Global section functor

Let X be a topological space, and $\mathrm {Sh}(X, \mathcal C)$ denote the category of sheaves with values in $\mathcal C$ . Then the map that associates to a sheaf $\mathcal F$ its global sections $\Gamma(X,\mathcal F)$ is a covariant functor to $\mathcal C$ .

If $\mathcal C$ 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 ${\underline{\mathbb Z}}_X$ be the locally constant sheaf. Then its global sections are given by $\Gamma (X, {\underline{\mathbb Z}}_X) = \mathbb Z^{\pi_0(X)}$ , i.e. the direct sum indexed by connected components of X

Let $\mathcal O_X$ be the sheaf of holomorphic functions on the compact connected complex manifold X, then by the maximum principle, global sections are constant, ie. $\Gamma (X, \mathcal O_X) = \mathbb C$

Let $\mathcal O (i), i\in \mathbb Z$ denote the twisting sheaves on the projective space $\mathbb P_k^n$ , then $\Gamma (X, \mathcal O(d)) = k_d[X_0, \ldots, X_n]$ for $d \ge 0$ , and 0 for $d <0$ .

See also