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
\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
\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
Comments (0)