\[
  0 \longrightarrow \Omega^{1}_{P/S} \longrightarrow g^{*}(E)(-1)
  \longrightarrow \mathcal{O}_{P} \longrightarrow 0
\]

\[
  \tag{1}
  \Omega^{r}_{P/S} \xrightarrow{\;\sim\;}
  g^{*}\bigl(\Lambda^{r+1}_{\mathcal{O}_S} E\bigr)(-r-1)
\]

\[
  0 \longrightarrow J/J^{2} \longrightarrow \Omega^{1}_{P/S} \otimes \mathcal{O}_X
  \longrightarrow \Omega^{1}_{X/S} \longrightarrow 0
\]

\[
  \tag{2}
  \Omega^{r}_{P/S} \otimes \mathcal{O}_X \simeq
  \Omega^{n}_{X/S} \otimes \Lambda^{p}_{\mathcal{O}_X}(J/J^{2})
\]

\[
  \struck{\Omega^{r}}\; f^{*}\bigl(\Lambda^{r+1}_{\mathcal{O}_S} E\bigr)(-r-1)
  \simeq \Omega^{n}_{X/S} \otimes \Lambda^{p}_{\mathcal{O}_X}(J/J^{2})
\]

\[
  g^{*}(I_d)(-d) \longrightarrow J
\]

\[
  \coprod g^{*}(M_d)(-d) \longrightarrow J
\]

\[
  \coprod f^{*}(M_d)(-d) \xrightarrow{\;\sim\;} J/J^{2} \simeq
  \mathrm{Conormal}_{P/X}
\]

\[
  \Lambda^{p}_{\mathcal{O}_X}\, J/J^{2} \simeq
  f^{*}\Bigl(\bigotimes_{i} \Lambda^{\alpha_i} M_{d_i}\Bigr)
  \bigl(-\textstyle\sum \alpha_i d_i\bigr)
\]

\[
  \Omega^{n}_{X/S} \otimes f^{*}\Bigl(\bigotimes_{i} \Lambda^{\alpha_i}
  M_{d_i}\Bigr)\bigl(-\textstyle\sum \alpha_i d_i\bigr)
  \xrightarrow{\;\sim\;} f^{*}\bigl(\Lambda^{r+1}_{S} E\bigr)(-r-1)
\]

\[
  \Omega^{n}_{X/S} \xrightarrow{\;\sim\;}
  f^{*}\Bigl(\bigl(\Lambda^{r+1}_{S} E\bigr) \otimes
  \bigotimes_{i} \Lambda^{\alpha_i} \check{M}_{d_i}\Bigr)
  \bigl(\textstyle\sum \alpha_i d_i - r - 1\bigr)
\]

\[
  \textstyle\sum \alpha_i d_i = r+1
\]

\[
  \Omega^{n}_{X/S} \xrightarrow{\;\sim\;} f^{*}(\ -\ )
\]

\[
  \Omega^{1}_{X/S} \xrightarrow{\;\sim\;} f^{*}\bigl(\Lambda^{3} E \otimes
  \check{M}\bigr)
\]

\[
  \delta_{r,k} \in \underline{\Phi}^{(k-1)(r+1)}\bigl(\Phi^{k}(P^{r})\bigr)
\]

\[
  J \simeq g^{*}(M)(-k)
\]

\[
  \Bigl[\ \struck{\mathcal{O}(-k) \otimes g^{*}(E)} \longrightarrow
  \mathcal{O}(-k) \otimes g^{*}\mathrm{Symm}_k(E) \longrightarrow
  \mathcal{O}_{P^{r}} \qquad
  \text{d'où} \quad \mathcal{O}(-k) \otimes g^{*}(M) \longrightarrow
  \mathcal{O}_{P^{r}} \quad -\ \Bigr]
\]

\[
  N(U) \;\underset{\log}{\overset{\exp}{\rightleftarrows}}\; 1_U + N(U)
\]

\[
  \begin{cases}
    \exp \xi = \sum_{n \geqslant 0} \frac{1}{n!}\,\xi^{n} \\[4pt]
    \log(1+\eta) = \sum_{n > 0} (-1)^{n+1} \frac{1}{n}\,\eta^{n}
  \end{cases}
\]

\[
  \begin{cases}
    \exp(\xi + \xi') = \exp \xi \, \exp \xi' \quad \text{si } [\xi, \xi'] = 0 \\[4pt]
    \struck{\log((1+\eta)(1+\eta'))} \\[2pt]
    \log g g' = \log g \; \log g' \quad \text{si } g g' = g' g
  \end{cases}
\]

