\[
\begin{array}{ccc}
X & \longleftarrow & X' \\
{\scriptstyle f}\downarrow & \text{cart.} & \downarrow{\scriptstyle f'} \\
Y & \xleftarrow[\text{plat}]{} & Y'
\end{array}
\qquad\qquad f_*(F)' \xrightarrow{\ \sim\ } f'_*(F')
\]

\[
\begin{array}{ccc}
Z' & \longrightarrow & Z \\
{\scriptstyle g'}\swarrow & & \swarrow{\scriptstyle g} \text{ quasi-compact} \\
S' & \longrightarrow & S
\end{array}
\qquad g(Z) = T
\]

\[
f^{-1}(\overline{T}) = \overline{f^{-1}(T)} .
\]

\[
\begin{array}{ccc} C' & \longleftarrow & C \\ \uparrow & & \uparrow{\scriptstyle u} \\ A' & \longleftarrow & A \end{array}
\]

\[
\text{Alors } \overline{T} = V(\mathfrak{J}) \ (\mathfrak{J} = \operatorname{Ker} u), \quad f^{-1}(\overline{T}) = V(\mathfrak{J}A'), \quad \overline{f^{-1}(T)} = V(K), \ K = \operatorname{Ker} u'
\]

\[
\begin{array}{l}
F(S) \to F(S') \rightrightarrows F(S'') \\
O(S) \to O(S') \rightrightarrows O(S'') \\
LF_{rc}(S) \to LF_{rc}(S') \rightrightarrows LF_{rc}(S'') \\
{}[LF(S) \to LF \rightrightarrows LF(S'')\ ?]
\end{array}
\quad
\left(\begin{array}{l}
\text{où } F = \text{parties fermées} \\
O = \text{parties ouvertes} \\
LF \text{ parties loc. fermées \uncertain{rétrocompactes}} \\
S'' = S' \times_S S'
\end{array}\right) \text{ est \emph{exact}}
\]

\[
\begin{array}{ccc}
X' & & X \\
\downarrow{\scriptstyle g'} & & \downarrow{\scriptstyle g} \\
Y' & \longrightarrow & Y \\
& \searrow \quad \swarrow & \\
S' & \longrightarrow & S
\end{array}
\]

\[
\begin{array}{ccc}
X' & \longrightarrow & X \\
{\scriptstyle g'}\downarrow & & \downarrow{\scriptstyle g} \\
Y' & \xrightarrow[\ f_Y\ ]{} & Y \\
& & \\
S' & \xrightarrow[\ f\ ]{} & S
\end{array}
\]

\[
\mathrm{Ssf}(F) \to \mathrm{Ssf}(F') \rightrightarrows \mathrm{Ssf}(F'')
\]

\[
\begin{array}{l}
\struck{\mathrm{Ssf}(S)} \\
\mathrm{Ssf}(S) \to \mathrm{Ssf}(S') \rightrightarrows \mathrm{Ssf}(S'') \\
\mathrm{Sso}(S) \to \mathrm{Sso}(S') \rightrightarrows \mathrm{Sso}(S'') \\
\mathrm{Ssrc}(S) \to \mathrm{Ssrc}(S') \rightrightarrows \mathrm{Ssrc}(S'')
\end{array}
\]

\[
\operatorname{Hom}_S(X, Y) \to \operatorname{Hom}_{S'}(X', Y') \rightrightarrows \operatorname{Hom}_{S''}(X'', Y'')
\]

\[
\begin{array}{ccccc}
X \times_S Y = Z & & Z' & & Z'' \\
\uparrow\downarrow & & \downarrow & & \downarrow \\
Y & & Y' & & Y''
\end{array}
\qquad S \to S' \rightrightarrows S''
\]

\[
\begin{array}{ccc}
X & & X' \\
\downarrow & & \downarrow \\
Y & \longleftarrow & Y' \\
\downarrow & & \\
S & \longleftarrow & S'
\end{array}
\qquad f \text{ fid. plat qu.-cpt}
\]

\[
\begin{array}{ccccccc}
\text{imm. fermée} & \Longrightarrow & \text{fini} & \Longrightarrow & \text{entier} & \Longrightarrow & \text{affine} \\
\Downarrow & & \Downarrow & & \Downarrow & & \Downarrow \\
\text{imm. qu.-cpte} & & \text{proj} & \Longrightarrow \text{ propre } \Longrightarrow & \text{univ. fermé} & & \\
\Downarrow & & \Downarrow & & & & \\
\text{quasi-affine de t.f.} & \Longrightarrow & \text{qu.-proj} & & & & \text{séparé}
\end{array}
\]

