trivial cokernel, but globally, we will have nontrivial cokernel.
§ Exponential sheaf sequence
0 > 2πiZ [α:incl]> O [β:exp(.)]> O* > 0

O
is the sheaf of the additive group of holomorphic functions. O*
is the sheaf of the group of nonzero holomorphic functions. 
α
, which embeds 2πin ∈2πiZ
as a constant function f(_) = 2πin
isinjective. 
e^(2πiZ) = 1
. So we have that the composition of the two maps β.α
isthe zero map, mapping everything in 2πiZ
to the identity of O*
.Thus, d^2 = 0
, ensuring that this is an exact sequence.  Let us consider the local situation. At each point
p
, we want to showthat β
is surjective. Pick any g ∈ O*p
. We have an open neighbourhood Ug
where g ≠ 0
. take the logarithm of g
to pull back g ∈ O*
to log g ∈ O
.Thus, β
is surjective at each local point p
.  On the other hand, the function
h(z) = z
cannot be in O*
. If it were,then there exists a homolorphic function called l ∈ O
[for log
] such thatexp(l(z)) = h(z) = z
everywhere on the complex plane.  Assume such a function exists. Then it must be the case that
d/dz exp(l(z)) = d/dz(z) = 1
. Thus, exp(l(z)) l'(z) = z l'(z) = 1
[use the fact that exp(l(z)) = z
]. This means that l'(z) = 1/z
.  Now, by integrating in a closed loop of
e^iθ
we have ∮l'(z) = l(1)  l(1) = 0
.  We also have that
∮l'(z) = ∮1/z = 2πi
.  This implies that
0 = 2πi
which is absurd.  Hence, we cannot have a function whose exponential gives
h(z) = z
everywhere.  Thus, the cokernel is nontrivial globally.