Démonstration. [01S7] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
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Démonstration.
Soit en effet une forme α ∈ 𝒜 c d − 1 , d ( X ) \alpha\in\mathscr{A}^{d-1,d}_{\text{c}}(X)
⟨ d ′ ( φ ∗ δ Y ) , α ⟩ = ⟨ φ ∗ δ Y , d ′ α ⟩ = ∫ Y φ ∗ ( d ′ α ) = ∫ Y d ′ ( φ ∗ α ) . \langle\mathop{\mathrm{d^{\prime}}}(\varphi_{*}\delta_{Y}),\alpha\rangle=\langle\varphi_{*}\delta_{Y},\mathop{\mathrm{d^{\prime}}}\alpha\rangle=\int_{Y}\varphi^{*}(\mathop{\mathrm{d^{\prime}}}\alpha)=\int_{Y}\mathop{\mathrm{d^{\prime}}}(\varphi^{*}\alpha).
Appliquant la formule de Stokes (théorème 3.12.1
⟨ d ′ ( φ ∗ δ Y ) , α ⟩ = − ∫ ∂ Y φ ∗ α = − ⟨ φ ∗ δ ∂ Y , α ⟩ . \langle\mathop{\mathrm{d^{\prime}}}(\varphi_{*}\delta_{Y}),\alpha\rangle=-\int_{\partial Y}\varphi^{*}\alpha=-\langle\varphi_{*}\delta_{\partial Y},\alpha\rangle.
Cela démontre la relation indiquée.
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