\[
X \xrightarrow{\ f\ } Y \xrightarrow{\ g\ } Z, \qquad L \text{ sur } X,\ M \text{ sur } Y, \quad g \text{ qu.-cpt.}
\]

\[
\left.\begin{array}{l} L \text{ ample rel}/Y \\ M \text{ ample rel}/Z \end{array}\right\}
\Longrightarrow \exists\, n_0 \text{ tel que } n \geqslant n_0 \text{ implique } L \otimes f^{*}(M^{\otimes n}) \text{ ample rel}/Z .
\]

\[
\begin{array}{ccc} X & \longleftarrow & X' \\ \downarrow & \text{cart.} & \downarrow \\ Y & \longleftarrow & Y' \end{array}
\qquad\qquad X \xrightarrow{\ f\ } Y \xrightarrow{\ g\ } Z
\]

\[
\begin{array}{lcl}
\operatorname{diag} g \in \mathcal{M} & \Longleftrightarrow & \forall S' \to S, \text{ les sections de } Y' \text{ sur } S' \text{ sont } \in \mathcal{M} \\
g : Y \to S & \Longleftrightarrow & \forall (u, v) : X \rightrightarrows Y \text{ sur } S,\ \operatorname{Ker}(u, v) \to X \text{ est } \in \mathcal{M} \\
 & \Longleftrightarrow & \forall f : X \to Y,\ \Gamma_f : X \to X \times_S Y \text{ est } \in \mathcal{M} \\
 & \Longleftrightarrow & gf \in \mathcal{M} \Rightarrow f \in \mathcal{M}
\end{array}
\]

\[
\begin{array}{ccc} X & \longleftarrow & X' \\ {\scriptstyle f}\downarrow & & \downarrow{\scriptstyle f'} \\ Y & \longleftarrow & Y' \end{array}
\qquad
\begin{array}{c} X \xrightarrow{\ f\ } Y \\ \searrow \quad \swarrow{\scriptstyle g} \\ S \end{array}
\]

\[
P(M) \simeq V(M)^* / \mathbb{G}_m \qquad (\text{isom. de faisceaux Zar.})
\]

\[
\operatorname{Hom}_k(\mathcal{X}, \underset{\substack{\wr\wr \\ \operatorname{Spec} A}}{Y}) \xrightarrow{\ \simeq\ } \operatorname{Hom}_{k\text{-alg}}(\underset{\substack{\wr\wr \\ \Gamma(Y, \mathcal{O}_Y)}}{A}, \Gamma(\mathcal{X}, \mathcal{O}_{\mathcal{X}}))
\]

\[
A_f \longrightarrow \Gamma(U_f, \mathcal{O}_{\mathcal{X}})
\]

\[
\Gamma(U_f, \mathcal{O}_Y) \simeq A_f \longrightarrow \Gamma(U_f, \mathcal{O}_{\mathcal{X}})
\]

\[
\begin{array}{ccccc}
\underset{\substack{\wr\wr \\ (\mathrm{Alg}_k)^{\circ}}}{\overset{\substack{(\mathrm{Sch}\text{-}\mathrm{Aff})^{\circ} \\ \wr\wr}}{(\mathrm{Aff}_k)}} & \xhookrightarrow{\ \mathrm{incl}\ } & (\mathrm{Sch})_k & \hookleftarrow & (\mathrm{Locan})_k
\end{array}
\]

\[
\begin{array}{lcl}
\widetilde{(\mathrm{Aff}_k)} \simeq \underline{\operatorname{Hom}}^{\sim}(\mathrm{Alg}_k, \mathrm{Ens}) & & \\
\widetilde{(\mathrm{Sch})_k} \xrightarrow{\ \mathrm{restr.}\ } \widetilde{\mathrm{Aff}_k} \simeq \underline{\operatorname{Hom}}^{\sim}(\mathrm{Alg}_k, \mathrm{Ens}) & & \text{(faisceaux pour la top. « de Zariski »)} \\
\qquad \downarrow \text{inc.} \qquad\qquad \downarrow \text{inc.} & & \\
\widehat{(\mathrm{Sch})_k} \xrightarrow{\ \mathrm{restr.}\ } \widehat{\mathrm{Aff}_k} \simeq \underline{\operatorname{Hom}}(\mathrm{Alg}_k, \mathrm{Ens}) & & \text{(pas plein\uncertain{fid.})} \\
\widehat{(\mathrm{Aff}_k)} \simeq \underline{\operatorname{Hom}}(\mathrm{Alg}_k, \mathrm{Ens}) & &
\end{array}
\]

