Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 113 / 4 / a

Table of contents
- Solution a

Solution

Choose an ordering of the index set

I

. The degree-

q

Čech cochain group is

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

For

s = (s_{i_{0} \dots i_{q}})

, the Čech coboundary is the alternating sum of restrictions

(δs)_{i_{0} \dots i_{q + 1}} = \sum_{j = 0}^{q + 1} (- 1)^{j} s_{i_{0} \dots i_{j} \dots i_{q + 1}}_{U_{i_{0}} \cap \dots \cap U_{i_{q + 1}}} .

The identity

δ^{2} = 0

makes this the Čech cochain complex, and the required Čech cohomology is

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

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version