Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 113 / 3 / a

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Solution

For an indexed open cover

U = (U_{i})_{i \in I}

and a sheaf

F

, the Čech cohomology cochain groups are

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

The differential is the alternating sum of restrictions:

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

Since

d^{2} = 0

, the cohomology

\overset{ˇ}{H}^{p} (U, F) = ker (d : \overset{ˇ}{C}^{p} \to \overset{ˇ}{C}^{p + 1}) / o p er a t or nam e im (d : \overset{ˇ}{C}^{p - 1} \to \overset{ˇ}{C}^{p})

is well defined.

Solved by gpt-5.6-sol high.

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