\[
\begin{array}{ccc}
(\mathrm{Sch})_k & \simeq & \mathrm{Sch}_{/\operatorname{Spec} k = S} \\
\cap & & \cap \\
\widehat{C}_{/S} & \approx & \widehat{C}_{/S}
\end{array}
\qquad C = \mathrm{Aff}_k
\]

\[
\pi(C, \widetilde{D}) = \pi(C', \widetilde{D}) \cap \pi(C, \widehat{D})
\qquad\qquad
\begin{array}{ccc}
& \pi(C, \widetilde{D}) & \\
\swarrow & & \searrow \\
\pi(C', \widetilde{D}) & & \pi(C, \widehat{D}) \\
\searrow & & \swarrow \\
& \pi(C', \widehat{D}) &
\end{array}
\]

\[
C \mathbin{\underset{\mathrm{Top}}{\times}} \widetilde{D} = \Bigl(C' \mathbin{\underset{\mathrm{Top}}{\times}} \widetilde{D}\Bigr) \cap \Bigl(C \mathbin{\underset{\mathrm{Top}}{\times}} \widehat{D}\Bigr)
\]

\[
(2) \qquad \widehat{C \times D} \overset{?}{\approx} \widehat{C} \times \widehat{D}
\]

\[
\underline{\operatorname{Homtop}}(E, \widehat{A}) \simeq \bigl(\underline{\operatorname{Hom}}^{*}_{\mathrm{gex}}(A, E)\bigr)^{\circ}, \qquad u \longmapsto u^{*}
\]

\[
i : C \to C' \qquad j : E \hookrightarrow \widehat{D} \quad (\text{i.e. } E \approx \widetilde{D} \text{ pour une top. conv. sur } D)
\]

\[
\pi(C, \widetilde{D}) \xrightarrow{\ \alpha_0\ } \underset{\substack{\wr\wr \\ \widehat{C \times D}}}{\pi(C, \widehat{D})} \xrightarrow{\ \alpha_1\ } \underset{\substack{\wr\wr \\ \widehat{C' \times D}}}{\pi(C', \widehat{D})}
\]

\[
\alpha_0 : \pi(C, \widetilde{D}) \longrightarrow \pi(C, \widehat{D}) \simeq \widehat{C \times D}
\]

\[
\begin{cases}
p_*(F) \in \widehat{C} & (\text{identifié à la sous-catégorie str. pleine image essentielle de } i_* : \widehat{C} \to \widehat{C'}) \\
q_*(F) \in \widetilde{D}
\end{cases}
\]

\[
X = \coprod_{i \in I} X_i \longleftarrow X_i
\]

\[
\begin{array}{ccc}
X' & \longrightarrow & X \\
\uparrow & & \uparrow \\
X'_i & \longrightarrow & X_i
\end{array}
\]

\[
\Bigl(\coprod_i X_i\Bigr) \times \Bigl(\coprod_j Y_j\Bigr) \simeq \coprod_{i,j} (X_i \times Y_j)
\]

\[
\Bigl[\; X \times \underbrace{\coprod Y_i}_{Y} \longleftarrow \coprod_i X \times Y_i \quad \ldots \;\Bigr]
\qquad
\begin{array}{ccccc}
X \times Y_i & \longrightarrow & X \times Y & & X \\
\downarrow & & \downarrow & & | \\
Y_i & \longrightarrow & Y & \text{---} & e
\end{array}
\]

\[
(\mathrm{Ens\ finis}) \longrightarrow \mathrm{Aff}_k, \qquad \Gamma \longmapsto \Gamma_k = \coprod_{\Gamma} e_k
\]

\[
\Gamma_k = V_{k^{\Gamma}} \qquad k \to k^{\Gamma} \ \text{contravariant en } \Gamma \ (\text{algèbre produit} \simeq \operatorname{Fonc}(\Gamma, k))
\]

\[
k[I] \to k[I \times I] \simeq k[I] \otimes k[I]
\]

