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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