\[
  N(U) \cap I \;\rightleftarrows\; 1_U + N(U) \cap I
\]

\[
  \begin{cases}
    \varphi \exp \xi = \exp \varphi(\xi) \\[2pt]
    \varphi \log g = \log \varphi(g)
  \end{cases}
\]

\[
  \Delta : U \longrightarrow U \,\widehat{\otimes}_{k}\, U
\]

\[
  \Delta(g) = g \otimes g
\]

\[
  \delta \xi = \xi \otimes \xi \qquad \text{si} \quad g = 1 + \xi
  \quad (\xi \in I(U))
\]

\[
  \Delta(g g') = \Delta(g)\,\Delta(g') = (g \otimes g)(g' \otimes g')
  = g g' \otimes g g'
\]

\[
  \Delta(g^{-1}) = \struck{\ill{}}\,(\Delta(g))^{-1}
  = (g \otimes g)^{-1} = g^{-1} \otimes g^{-1}
\]

\[
  \pi(U) \;\underset{\log}{\overset{\exp}{\rightleftarrows}}\; \gamma(U)
\]

\[
  \boxed{\ I(U) \ \ill{} \ \exp\bigl\{\ \ill{} \ \bigr\}\ }
\]

\[
  \Bigl[\ \struck{I(U) \otimes_{k} I(\Lambda)} \Bigr] \qquad
  \boxed{U \otimes_{k} I(\Lambda)} \longrightarrow
  1 + I(U) \otimes_{k}
\]

\[
  U \otimes I(\Lambda)
  \;\underset{\log}{\overset{\exp}{\rightleftarrows}}\;
  1 + U \otimes I(\Lambda)
\]

\[
  I(U) \otimes I(\Lambda)
  \;\underset{\log}{\overset{\exp}{\rightleftarrows}}\;
  1 + I(U) \otimes I(\Lambda)
\ \Bigr]
\]

\[
  \pi_{0}(U \otimes \Lambda)
  \;\underset{\log}{\overset{\exp}{\rightleftarrows}}\;
  \gamma_{0}(U \otimes_{k} \Lambda)
\]

\[
  \pi(U) \otimes_{k} I(\Lambda) \xrightarrow{\;\alpha\;}
  \pi_{0}(U \otimes_{k} \Lambda)
\]

\[
  0 \longrightarrow \pi(U) \longrightarrow \struck{\ill{}}
  \uncertain{I(U)} \xrightarrow{\;\delta\;} I(U) \otimes I(U)
\]

\[
  \pi(U) \otimes_{k} I(\Lambda) \xrightarrow{\;\exp\circ\alpha\;}
  1 + \pi_{0}(U \otimes_{k} \Lambda)
\]

\[
  \xi . a = 0 \;\Longleftrightarrow\; g a = a
\]

\[
  \xi \ \ill{} \ \Longleftrightarrow\ g \ \ill{}
\]

\[
  M' \ \uncertain{\text{stable}} \ \ill{} \ \xi
  \;\Longleftrightarrow\;
  M' \ \uncertain{\text{stable}} \ \ill{} \ g \qquad (M' \subset M)
\]

\[
  \xi(x . y) = (\xi x) y + x (\xi y)
  \;\Longleftrightarrow\;
  g(x . y) = (g x)(g y)
\]

\[
  \pi \;\underset{\log}{\overset{\exp}{\rightleftarrows}}\; \gamma
  \qquad (\uncertain{\text{multiplicatif}})
\]

\[
  B = A \otimes_{k} \Lambda
\]

\[
  \mathcal{T}_{B/\Lambda} \struck{(B/A)}
  = \operatorname{Hom}_{B}(\Omega^{1}_{B/\Lambda}, B)
  \simeq \operatorname{Hom}_{B}(\Omega^{1}_{A/k} \otimes_{k} \Lambda,
    A \otimes_{k} \Lambda)
  \simeq \operatorname{Hom}_{A}(\Omega^{1}_{A/k}, A \otimes_{k} \Lambda)
  \simeq \text{k-dérivations de } A \text{ dans } A \otimes_{k} \Lambda
\]

\[
  \mathcal{T}_{A/k} \otimes_{k} I(\Lambda)
  \simeq \pi_{0}\bigl(\uncertain{\mathcal{T} \otimes_{k} U}\bigr),
\]

