Cote n° 21 · pages 4–28
· 54 displayed formulas · [Chapitre] 0 III : tapuscrit et copies de tapuscrit annotés (s.d.), notes manuscrites (s.d.).
Inventory dating : [vers 1960-1967]
Édition de démonstration
\[\bigcup_i \mathrm{supp}\, M_i \;=\; Y ,\]
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\[
\bigcup_i \mathrm{supp}\, M_i \;=\; Y ,
\]\[A^{o} \longrightarrow \mathrm{End}(P) ,\]
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\[
A^{o} \longrightarrow \mathrm{End}(P) ,
\]\[P \otimes_A M ,\]
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\[ P \otimes_A M , \]
\[\mathrm{Hom}_{\underline{C}'}(P \otimes_A M,\, Q) \;=\; \mathrm{Hom}_A(M,\, \mathrm{Hom}_{\underline{C}'}(P,Q)) ,\]
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\[
\mathrm{Hom}_{\underline{C}'}(P \otimes_A M,\, Q) \;=\; \mathrm{Hom}_A(M,\, \mathrm{Hom}_{\underline{C}'}(P,Q)) ,
\]\[M'' \longrightarrow M' \longrightarrow M \longrightarrow 0\]
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\[ M'' \longrightarrow M' \longrightarrow M \longrightarrow 0 \]
\[P \otimes_A M'' \longrightarrow P \otimes_A M' \longrightarrow P \otimes_A M \longrightarrow 0\]
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\[ P \otimes_A M'' \longrightarrow P \otimes_A M' \longrightarrow P \otimes_A M \longrightarrow 0 \]
\[P \otimes_A A_g \;\simeq\; P .\]
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\[ P \otimes_A A_g \;\simeq\; P . \]
\[\mathrm{Hom}_A(M,P)\]
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\[
\mathrm{Hom}_A(M,P)
\]\[\mathrm{Hom}_{\underline{C}_1}(Q,\, \mathrm{Hom}_A(M,P)) \;=\; \mathrm{Hom}_A(M,\, \mathrm{Hom}_{\underline{C}_1}(Q,P)) .\]
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\[
\mathrm{Hom}_{\underline{C}_1}(Q,\, \mathrm{Hom}_A(M,P)) \;=\; \mathrm{Hom}_A(M,\, \mathrm{Hom}_{\underline{C}_1}(Q,P)) .
\]\[P \otimes_A M \longrightarrow \ill{}\]
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\[
P \otimes_A M \longrightarrow \ill{}
\]\[T : \underline{C}^{o} \longrightarrow \underline{C}' ,\]
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\[
T : \underline{C}^{o} \longrightarrow \underline{C}' ,
\]\[T(M) \longrightarrow \mathrm{Hom}_A(M, T(A)) ,\]
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\[
T(M) \longrightarrow \mathrm{Hom}_A(M, T(A)) ,
\]\[T : \underline{C} \longrightarrow \underline{C}'\]
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\[
T : \underline{C} \longrightarrow \underline{C}'
\]\[T_n : \underline{C}_n \longrightarrow \underline{C}' ,\]
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\[
T_n : \underline{C}_n \longrightarrow \underline{C}' ,
\]\[T_n(A_n) \otimes_{A_n} M \longrightarrow T_n(M) = T(M)\]
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\[
T_n(A_n) \otimes_{A_n} M \longrightarrow T_n(M) = T(M)
\]\[\struck{\ill{}} \qquad T(A_n) \otimes_A M \longrightarrow T_n(M) \quad \add{(n \geqslant n_o)} .\]
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\[
\struck{\ill{}} \qquad T(A_n) \otimes_A M \longrightarrow T_n(M) \quad \add{(n \geqslant n_o)} .