\[
\operatorname{Spec} I_k = \operatorname{Spec} k^{I} = \coprod_{I} \underbrace{\operatorname{Spec}(k)}_{S} \qquad \text{\uncertain{envoyé} dans $S$ \uncertain{à} \uncertain{façon} \uncertain{évidente}.}
\]

\[
\begin{array}{ccc}
\mathfrak{X} & \dashrightarrow & \mathfrak{X}' \\
\downarrow & & \vdots \\
\mathfrak{Y} & \longleftrightarrow & \mathfrak{Y}'
\end{array}
\qquad\qquad
\begin{array}{ccc}
X & \longrightarrow & X' \\
\downarrow & & \downarrow \\
Y & \longrightarrow & Y'
\end{array}
\]

\[
\begin{array}{ccc}
A & \xrightarrow{\ u\ } & B \\
Y & \xleftarrow[\ \varphi\ ]{} & X
\end{array}
\qquad \text{formule pour } \overline{\varphi(Z)}, \ Z = V(J)
\]

\[
\operatorname{Ker} \varphi \subset \operatorname{Nil} A
\]

\[
\begin{cases}
\varphi(x_{\mathfrak{p}}) = y_{\varphi^{-1}(\mathfrak{p})} \\
\varphi^{-1}(V(J)) = V(u(J)B) \\
\varphi^{-1}(Y_f) = X_{u(f)}
\end{cases}
\]

\[
\lvert \operatorname{Spec} \mathfrak{X} \times_S \mathfrak{Y} \rvert \longrightarrow \lvert \operatorname{Spec}(\mathfrak{X}) \rvert \times_{\lvert \operatorname{Spec} S \rvert} \lvert \operatorname{Spec} \mathfrak{Y} \rvert
\]

\[
P_I = k[(T_i)_{i \in I}] \qquad\qquad I, J \text{ finis ou infinis}
\]

\[
k^I \simeq \operatorname{Hom}_{k\text{-alg}}(P_I, k) \qquad \struck{\text{l'esp. affine sur } k \text{ de dim } I \ \ill{}}
\]

\[
f \in P_I, \quad x = (x_i)_{i \in I} \in k^I \qquad f(x) = \varepsilon_x(f)
\]

\[
(1) \qquad F_j(t) = 0 \quad j \in J \qquad\qquad V((F_j))
\]

\[
\mathbb{E}^I_k : k' \longmapsto k'^I \qquad \mathbb{E}^I_k(k') = k'^I
\]

\[
V_S(k') \subset \mathbb{E}^I_k(k') \qquad\qquad V_S \subset \mathbb{E}^I_k
\begin{cases}
\text{étude indépendante du plongement} \\
\text{étude du plongement}
\end{cases}
\]

\[
A_J = P_I / J
\]

\[
\begin{array}{ccc}
k'^I & \simeq & \operatorname{Hom}_{k\text{-alg}}(P_I, k') \\
\cup & & \cup \\
V_J(k') & \simeq & \{ u : P_I \to k' \mid u(J) = 0 \ \struck{\ill{}} \ (\text{i.e. s'annule sur les } F_j) \} \\
& & \wr\! \mid \\
& & \operatorname{Hom}_k(A_J, k')
\end{array}
\]

\[
V_J \subset V_{J'} \Longleftrightarrow J' \subset J
\qquad \Longleftarrow \quad
\Bigl\{ J = \{ f \in P_I \mid f(x) = 0 \ \forall\, k'/k \text{ et } x \in V_J(k') \} \Bigr.
\]

\[
V'_{S} = V'_{J} = V'_{\widetilde{J}}
\]

\[
\widetilde{J} = \{ f \in P_I \mid f(x) = 0 \ \forall x \in V_J(k'),\ k'/k \text{ un corps} \}
\]

\[
\widetilde{J} \subset \widetilde{J}' \Longleftrightarrow V(J') \subset V(J)
\]

\[
A \simeq \operatorname{Hom}_{k\text{-alg}}(k[T], A) \simeq \operatorname{Hom}(V_A, \mathbb{E}^1_k)
\]

\[
/ \qquad A \text{ l'\emph{anneau des fonctions sur l'espace algébrique affine}} \ X.
\]

\[
V_A(k') \xrightarrow{\ f_{k'}\ } k' \quad \text{est induit par} \quad \mathbb{E}^I(k') = k'^I \xrightarrow{\ g_{k'}\ } \quad \text{défini par } g ]
\]