\[
  \Lambda_{n} = \Lambda / I^{n+1}, \qquad
  T_{n} = \operatorname{Spec}(\Lambda_{n}) \quad \text{donc}
\]

\[
  T_{0} \simeq S \subset T_{1} \subset T_{2} \cdots \subset T_{n}
  \subset \cdots T_{N} = T
\]

\[
  \xi = X_{0} \struck{\times_{S}} T = X_{0} \otimes_{\underline{\mathcal{O}}_S} \Lambda
  \qquad\Big/\quad \text{Donc } \xi_{n} = X_{0} \times_{S} T_{n},
  \ \text{en particulier } \xi_{0} = X_{0}
\]

\[
  |X| = |\xi| = |X_{0}| = |\xi_{0}|
\]

\[
  \underline{\mathrm{Aut}}^{X_0}_{T}(X) \ \ill{} \ =
  \underline{\mathrm{Aut}}^{X_0}_{\Lambda}(X)
\]

\[
  \overline{X} = \underline{\operatorname{Hom}}_{T/S}(T, X)
\]

\[
  \underline{\operatorname{Hom}}_{T/S}(T, X) \longrightarrow
  \underline{\operatorname{Hom}}_{T_0/S}(T_{0}, X \times_{T} T_{0})
  \simeq X_{0},
\]

\[
  \overline{X} \xrightarrow{\;\pi^{X}_{\Lambda}\;} X_{0}
\]

\[
  \struck{U}\ \overline{U} \simeq (\pi^{X}_{\Lambda})^{-1}(U)
\]

\[
  u \longmapsto u(s_{0})
\]

\[
  \underline{\mathrm{Is}}^{X_0}_{T}(\xi, X) \longrightarrow
  \underline{S}(\overline{X}/X_{0})
\]

\[
  \underline{S}(\overline{X}/X_{0}) \longrightarrow
  \underline{\mathrm{Is}}_{X_{0}}(\overline{\xi}, \overline{X})
\]

\[
  \boxed{\;\underline{\mathrm{Aut}}^{X_0}_{T}(\xi) =
  \underline{G}^{\Lambda}_{X_{0}}\;}
\]

\[
  \xi = X_{0} \times_{S} T = X_{0} \otimes_{\underline{\mathcal{O}}_{S}}
  \underline{\Lambda}
\]

\[
  X_{0} \times_{S} T = X_{0} \otimes_{\underline{\mathcal{O}}_{S}}
  \underline{\Lambda}
\]

\[
  P^{n}(Y\,/\,X\,/\,S)
\]

\[
  (pr_{2}\, i^{(n)}_{X})^{*}(Y/X) = \struck{P^{(n)}(Y/X/S)}\;
  \Delta^{(n)}_{X/S} \times_{X \times_{S} X} (X \times_{S} Y)
  = \struck{\ill{}}\; \pi^{(n)}(Y/X/S)
\]

\[
  X \times_{\Delta^{(n)}_{X/S}} \pi^{n}(Y/X/S) = Y
\]

\[
  d^{(n)}_{Y/X/S} : \pi^{n}(Y/X/S) \longrightarrow X \times_{S} Y
  \longrightarrow Y
\]

\[
  \boxed{\;(\delta^{n}_{X/S})_{*}\bigl(\pi^{n}(Y/X/S) /
  \Delta^{(n)}_{X/S}\bigr) = P^{(n)}(Y/X/S).\;}
\]

\[
  \underline{\Gamma}\bigl(\pi^{n}(Y/X/S)/\Delta^{(n)}_{X/S}\bigr) \simeq
  \underline{\Gamma}\bigl(P^{(n)}(Y/X/S)/X\bigr)
\]

\[
  \operatorname{Hom}_{X}\bigl(X', P^{(n)}(Y/X/S)\bigr) \simeq
  \operatorname{Hom}_{\Delta^{(n)}_{X/S}}\bigl(X' \times_{X}
  \Delta^{n}_{X/S},\ \pi^{(n)}(Y/X/S)\bigr)
\]

\[
  \Gamma_{\varphi} = \struck{\ill{}}\ \varphi \times_{S} \mathrm{id}_{X} :
  X' \longrightarrow X' \times_{S} X
\]

\[
  \Gamma_{\varphi}^{(n)} \longrightarrow X' \times_{S} X
\]

