For any open \(U \subseteq X\) show that the functor \[ \Gamma\qty{U, {-}}: \Sh(X) \to \mathsf{Ab}\mathsf{Grp} \] is left exact, but need not be exact.
Solution
Let \[ \xi: 0 \to \mcf_1 \xrightarrow{f} \mcf_2 \xrightarrow{g} \mcf_3 \to 0 \] be an exact sequence of sheaves of abelian groups on \(X\), where a morphism such as \(f\) is data of the form
Applying \(\Gamma(X; {-})\) gives \[ \xi_X: 0 \to \mcf_1(X) \xrightarrow{f_X} \mcf_2(X) \xrightarrow{g_X} \mcf_3(X) ,\] which is exact at \(\mcf_1(X)\) and at \(\mcf_2(X)\).
Exactness at \(\mcf_1(X)\).
Since \(f\) is a monomorphism, \(\ker f = \mathbf 0\) as a sheaf, so \[ (\ker f)(U) = (\mathbf 0)(U) = 0 .\]
Since \[ (\ker f)(U) &\da \ker f_U \da \ker( \mcf_1(U) \xrightarrow{f_U} \mcf_2(U)) \\ \implies \ker f_X &= (\ker f)(X) = (\mathbf 0)(X) = 0 .\]
The first equality holds because the kernel presheaf \(U\mapsto\ker f_U\) is a sheaf. The image presheaf \(U\mapsto\im g_U\) is in general not a sheaf; the image sheaf is its sheafification.
A second proof that \(f_X\) is injective uses stalks. A sequence of sheaves is exact if and only if it is exact on stalks, and restriction to stalks gives commutative squares for all \(p\in X\):
Suppose \(s\in \mcf_1(X)\) and \(f_X(s) = 0\) in \(\mcf_2(X)\). Then \[ (\mcf_2 \mid^X_p \circ f_X )(s) &= \mcf_2\mid^X_p( 0) = 0 \in (\mcf_2)_p \quad\text{group homomorphisms send $0$ to $0$}\\ \implies (f_p \circ \mcf_1 \mid^X_p)(s) &= (\mcf_2 \mid^X_p \circ f_X)(s) = 0 \quad\text{by commutativity} \\ \implies \mcf_1 \mid^X_p (s) &= 0 \quad\text{left-cancel $f_p$ since it is mono} ,\] which holds for all \(p\).
By the sheaf condition on \(\mcf_1\), \(s= 0 \in \mcf_1(X)\):
- Fix \(p\). For \(s\in \mcf_1(X)\), write a representative \(\mcf_1\mid^X_p(s) = [U, \tilde s\in \mcf_1(U)]\).
Recall \((U_1, s_1) \sim (U_2, s_2) \in \mcf_p \iff\) they are both equivalent to \((W, t)\) where \(W \subseteq U_1 \intersect U_2\) and \[ \mcf_1 \mid^{U_1}_W (s_1) = t = \mcf_1\mid^{U_2}_W (s_2) .\]
Then \(s_p \da \mcf_1 \mid^X_p(s) = 0 \sim (W, 0) \in (\mcf_1)_p\) means there is some \(W_p\) and a lift \(\tilde s(p) = 0 \in \mcf_1(W_p)\) with \(\mcf_1\mid^{W_p}_p(\tilde s(p)) = s_p\).
This holds for all \(p\), and \(\ts{W_p}_{p\in X} \covers X\), so by the gluing axiom for \(\mcf_1\) the sections \(\ts{ \tilde s(p) \in \mcf_1(W_p) \st p\in X}\) glue to a unique \(\tilde s\in \mcf_1(X)\); by uniqueness \(\tilde s = s = 0 \in \mcf_1(X)\).
Exactness at \(\mcf_2(X)\).
- \(\im f_X \subseteq \ker g_X\). Let \(s \in \im f_X \subseteq \mcf_2(X)\); restriction to the stalk at \(p\) gives the diagram
Choose \(r\in\mcf_1(X)\) with \(f_X(r)=s\). By commutativity, the image of \(s\) in \((\mcf_2)_p\) is \(f_p\) applied to the image of \(r\) in \((\mcf_1)_p\), so \[ \mcf_2\mid^X_p(s) \in \im f_p = \ker g_p .\]
Put \(\ell\da g_X(s)\). By commutativity, \(\mcf_3 \mid^X_p(\ell) = g_p(\mcf_2\mid^X_p(s)) = 0\). Since this is true at all stalks, \(\ell = 0\in \mcf_3(X)\), so \(s \in \ker g_X\).
\(\ker g_X \subseteq \im f_X\). Let \(s\in\mcf_2(X)\) with \(g_X(s)=0\). For each \(p\in X\), the germ \(s_p\) lies in \(\ker g_p = \im f_p\), so there are an open neighborhood \(W_p\) of \(p\) and \(t_p\in\mcf_1(W_p)\) with \(f_{W_p}(t_p) = s|_{W_p}\) (shrinking \(W_p\) so that the germ equality holds on \(W_p\)). On \(W_p\cap W_q\), \(f(t_p|_{W_p\cap W_q} - t_q|_{W_p\cap W_q}) = 0\), and \(f_{W_p\cap W_q}\) is injective by exactness at \(\mcf_1\) applied to the open set \(W_p\cap W_q\). Hence the \(t_p\) agree on overlaps and glue to \(t\in\mcf_1(X)\). Then \(f_X(t)-s\) restricts to \(0\) on every \(W_p\), so \(f_X(t)=s\) by the sheaf axiom for \(\mcf_2\).
The arguments above use only the sheaf axioms on the open sets involved, so they apply verbatim with \(X\) replaced by an arbitrary open \(U\subseteq X\); hence \(\Gamma(U,-)\) is left exact.
Failure of right exactness.
The exponential \(z\mapsto e^{2\pi i z}\) gives the exact sequence of groups \[ 0 \to \ZZ \to \GG_a(\CC) \xrightarrow{\exp: z\mapsto e^{2\pi i z}} \GG_m(\unitsof{\CC}) \to 0 \in \Grp ,\] and, on \(X\da \unitsof{\CC}\), the exact sequence of sheaves \[ 0 \to \underline{\ZZ} \to \Hol_X({-}) \xrightarrow{\exp} \unitsof{\Hol_X({-})} \to 0 \in \Sh(X, \Grp) .\] Here \(\unitsof{\Hol_X({-})}\) denotes the multiplicatively invertible functions, i.e. the nonvanishing functions.
Applying \(\Gamma(X; {-})\) gives \[ 0 \to \ZZ \to \Hol_X(X) \xrightarrow{\exp} \unitsof{\Hol_X(X)} \to 0 .\] If \(\exp\colon\Hol_X(X)\to\unitsof{\Hol_X(X)}\) were surjective, every nonvanishing holomorphic function on \(\unitsof{\CC}\) would have a logarithm on all of \(\unitsof{\CC}\). The identity function \(z\mapsto z\) has none, since \(\oint_{\abs{z}=1} dz/z = 2\pi i \neq 0\).