\[
\struck{\textstyle\prod} \ \operatorname{Hom}(X', I_{X'})^E \qquad\qquad \operatorname{Hom}(X', I_{X'}^E)
\]

\[
(P \amalg P)^E \qquad \struck{\ill{}}
\]

\[
A = \bigotimes_{i \in I} A_i
\]

\[
\operatorname{Ker}(u, v) \simeq X \times_{(Y \times Y)} \Delta_Y
\]

\[
\simeq B / \text{idéal engendré par les } u(x) - v(x)
\]

\[
G \longrightarrow G'
\]

\[
g \longmapsto g.x
\]

\[
X_i \longrightarrow X \qquad (i \in I)
\]

\[
\coprod_i \operatorname{Hom}_{k\text{-alg}}(A_i, k') \longrightarrow \operatorname{Hom}_{k\text{-alg}}\Bigl(A = \prod A_i,\ k'\Bigr)
\]

\[
\emptyset_k = V_0 \quad (\text{algèbre nulle sur } k)
\]

\[
\emptyset_k(k') =
\begin{cases}
\emptyset & \text{si } k' \neq 0 \quad \text{i.e. } V_{k'} \not\simeq \emptyset_k \\
\{e\} & \text{si } k' = 0
\end{cases}
\qquad\qquad \emptyset_k(k') = \operatorname{Hom}(V_{k'}, \emptyset_k)
\]

\[
A \longleftrightarrow \mathfrak{X} = V_A
\]

\[
V(S) = \text{ensemble des idéaux premiers de } A \text{ qui contiennent } S.
\]

\[
\struck{V(\textstyle\sum J)} \quad V\Bigl(\bigcup_{i \in I} S_i\Bigr) = V\Bigl(\sum J_i\Bigr) = \bigcap_i V(S_i) = \bigcap_i V(J_i) \qquad I \text{ quelc.}
\]

\[
V\Bigl(\bigcap_i J_i\Bigr) = V\Bigl(\prod J_i\Bigr) = \bigcup_i V(J_i) \qquad I \text{ fini}
\]

\[
\overset{\text{déf}}{\Longleftrightarrow}
\begin{cases}
X \neq \emptyset \\
X = X' \cup X'' \Rightarrow X' = X \text{ ou } X'' = X \quad (X', X'' \text{ fermés})
\end{cases}
\]

\[
\text{i.e.} \quad
\begin{cases}
A \neq 0 \\
\text{i.e. } J' . J'' = 0 \Rightarrow J' \text{ ou } J'' \text{ nul}
\end{cases}
\]

\[
\begin{aligned}
&\text{i.e. } V(f) \supset V(g) \\
&\text{i.e. } \widetilde{fA} \subset \widetilde{gA} \\
&\text{i.e. } f \in \widetilde{gA} \\
&\text{i.e. } \begin{cases} \exists n \in \mathbb{N}, \\ h \in A \end{cases} f^n = gh \qquad \Big| \text{ écrivons } f \prec g
\end{aligned}
\]

\[
\begin{aligned}
&X_f = \bigcup X_{f_i} \\
&\text{i.e. } V(f) = \bigcap V(f_i) \\
&\text{i.e. } \widetilde{fA} = \widetilde{\textstyle\sum f_i A} \\
&\text{i.e. } \begin{cases} \forall i \ f_i \prec f \\ \exists n \in \mathbb{N} \text{ et } (h_i)_{i \in I} \text{ avec } f^n = \sum f_i h_i \end{cases}
\end{aligned}
\]

\[
P(f) \longrightarrow \prod_i P(f_i) \rightrightarrows \prod_{ij} P(f_i f_j)
\]

\[
M_f = \varinjlim_n (M, n) \qquad \text{où} \qquad (M, n) \xrightarrow{\ f^{n'-n}\ } (M, n') \quad \text{si } n' \geqslant n, \qquad (M,n) = M
\]

\[
M \xrightarrow{\ i_f\ } M_f
\]

\[
M_f \longrightarrow \prod M_{f_i} \rightrightarrows \prod M_{f_i f_j} \qquad \text{exact si } f_i \prec f \text{ et } f^n = \sum f_i h_i
\]

\[
A_{\mathfrak{p}} = \underline{\mathcal{O}}_{X,x} \quad \text{anneau local}
\]