\[
  d^{n}(Y/X/S) : \underline{\Gamma}(Y/X) \longrightarrow
  \underline{\Gamma}\bigl(P^{(n)}(Y/X/S)/X\bigr)
\]

\[
  \varepsilon^{n}(Y/X/S) : \underline{\Gamma}\bigl(P^{(n)}(Y/X/S)/X\bigr)
  \longrightarrow \underline{\Gamma}(Y/X)
\]

\[
  \underline{\Gamma}(Y/X) \longrightarrow
  \underline{\Gamma}\bigl(P^{n}(Y/X/S)/X\bigr)
\]

\[
  \underline{\Gamma}(Y/X) \longrightarrow \struck{\operatorname{Hom}}\;
  \underline{\Gamma}\bigl(Y \times_{X} \Delta^{(n)}_{X/S} \,/\,
  \Delta^{(n)}_{X/S}\bigr)
\]

\[
  \operatorname{Hom}_{X}\bigl(\Delta^{n}_{X/S}, Y\bigr) =
  \underline{\Gamma}\bigl(Y \times_{X} \Delta^{(n)}_{X/S} \,/\,
  \Delta^{(n)}_{X/S}\bigr)
\]

\[
  \Delta^{(m)}_{X/S} \longrightarrow \Delta^{(n)}_{X/S}
\]

\[
  \pi^{(n)}(Y/X/S) = \pi^{n'}(Y/X/S) \times_{\Delta^{(n)}_{X/S}}
  \Delta^{(m)}_{X/S}
\]

\[
  P^{n'}(Y/X/S) \xrightarrow{\;\varphi_{nn'}\;} P^{n}(Y/X/S)
\]

\[
  \begin{cases}
    \varepsilon\, \varphi_{on} = \varepsilon^{n} \\
    \varepsilon^{(n)}(Y/X/S) \circ \varphi_{nn'} = \varepsilon^{(n')} \\
    \varphi_{nn'}\, d^{(n')} = d^{(n)}.
  \end{cases}
\]

\[
  m \longrightarrow 1 \otimes m
\]

\[
  \underline{\Gamma}\bigl(Y^{\{\Delta^{n}_{X/S}\}}/Y\bigr)
\]

\[
  \underline{\Gamma}\bigl(P^{n}(Y/X/S)/Y\bigr)
\]

\[
  P^{n}(Y/X) \longrightarrow f^{*}\bigl(P^{n}(X/S)\bigr)
\]

\[
  P^{n}(Y/S) \longleftarrow f^{*}P^{n}(X/S)
\]

\[
  \varphi\bigl(g\, d^{n}f \otimes m\bigr) = m \otimes f\, d^{n}g
\]

\[
  \varphi\bigl(g\, d^{n}(f\,\ill{})\bigr) = f m \otimes \ill{}
\]

\[
  \ill{} \otimes_{A} (A \otimes_{\Lambda} A)
\]

\[
  M \otimes_{\Lambda} A
\]

\[
  \ill{}(A \otimes_{\Lambda} M) \qquad
  \underline{A} \otimes_{\Lambda} S_{\Lambda}(M) \qquad
  \struck{A} \otimes_{\Lambda} S_{A}(M)
\]

\[
  \struck{\operatorname{Hom}_{B}(M, B)}
\]

\[
  \widetilde{w}(g) = u\, g_{S'}\, v^{-1}
\]

\[
  \underline{\Gamma}\bigl(X_{Y}^{\{Y_{S'}\}} / Y\bigr)
\]

\[
  \underline{\Gamma}(X_{S'}/Y_{S'})
\]

\[
  \underline{\Gamma}(X/Y) \longrightarrow
  \underbrace{\underline{\Gamma}(X_{S'}/Y_{S'})}_{\text{ensemble :
  groupe d'opérateurs}} \simeq
  \underline{\Gamma}\bigl(X_{Y}^{\{Y_{S'}\}} / Y\bigr)
\]

\[
  \operatorname{Aut}^{\{\Lambda\}}_{S}(X'_{S}, Y_{S})
\]

\[
  w = (u, v), \qquad w' = (u', v')
\]

\[
  h^{\{Y_{S'}\}}\bigl(\widetilde{w}(g)\bigr) = \widetilde{w}'(h \circ g)
\]

\[
  w = (u, v), \qquad w' = (u', v')
\]

\[
  \bigl(X \times_{Y} X'\bigr)_{S'} = X_{S'} \times_{Y_{S'}} X'_{S'}
\]

