Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 2 / a

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- Solution a

Solution

For an ordered open cover

U = (U_{i})

and a sheaf of sets

F

, the Čech cochain group is

\overset{ˇ}{C}^{p} (U, F) = \prod_{i_{0} < \dots < i_{p}} F (U_{i_{0}} \cap \dots \cap U_{i_{p}}) .

Its Čech coboundary is

(δc)_{i_{0} \dots i_{p + 1}} = \sum_{r = 0}^{p + 1} (- 1)^{r} c_{i_{0} \dots i_{r} \dots i_{p + 1}} ∣.

Terms in

δ^{2}

cancel in pairs. Hence

\overset{ˇ}{H}^{j} (U, F) = ker (δ : \overset{ˇ}{C}^{j} \to \overset{ˇ}{C}^{j + 1}) / im (δ : \overset{ˇ}{C}^{j - 1} \to \overset{ˇ}{C}^{j}) .

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