Exercise 111.48.6. Let $k$ be a field. Let $X = \mathbf{P}^ d_ k$. Set $\omega _{X/k} = \mathcal{O}_ X(-d - 1)$. Prove that for finite locally free modules $\mathcal{E}$, $\mathcal{F}$ the cup product on Ext combined with the trace map on Ext
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{E}, \mathcal{F} \otimes _{\mathcal{O}_ X} \omega _{X/k}) \times \mathop{\mathrm{Ext}}\nolimits ^{d - i}_ X(\mathcal{F}, \mathcal{E}) \to \mathop{\mathrm{Ext}}\nolimits _ X^ d(\mathcal{F}, \mathcal{F} \otimes _{\mathcal{O}_ X} \omega _{X/k}) \to H^ d(X, \omega _{X/k}) = k \]
produces a nondegenerate pairing. Hint: you can either reprove duality in this setting or you can reduce to cohomology of sheaves and apply the Serre duality theorem as proved in the lectures.
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