\[
  \widetilde{w}''\bigl(g \times_{Y} g'\bigr) = \widetilde{w}(g) \times_{Y}
  \widetilde{w}'(g')
\]

\[
  \bigl(X \times_{Y} X'\bigr)_{Y}^{\{Y_{S'}\}} \simeq
  X_{Y}^{\{Y_{S'}\}} \times_{Y} X'^{\,\{Y_{S'}\}}_{Y}
\]

\[
  F = a X^{p} + b Y^{p} - \struck{1}
\]

\[
  k[X, Y]/(F) = A \;\simeq\; k[Z]/(Z^{p} - c)\,[X]
\]

\[
  k[Z]\,[T]/(T^{p})
\]

\[
  F = Y^{2} - X^{p} + a \qquad (p \neq 2)
\]

\[
  k[X, Y]/(F) = A
\]

\[
  F = Y^{2} - (X - \lambda)^{p} = 0 \qquad (\lambda = a^{1/p}).
\]

\[
  Y^{2} - Z^{p} \longrightarrow 0
\]

\[
  \Omega^{1}_{k}(A) \simeq A^{2} \Big/ A\Bigl(\frac{\partial F}{\partial X},
  \frac{\partial F}{\partial Y}\Bigr) \simeq A \;\struck{\ill{}}\;
  \simeq A + A/2Y
\]

\[
  \bigl(Y,\ Y^{2} - X^{p} + a\bigr) = \bigl(Y,\ X^{p} - a\bigr) \;;\qquad
  k[X, Y]/\bigl(Y,\ X^{p} - a\bigr) \simeq k[X]/(X^{p} - a)
\]

\[
  \mathfrak{m}'' = \bigl(Y^{2},\ Y(X^{p} - a),\ (X^{p} - a)^{2},\
  Y^{2} - X^{p} + a\bigr)
\]

\[
  = \bigl((X^{p} - a),\ Y^{2},\ Y(X^{p} - a)\bigr)
\]

\[
  \struck{d\,\delta f = \delta f \;?}
\]

\[
  \delta f = 0 \iff f \in A[B^{p}]
\]

\[
  \delta^{(\infty)} f = 0 \iff f \in A[B^{p^{k}}]
\]

\[
  \Omega^{1}_{B/A} = 0
\]

\[
  \Omega^{1}_{B/A} = 0
\]

\[
  \Omega^{1}(B/A)
\]

\[
  F : \mathcal{C} \longrightarrow \Phi^{\mathcal{C}}
\]

\[
  F_{\xi} \quad \ill{} \quad \ill{}
\]

\[
  \struck{\ill{} = \ill{}}\qquad \struck{X = \ill{}}
\]

\[
  S^{d,n} = \operatorname{Spec}\ \ill{}\, \bigl(A[t_{1}, \ldots, t_{n}] /
  (t_{i} - \ill{})\bigr) \ill{}
\]

\[
  G^{\mathrm{dif}}_{\ill{}}(\ill{})
\]

\[
  E^{n}_{S^{d,n}} \qquad E^{n} \ill{}
\]

\[
  E^{n}_{S^{d,n}} \times_{S^{d,n}} S.
\]

\[
  \bigl\{ E^{n}_{S^{d,n}} \bigr\} = \bigl\{ E^{n} \times_{S} S^{d,n}
  \bigr\}, \quad \ill{}
\]

\[
  (21.2.9.3) \qquad \operatorname{div}(s_D) = D .
\]

\[
  \dim\operatorname{proj}(M) = \operatorname{prof} A - \operatorname{prof} M .
\]

\[
  R\operatorname{Hom}(K, L) = R\operatorname{Hom}(K, A) \overset{L}{\otimes} L
\]

\[
  E_2^{pq} = \operatorname{Tor}_{-p}\big(\struck{\ill{}}\operatorname{Ext}^q(k, A), M\big)
  \Rightarrow \operatorname{Ext}^{p+q}(k, M) . \qquad \text{(dessin ci-contre)}
\]

\[
  \operatorname{Ext}^n(k, M) = 0 \text{ si } n < a - d \text{ et}
\]

\[
  \operatorname{Ext}^{a-d}(k, M) = \operatorname{Tor}_d\big(\operatorname{Ext}^a(k, A), M\big),
  \text{ non nul}
\]