\]\[\varprojlim T(A_n) \otimes_A M \longrightarrow T(M)\]
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\[ \varprojlim T(A_n) \otimes_A M \longrightarrow T(M) \]
\[\left(\varprojlim T(A_n)\right) \otimes_A M \longrightarrow T(M) ,\]
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\[ \left(\varprojlim T(A_n)\right) \otimes_A M \longrightarrow T(M) , \]
\[T : \underline{C}^{o} \longrightarrow \underline{C}' ,\]
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\[
T : \underline{C}^{o} \longrightarrow \underline{C}' ,
\]\[(\%) \qquad T_n(M) \longrightarrow \mathrm{Hom}_A(M, T_n(A_n)) \qquad (n \geqslant n_o) ,\]
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\[
(\%) \qquad T_n(M) \longrightarrow \mathrm{Hom}_A(M, T_n(A_n)) \qquad (n \geqslant n_o) ,
\]\[T(M) \longrightarrow \varinjlim_n \mathrm{Hom}_A(M, T(A_n)) \longrightarrow \mathrm{Hom}_A\!\left(M, \varinjlim_n T(A_n)\right)\]
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\[
T(M) \longrightarrow \varinjlim_n \mathrm{Hom}_A(M, T(A_n)) \longrightarrow \mathrm{Hom}_A\!\left(M, \varinjlim_n T(A_n)\right)
\]\[T(M) \longrightarrow \mathrm{Hom}_A\!\left(M, \varinjlim T(A_n)\right)\]
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\[
T(M) \longrightarrow \mathrm{Hom}_A\!\left(M, \varinjlim T(A_n)\right)
\]\[H^i = \varinjlim_m T^i(A_m) \qquad (\text{où } A_m = A/I^{m+1}) .\]
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\[
H^i = \varinjlim_m T^i(A_m) \qquad (\text{où } A_m = A/I^{m+1}) .
\]\[T^{n}(M) = \mathrm{Hom}_A(M, H^{n})\]
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\[
T^{n}(M) = \mathrm{Hom}_A(M, H^{n})
\]\[\mathrm{Ass}\, T(M) = \mathrm{supp}\, M \cap S ,\]
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\[
\mathrm{Ass}\, T(M) = \mathrm{supp}\, M \cap S ,
\]\[T(M) \simeq \mathrm{Hom}(M,H)\]
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\[
T(M) \simeq \mathrm{Hom}(M,H)
\]\[(\%) \qquad \mathrm{Ass}\, T(M) = \mathrm{supp}\, M \cap \mathrm{Ass}\, H ,\]
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\[
(\%) \qquad \mathrm{Ass}\, T(M) = \mathrm{supp}\, M \cap \mathrm{Ass}\, H ,
\]\[\mathrm{Ass}\, H = S .\]
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\[
\mathrm{Ass}\, H = S .
\]\[u : N \longrightarrow H\]
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\[ u : N \longrightarrow H \]
\[0 \longrightarrow N/N \cap I^m M \longrightarrow M/I^m M\]
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\[ 0 \longrightarrow N/N \cap I^m M \longrightarrow M/I^m M \]
\[u' : N/N\, I^m M \longrightarrow H \; ;\]
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\[ u' : N/N\, I^m M \longrightarrow H \; ; \]
\[\mathrm{Hom}(M, H^{o}_{I}(K)) \xrightarrow{\;\sim\;} \mathrm{Hom}(M,K) ,\]
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\[
\mathrm{Hom}(M, H^{o}_{I}(K)) \xrightarrow{\;\sim\;} \mathrm{Hom}(M,K) ,
\]\[F \longrightarrow i_{*} i^{*}(F) = i_{*}(F|U)\]
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\[
F \longrightarrow i_{*} i^{*}(F) = i_{*}(F|U)
\]\[\Gamma(V,F) \longrightarrow \Gamma(V \cap U, F)\]
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\[ \Gamma(V,F) \longrightarrow \Gamma(V \cap U, F) \]
\[E_{(x)} \supset E'_{(x)} .\]
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\[
E_{(x)} \supset E'_{(x)} .
\]\[(*) \qquad \Gamma(V_{(x)}, F_{(x)}) \;\simeq\; \varinjlim_{W \text{ voisinage ouvert de } x} \Gamma(W \cap V, F)\]
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\[
(*) \qquad \Gamma(V_{(x)}, F_{(x)}) \;\simeq\; \varinjlim_{W \text{ voisinage ouvert de } x} \Gamma(W \cap V, F)
\]\[\Gamma(X,F) \longrightarrow \Gamma(U,F) \qquad (U = X - Y)\]
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\[ \Gamma(X,F) \longrightarrow \Gamma(U,F) \qquad (U = X - Y) \]
\[W_{ij} \cap V_{(x)} = W_{ij,x}\]
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\[
W_{ij} \cap V_{(x)} = W_{ij,x}
\]\[W'_{ij} = W_{ij} .\]
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\[
W'_{ij} = W_{ij} .
