trivial cokernel, but globally, we will have non-trivial 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 non-zero 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
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
g ≠ 0. take the logarithm of
g to pull back
g ∈ O* to
log g ∈ O.Thus,
β is surjective at each local point
- 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 that
exp(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.