\[
  \varprojlim \operatorname{Tor}_p^A(N / \mathfrak{m}^k N, M)
  = \operatorname{Tor}_p^A(N, M)^{\wedge}
\]

\[
  \underset{\longleftarrow}{\text{“lim”}}\;
  \operatorname{Tor}_p^A(N, M) \big/ \mathfrak{m}^k \operatorname{Tor}_p^A(N, M)
  \;\xrightarrow{\ \sim\ }\;
  \underset{\longleftarrow}{\text{“lim”}}\;
  \operatorname{Tor}_p^A(N / \mathfrak{m}^k N, M)
\]

\[
  \mathfrak{m}' / (\mathfrak{m}'^2 + \mathfrak{m} A')
  \;\xrightarrow{\ \sim\ }\;
  \mathfrak{n} / (\mathfrak{n}^2 + \mathfrak{m} B) .
\]

\[
      (x_n^\alpha)^p = P_n^\alpha(x_m^\beta,\ m < n) + q_n^\alpha
      \quad \text{avec } q_n^\alpha \text{ dans l'idéal maximal}
      \ (P \text{ à coeff ds } A)
    \]

\[
      (x_n^\alpha)^p = P_n^\alpha(x_m^\beta,\ m < n) + Q_{n,1}^\alpha(x_m^\beta) + r_n^\alpha
      \quad \text{avec } P \text{ (resp } Q) \text{ à coefficients dans } A \text{ (resp } \mathfrak{m})
    \]

\[
      \boxed{\,(x_n^\alpha)^p = P_n^\alpha(x_m^\beta,\ m < n) + \sum_{i=1}^{\infty} Q_{n,i}^\alpha(x_m^\beta)\,}
    \]

\begin{equation}
      (x_n^\alpha)^p = P_n^\alpha(x_m^\beta,\ m < n) + \sum_{i=1}^{k-1} Q_{n,i}^\alpha(x_m^\beta)
    \end{equation}

\begin{gather*}
  f_1 \dots f_n \text{ régulier} \Longleftrightarrow f_1 \dots f_n \text{ quasi-régulier} \Longleftrightarrow \\
  A/I[T_1 \dots T_n] \otimes_{A/I} M/I \to \operatorname{gr}_I(A) \otimes_{\operatorname{gr}^0_I(A)} \operatorname{gr}^0_I(M) \to \operatorname{gr}_I(M) \text{ bijectif} \\
  \Longleftrightarrow \operatorname{gr}_I(A) \otimes_{A/I} M/I \to \operatorname{gr}_I(M) \text{ bijectif et} \\
  A/I[T_1 \dots T_n] \otimes A/\operatorname{Ann}(M/I) \to \operatorname{gr}_I(A) \otimes A/\operatorname{Ann}(M/I) \text{ bijectif.} \\
  \Longleftrightarrow \operatorname{gr}_I(A) \otimes M/I \xrightarrow{\sim} \operatorname{gr}_I(M),
  \text{ et modulo } \operatorname{Ann}(M/I),\ I/I^2 \text{ libre de rg } n \text{ et}
  \operatorname{Sym}_{A/I}(I/I^2) \simeq \operatorname{gr}_I(A).
\end{gather*}

\[
  \text{« } \varprojlim \text{ »}\ \operatorname{Tor}^A_1(M, k) \big/ \mathfrak{n}^k \operatorname{Tor}^A_1(\dots)
\]

\[
  G(Y) = \operatorname{Ker}\big(F'(Y') \rightrightarrows F''(Y'')\big).
\]

\[
  A_{0\,T_0'} \to B_{0\,T_0'}
\]

\[
  i_*(\Lambda_V) = \Lambda_C, \qquad
  R^1 i_*(\Lambda_V) = \uncertain{\mathcal{H}^{-1}_{Z,C}}
\]

\[
  \uncertain{u^*}\big(R^i p_*(\bar F)\big) \to R^i p_{Y*}\big(u'^*(\bar F)\big)
\]

\[
  \operatorname{Codim}(Y_i, V(x)) = \dim \mathcal{O}_{V(x), y_i}
  \geq \dim \mathcal{O}_{X, y_i} - 1 = \operatorname{Codim}(Y_i, X) - 1
  \geq \operatorname{Codim}(Y, X) - 1 ,
\]

\[
  \begin{cases}
    \operatorname{Codim}(Y_i, V(x)) = \struck{\dim} \operatorname{Codim}(Y_i, X) - 1 \\
    \operatorname{Codim}(Y_i, X) = \operatorname{Codim}(Y, X)
  \end{cases}
\]

