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The Stacks project

Exercise 111.48.4. Let $X$ be a ringed space. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module with dual $\mathcal{E}^\vee = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{O}_ X)$. Prove the following statements

  1. $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ i_{\mathcal{O}_ X}( \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{E}, \mathcal{G}) = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ i_{\mathcal{O}_ X}( \mathcal{F}, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \mathcal{G}) = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ i_{\mathcal{O}_ X}( \mathcal{F}, \mathcal{G}) \otimes _{\mathcal{O}_ X} \mathcal{E}^\vee $, and

  2. $\mathop{\mathrm{Ext}}\nolimits ^ i_ X( \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{E}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{F}, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \mathcal{G})$.

Here $\mathcal{F}$ and $\mathcal{G}$ are $\mathcal{O}_ X$-modules. Conclude that

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{E}, \mathcal{G}) = H^ i(X, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \mathcal{G}) \]

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View Exercise 111.48.4 as pdf