The Stacks project

Exercise 111.63.7. Let $k$ be an algebraically closed field. Let $X$ be a proper scheme of dimension $d$ over $k$ with dualizing module $\omega _ X$. You are given the following information:

1. $\text{Ext}^ i_ X(\mathcal{F}, \omega _ X) \times H^{d - i}(X, \mathcal{F}) \to H^ d(X, \omega _ X) \xrightarrow {t} k$ is nondegenerate for all $i$ and for all coherent $\mathcal{O}_ X$-modules $\mathcal{F}$, and

2. $\omega _ X$ is finite locally free of some rank $r$.

Show that $r = 1$. (Hint: see what happens if you take $\mathcal{F}$ a suitable module supported at a closed point.)