\[
  \operatorname{Hom}_1(X, Y) \xrightarrow{\ \sim\ } \operatorname{Hom}_0(X, Y(1)) .
\]

\[
  \alpha_X : X(1) \xrightarrow{\ 1\ } X
\]

\[
  \beta_X : X(-1) \xrightarrow{\ -1\ } X .
\]

\[
  \text{I}
  \begin{cases}
    \operatorname{Hom}_n(X, Y) \xrightarrow{\ \sim\ } \operatorname{Hom}_{n+1}(X(1), Y) \\
    \operatorname{Hom}_n(Y, X(1)) \xrightarrow{\ \sim\ } \operatorname{Hom}_{n+1}(Y, X)
  \end{cases}
  \qquad
  \text{II}
  \begin{cases}
    \operatorname{Hom}_n(X, Y) \xrightarrow{\ \sim\ } \operatorname{Hom}_{n-1}(X(-1), Y) \\
    \operatorname{Hom}_n(Y, X(-1)) \xrightarrow{\ \sim\ } \operatorname{Hom}_{n-1}(Y, X) \struck{\ill{}}
  \end{cases}
\]

\[
  \operatorname{prof}_{A_{\mathfrak{q}}} M_{\mathfrak{q}}
  = \operatorname{Inf}_{\mathfrak{p} \to \mathfrak{q}}
    \operatorname{prof}_{B_{\mathfrak{p}}} M_{\mathfrak{p}} ,
  \qquad
  \dim_{A_{\mathfrak{q}}} M_{\mathfrak{q}}
  = \operatorname{Sup}_{\mathfrak{p} \to \mathfrak{q}}
    \dim_{B_{\mathfrak{p}}} M_{\mathfrak{p}} ,
\]

\[
  \operatorname{prof}_{B_{\mathfrak{p}}} M_{\mathfrak{p}}
  \geq \operatorname{prof}_{A_{\mathfrak{q}}} M_{\mathfrak{q}}
  \geq \operatorname{Inf}(k, \dim_{A_{\mathfrak{q}}} M_{\mathfrak{q}})
  \geq \operatorname{Inf}(k, \dim_{B_{\mathfrak{p}}} M_{\mathfrak{p}}) ,
  \quad \text{cqfd.}
\]

\[
  \operatorname{prof} M_{\mathfrak{p}} \geq \operatorname{Inf}(k, \dim M_{\mathfrak{p}})
  \iff
  \operatorname{prof} \hat{M}_{\mathfrak{p}} \geq \operatorname{Inf}(k, \dim \hat{M}_{\mathfrak{p}}) ,
  \qquad
  \operatorname{prof} \hat{M}_{\mathfrak{q}_i} \geq \operatorname{Inf}(k, \dim \hat{M}_{\mathfrak{q}_i}) .
\]

\[
  u' : D \to X' , \qquad u'' : D \to X'' , \qquad u'(D) \not\supset Z'_0 ,
\]

\[
  X = X' \coprod_D X'' .
\]

\[
  V(f) \simeq V(f') \coprod_{V(f_0)} V(f'') .
\]

\[
  (V(f') \cap Z'_0 =)\ V(f') \cap Z'_0 = \{x\} , \qquad
  (V(f'') \cap Z''_0 =)\ V(f'') \cap Z''_0 \subset \{x\} .
\]

\[
  Z = \struck{\ill{}} Z'_0 \cup Z''_0 \cup D ,
\]

\[
  A \xrightarrow{\ \psi\ } B \xrightarrow{\ \varphi\ } C
\]

\[
  v^n : \sum_{i + j = n} A^i \otimes_K C^j \longrightarrow B^n
\]

\[
  P_{A, x}(t) = \sum_i \operatorname{rg}_{k(x)}(A^i(x))\, t^i \in \mathbb{Z}[[t]]
\]

\[
  (\ast) \qquad P_{B, x} \ll P_{A, x} P_{C, x}
\]

\[
  \begin{cases}
    B \simeq A[t_1, \dots, t_r] \\
    \psi \simeq \text{hom. can. } A \to A[t_1, \dots, t_r] \\
    \varphi \simeq \text{hom. \uncertain{projection} \uncertain{canonique} }
      A[t_1, \dots, t_r] \to \underbrace{A / A^+}_{K}[t_1, \dots, t_r]
  \end{cases}
\]