U = (U_{i})

and a sheaf

F

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

with the alternating sum of restriction maps as differential. Its cohomology is the Čech cohomology

\overset{ˇ}{H}^{p} (U, F)

Cover

P_{k}^{1}

U_{0} = D_{+} (x_{0})

and

U_{1} = D_{+} (x_{1})

, and write

t = x_{1} / x_{0}

O (- 1)

. In compatible trivializations, its degree-one Čech quotient is

H^{1} (P_{k}^{1}, O (- 1)) ≅ \frac{k [ t , t ^{- 1} ]}{k [ t ] + t ^{- 1} k [ t ^{- 1} ]} .

The first summand contains every nonnegative power of

t

and the second every negative power, so

H^{1} (P_{k}^{1}, O (- 1)) = 0.

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