = \struck{\ill{}}\; \frac{(\uncertain{e}\,p^{\nu+1})!}{(p^{\nu}!)^{p}}\; \Gamma^{\uncertain{e}\,p^{\nu+1}}(\lambda)\ }
\]

\[
  \begin{cases}
    k[[x]][Y] / (Y^{p} - x^{q}) \\
    k[[x]][Y] / (Y^{2} + P(x)\,Y + Q(x)) \\
    k[x_{1}, \ldots, x_{n}] / (x_{1}^{p_{1}}, x_{2}^{p_{2}}, \ldots, x_{n}^{p_{n}}) \qquad \text{p. ex. } k[x]/(x^{p}) \\
    k[x, y] / \mathfrak{m}^{2} \\
    k[x, y] / (x^{3}, y^{2}, x^{2}y) \\
    k[x, y] / (x^{3}, y^{2}, xy)
  \end{cases}
\]

\[
    \boxed{\ \pi_{m+n}(x) = \binom{m+n}{n}^{-1}\, \pi_{m}(x)\,\pi_{n}(x)\ }
  \]

\[
    c_{n} = \frac{(pn)!}{(n!)^{p}\; p!}
  \]

\begin{align*}
    v_{p}(c_{n}) &= v_{p}\big((pn)!\big) - p\, v_{p}(n!) - 1 \\
      &= \frac{pn - \sum a_{i}}{p-1} - p\,\frac{n - \sum a_{i}}{p-1} - 1 \\
      &= \struck{\ill{}}\ \Big(\sum a_{i}\Big) - 1
  \end{align*}

\[
    d_{m} = \frac{(pm)!}{(p!)^{m}\; m!}
  \]

\begin{align*}
    v_{p}(d_{m}) &= v_{p}\big((pm)!\big) - m\, v_{p}(p!) - v_{p}(m!) \\
      &= \frac{pm - \sum b_{i}}{p-1} - m - \frac{m - \sum b_{i}}{p-1} \\
      &= 0
  \end{align*}

\[
    \pi_{pm} = \Big(\frac{(pm)!}{(p!)^{m}\, m!}\Big)^{-1}\, \pi_{m} \circ \pi_{p}\ ]
  \]

\[
  \begin{cases}
    \pi(x+y) = \pi x + \pi y + \displaystyle\sum_{0 < i < p} \struck{\ill{}}\ \frac{1}{i!\,(p-i)!}\; x^{i} y^{p-i} \\[8pt]
    \pi(fx) = f^{p}\, \pi x
  \end{cases}
\]

\[
  \pi_{n}(x) = c_{n}\; x^{a_{0}}\, \pi(x)^{a_{1}}\, \pi^{2}(x)^{a_{2}} \cdots \pi^{r}(x)^{a_{r}}
\]

\[
\Phi\Bigl(\sum_i t_i x_i\Bigr)
= \sum_{p_1 + \cdots + p_r = n} \Phi_{p_1 \ldots p_r}(x_1, \ldots, x_r)\,
t_1^{p_1} \cdots t_r^{p_r}
\]

\[
\Phi_{p_1 \ldots p_r} : M \times \cdots \times M \to N
\]

\[
\varphi_{p_1, \ldots, p_r} : \underbrace{M \times \cdots \times M}_{r} \to N
\]

\[
\varphi_{p_1, \ldots, p_r}\Bigl(\sum_{1 \leq i \leq s_1} x_{1,i},\
\sum_{1 \leq i \leq s_2} x_{2,i},\ \ldots,\ \sum_{i \leq s_r} x_{r,i}\Bigr)
= \sum_{\substack{\sum_{1 \leq i_1 \leq s_1} q_{1,i_1} = p_1 \\ \sum_{1 \leq i_2 \leq s_2} q_{1,i_2} = p_2 \\ \cdots \\ \sum_{1 \leq i_r \leq s_r} q_{r,s_r} = p_r}}
\varphi_{q_{1,i_1}, \ldots, q_{r,s_r}}(x_{1,1}, \ldots, x_{r,s_r})
\]

\[
\frac{(\pi y)^n}{n!} = \frac{\pi^n}{n!}\, y^n = \pi^{(n)} y^n
\]

\[
\pi^{(n)} \in \pi A \qquad (n \geq 1)
\]

\[
(1)\qquad
\begin{cases}
\pi^{(1)} = \pi \\ \pi^{(m)} \pi^{(n)} = \dfrac{(m+n)!}{m!\,n!}\, \pi^{(m+n)} \\ \pi^{(p)} \Bigl(\dfrac{\pi^{(q)}}{\pi}\Bigr)^{p} = \dfrac{(pq)!}{p!\,(q!)^p}\, \pi^{(pq)}
\end{cases}
\]

\[
x^{(n)} = (\pi y)^{(n)} = \pi^{(n)} y^n \qquad \text{pour } x = \pi y \in J,
\]

\[
\pi^n = n!\, \pi^{(n)}
\]

\[
\pi^n \in n!\, A \qquad \forall n \geq 2 .
\]

\[
v(\pi^n/n!) \geq 0 \qquad \text{pour tout } n
\]

\[
n\, v(\pi) - v(n!) \geq 0 \qquad \text{pour tout } n .
\]

\[
v(\pi) \geq e\, \frac{v_p(n!)}{n} \qquad \text{pour tout } n
\]

\[
v(\pi) \geq e\, \lambda(p) \qquad \text{où } \lambda(p) = \sup_n
\frac{v_p(n!)}{n} = \frac{1}{p-1}
\]

\[
v(\pi) \geq \frac{e}{p-1} .
\]

\[
\boxed{e \leq p-1}
\]

\[
(18.13.4.1)\qquad \mathrm{Hom}_B(\overline{\Omega}^1_{B/A}, L)
\xrightarrow{\ \sim\ } \mathrm{D\acute{e}r}_A(B, L) =
\mathrm{D\acute{e}r}.\mathrm{cont}_A(B, L)
\]

\[
f^{!}(K^{\bullet}) = R\,\underline{\mathrm{Hom}}^{\bullet}_{f^{-1}(A)}
\bigl(B,\ f^{-1}(K) \otimes T_{X/Y}\,\supplied{[}2d\supplied{]}\bigr) .
\]

\[
H^{*}(X) \Leftarrow H^p\bigl(X, \underline{H}^q(\underline{\Omega})\bigr) .
\]