The Stacks project

Exercise 111.48.5. Let $X$ be a ringed space. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Construct a trace map

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

for all $i$. Generalize to a trace map

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

for any $\mathcal{O}_ X$-module $\mathcal{F}$.

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View Exercise 111.48.5 as pdf