Global section functor
From Infogalactic: the planetary knowledge core
Lua error in package.lua at line 80: module 'strict' not found. Let X be a topological space, and denote the category of sheaves with values in
. Then the map that associates to a sheaf
its global sections
is a covariant functor to
.
If is the category of abelian groups, then this functor is left exact. This important remark leads to the notion of sheaf cohomology, via derived functors.
Examples
- Let
be the locally constant sheaf. Then its global sections are given by
, i.e. the direct sum indexed by connected components of X
- Let
be the sheaf of holomorphic functions on the compact connected complex manifold X, then by the maximum principle, global sections are constant, ie.
- Let
denote the twisting sheaves on the projective space
, then
for
, and 0 for
.