\]\[\Gamma(W,F) \longrightarrow \Gamma(W \cap V, F)\]
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\[ \Gamma(W,F) \longrightarrow \Gamma(W \cap V, F) \]
\[\varinjlim_W \Gamma(W,F) \longrightarrow \Gamma(W \cap V, F)\]
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\[ \varinjlim_W \Gamma(W,F) \longrightarrow \Gamma(W \cap V, F) \]
\[\Gamma(X_{(x)}, F_{(x)}) \longrightarrow \Gamma(X'_{(x)}, F_{(x)})\]
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\[
\Gamma(X_{(x)}, F_{(x)}) \longrightarrow \Gamma(X'_{(x)}, F_{(x)})
\]\[\Gamma\mathcal{F}(V) \longrightarrow \mathcal{F}(V \cap \mathcal{U})
\qquad (\mathcal{U} = X - Y)\]
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\[
\Gamma\mathcal{F}(V) \longrightarrow \mathcal{F}(V \cap \mathcal{U})
\qquad (\mathcal{U} = X - Y)
\]\[\mathrm{Pseudolim}_{W}\ \mathcal{F}(W) \longrightarrow
\mathrm{Pseudolim}_{W}\ \mathcal{F}(W - W \cap \bar{x})\]
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\[
\mathrm{Pseudolim}_{W}\ \mathcal{F}(W) \longrightarrow
\mathrm{Pseudolim}_{W}\ \mathcal{F}(W - W \cap \bar{x})
\]\[\mathcal{F}(X) \longrightarrow \mathcal{F}(X - \{x\})\]
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\[
\mathcal{F}(X) \longrightarrow \mathcal{F}(X - \{x\})
\]\[\mathcal{F}_{(x)}(W) \;=\; \mathrm{Pseudolim}_{W}\ \mathcal{F}(W)\]
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\[
\mathcal{F}_{(x)}(W) \;=\; \mathrm{Pseudolim}_{W}\ \mathcal{F}(W)
\]\[\underline{\mathrm{Hom}}(f,g)_{(x)} \;\Longrightarrow\;
\underline{\mathrm{Hom}}(f_{(x)}, g_{(x)})\]
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\[
\underline{\mathrm{Hom}}(f,g)_{(x)} \;\Longrightarrow\;
\underline{\mathrm{Hom}}(f_{(x)}, g_{(x)})
\]\[\text{(ii)} \Longrightarrow \text{(i)}\]
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\[
\text{(ii)} \Longrightarrow \text{(i)}
\]\[\mathcal{F}(W) \longrightarrow \mathcal{F}(W - W \cap \bar{x})\]
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\[
\mathcal{F}(W) \longrightarrow \mathcal{F}(W - W \cap \bar{x})
\]\[\mathcal{H}(\mathcal{F}(W)) \;\xrightarrow{\ \sim\ }\;
\mathcal{H}(\mathcal{F}(W - W \cap \bar{x}))\]
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\[
\mathcal{H}(\mathcal{F}(W)) \;\xrightarrow{\ \sim\ }\;
\mathcal{H}(\mathcal{F}(W - W \cap \bar{x}))
\]\[\mathcal{H}(\mathcal{F}_{(x)}(X_{x})) \;\xrightarrow{\ \sim\ }\;
\mathcal{H}(\mathcal{F}_{x}(X'_{\alpha_1}))\]
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\[
\mathcal{H}(\mathcal{F}_{(x)}(X_{x})) \;\xrightarrow{\ \sim\ }\;
\mathcal{H}(\mathcal{F}_{x}(X'_{\alpha_1}))
\]\[\underline{H}^{2}_{Y}(F) \;=\; \text{faisceau associé au préfaisceau}\]
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\[
\underline{H}^{2}_{Y}(F) \;=\; \text{faisceau associé au préfaisceau}
\]\[V \rightsquigarrow H^{\ill}(V \cap \mathcal{U}, F).\]
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\[
V \rightsquigarrow H^{\ill}(V \cap \mathcal{U}, F).
\]\[F \longrightarrow i_{*} i^{*} F .\]
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\[
F \longrightarrow i_{*} i^{*} F .
\]