For any open subset \(U \subseteq X\), show that the functor \(\Gamma(U, \wait)\) from sheaves on \(X\) to abelian groups is left exact. That is, if \[ 0 \to \mcf' \to \mcf \to \mcf'' \] is an exact sequence of sheaves, then \[ 0 \to \Gamma(U, \mcf') \to \Gamma(U, \mcf) \to \Gamma(U, \mcf'') \] is an exact sequence of groups.
The functor \(\Gamma(U, \wait)\) is not exact in general; Hartshorne II.1.21 supplies a counterexample on \(\PP^1\).
Solution
Let \[ 0\longrightarrow\mcf'\xrightarrow{i}\mcf\xrightarrow{p}\mcf'' \] be an exact sequence of sheaves on \(X\), and fix an open subset \(U\subseteq X\).
1. The map \[ \Gamma(U,\mcf')\xrightarrow{\Gamma(U,i)}\Gamma(U,\mcf) \] is injective.
Exactness at \(\mcf'\) says that \(i\) is an injective morphism of sheaves. By definition of injectivity for sheaf morphisms, for every open set \(V\subseteq X\) the map \[ i(V):\mcf'(V)\longrightarrow\mcf(V) \] is injective. Taking \(V=U\) gives the claim.
2. After identifying \(\mcf'\) with the kernel subsheaf \(\ker p\subseteq\mcf\), one has \[ \Gamma(U,\mcf') = \ker\bigl(\Gamma(U,\mcf)\to\Gamma(U,\mcf'')\bigr). \]
Exactness at \(\mcf\) gives \[ \operatorname{im}i=\ker p. \] By Exercise II.1.6, the injection \(i\) identifies \(\mcf'\) with the subsheaf \(\operatorname{im}i\), hence with \(\ker p\).
The kernel sheaf is computed sectionwise: \[ (\ker p)(U) = \ker\bigl(p(U):\mcf(U)\to\mcf''(U)\bigr). \] Therefore \[ \Gamma(U,\mcf') =\Gamma(U,\ker p) =\ker\bigl(\Gamma(U,\mcf)\to\Gamma(U,\mcf'')\bigr). \]
3. The sequence \[ \boxed{ 0\longrightarrow\Gamma(U,\mcf') \longrightarrow\Gamma(U,\mcf) \longrightarrow\Gamma(U,\mcf'')} \] is exact.
4. QED Step 3 is the required left-exact sequence.