Cote n° 155 · pages 1–9
· 23 displayed formulas · [Rigidité de catégories tensorielles ?] : notes manuscrites (s.d.).
Inventory dating : [à partir de 1984]
Édition de démonstration
\[F : C \longrightarrow \operatorname{Mod}_{f}(k')\]
LaTeX source
\[
F : C \longrightarrow \operatorname{Mod}_{f}(k')
\]\[A \longrightarrow A \mathbin{\hat{\otimes}}_{k'} A\]
LaTeX source
\[
A \longrightarrow A \mathbin{\hat{\otimes}}_{k'} A
\]\[\widetilde{\mathcal{J}}/\mathcal{J} \subset A \mathbin{\hat{\otimes}}_{k} A
\qquad
\bigl(\ \mathcal{J} = J\cdot(A \mathbin{\hat{\otimes}}_{k} A),\ \text{idéal} \ \bigr)\]
LaTeX source
\[
\widetilde{\mathcal{J}}/\mathcal{J} \subset A \mathbin{\hat{\otimes}}_{k} A
\qquad
\bigl(\ \mathcal{J} = J\cdot(A \mathbin{\hat{\otimes}}_{k} A),\ \text{idéal} \ \bigr)
\]\[J = \operatorname{Ker}(k' \otimes_{k} k' \to k'),
\qquad
\widetilde{\mathcal{J}} = \{\, \lambda \in A \mathbin{\hat{\otimes}}_{k} A
\mid \lambda\mathcal{J} \subset \mathcal{J} \,\}\]
LaTeX source
\[
J = \operatorname{Ker}(k' \otimes_{k} k' \to k'),
\qquad
\widetilde{\mathcal{J}} = \{\, \lambda \in A \mathbin{\hat{\otimes}}_{k} A
\mid \lambda\mathcal{J} \subset \mathcal{J} \,\}
\]\[\check{U}^{(k')} = A \;\simeq\; k' \otimes_{k} \check{k}'^{(k)}
\;\simeq\; \operatorname{End}_{k}(k'),\]
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\[
\check{U}^{(k')} = A \;\simeq\; k' \otimes_{k} \check{k}'^{(k)}
\;\simeq\; \operatorname{End}_{k}(k'),
\]\[M' \longrightarrow M' \otimes_{k' \otimes_{k} k'} k'\]
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\[
M' \longrightarrow M' \otimes_{k' \otimes_{k} k'} k'
\]\[M \longmapsto \operatorname{Hom}_{A}(M, I)\]
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\[
M \longmapsto \operatorname{Hom}_{A}(M, I)
\]\[I \;=\; \struck{\ill{}}\, D(A) \;\simeq\; \check{A}_{g}^{(k')} \;=\; U\]
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\[
I \;=\; \struck{\ill{}}\, D(A) \;\simeq\; \check{A}_{g}^{(k')} \;=\; U
\]\[A \otimes A\]
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\[ A \otimes A \]
\[\operatorname{End}_{A}(U) \;\simeq\; \operatorname{End}_{A}(A_{d})^{\circ}
\;\simeq\; A^{\circ}\]
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\[
\operatorname{End}_{A}(U) \;\simeq\; \operatorname{End}_{A}(A_{d})^{\circ}
\;\simeq\; A^{\circ}
\]\[A \longrightarrow A^{\circ}\]
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\[
A \longrightarrow A^{\circ}
\]\[D(M) \;=\; \operatorname{Hom}_{A}(M, U_{g})
\;=\; \operatorname{Hom}_{A}\bigl(M_{g}\,;\, \operatorname{Hom}_{k'}(A_{g}, k')\bigr)\]
LaTeX source
\[
D(M) \;=\; \operatorname{Hom}_{A}(M, U_{g})
\;=\; \operatorname{Hom}_{A}\bigl(M_{g}\,;\, \operatorname{Hom}_{k'}(A_{g}, k')\bigr)
\]\[\varphi : A_{g} \times M_{g} \longrightarrow k'\]
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\[
\varphi : A_{g} \times M_{g} \longrightarrow k'
\]\[\begin{align*}
\varphi(\lambda u, x) &= \lambda\,\varphi(u,x), \\
\varphi(u \cdot v, x) &= \varphi(u, v.x),
\end{align*}\]
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\begin{align*}
\varphi(\lambda u, x) &= \lambda\,\varphi(u,x), \\
\varphi(u \cdot v, x) &= \varphi(u, v.x),
\end{align*}\[A_{d} \otimes_{A} M \longrightarrow k'\]
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\[
A_{d} \otimes_{A} M \longrightarrow k'
\]\[A_{d} \otimes_{A} M \;\simeq\; M \qquad \text{(comme du $k'$-module)}\]
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\[
A_{d} \otimes_{A} M \;\simeq\; M \qquad \text{(comme du $k'$-module)}
\]\[1 \otimes x \longleftarrow x,
\qquad
\lambda(1 \otimes x) = \lambda \otimes x = 1 \otimes \lambda x \longleftarrow \lambda x\]
LaTeX source
\[ 1 \otimes x \longleftarrow x, \qquad \lambda(1 \otimes x) = \lambda \otimes x = 1 \otimes \lambda x \longleftarrow \lambda x \]
\[F'(D(M)) \;=\; \check{M}^{(k')}\]
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\[
F'(D(M)) \;=\; \check{M}^{(k')}
\]\[\underbrace{F'(M) \otimes F'(D(M))}_{(M^{k'})^{\vee}}
\longrightarrow k' = F'(1_{C})\]
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\[
\underbrace{F'(M) \otimes F'(D(M))}_{(M^{k'})^{\vee}}
\longrightarrow k' = F'(1_{C})
\]\[M \otimes D(M) \longrightarrow 1_{C}\]
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\[
M \otimes D(M) \longrightarrow 1_{C}
\]\[M^{k'} \otimes_{k'} (M^{k'})^{\vee} \longrightarrow k'\]
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\[
M^{k'} \otimes_{k'} (M^{k'})^{\vee} \longrightarrow k'
\]\[M = A\]
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\[ M = A \]
\[A^{k'} \otimes_{k'} U \xrightarrow{\ \text{acc. can.}\ } k'\]
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\[
A^{k'} \otimes_{k'} U \xrightarrow{\ \text{acc. can.}\ } k'
\]