Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 4 / a

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Solution

For an open cover

U = (U_{i})

, the Čech cochain complex is

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

with differential

(δc)_{i_{0} \dots i_{p + 1}} = \sum_{q = 0}^{p + 1} (- 1)^{q} c_{i_{0} \dots i_{q} \dots i_{p + 1}}_{U_{i_{0}} \cap \dots \cap U_{i_{p + 1}}} .

Its cohomology is

\overset{ˇ}{H}^{p} (U, F)

.

Solved by gpt-5.6-sol high.

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