Let \(\mcf\) and \(\mcg\) be sheaves on \(X\). Show that the presheaf \(U \mapsto \mcf(U) \oplus \mcg(U)\) is a sheaf. It is called the direct sum of \(\mcf\) and \(\mcg\) and is denoted \(\mcf \oplus \mcg\).
Show that it plays the role of both the direct sum and the direct product in the category of sheaves of abelian groups on \(X\).
Solution
Define a presheaf \(\mcf\oplus\mcg\) by \[ (\mcf\oplus\mcg)(U)=\mcf(U)\oplus\mcg(U), \] with restriction maps taken componentwise.
1. The presheaf \(\mcf\oplus\mcg\) is a sheaf.
Let \(U=\bigcup_iU_i\) be an open cover. Suppose \[ (s_i,t_i)\in\mcf(U_i)\oplus\mcg(U_i) \] agree on every overlap \(U_i\cap U_j\). Then the \(s_i\) agree pairwise in \(\mcf\), and the \(t_i\) agree pairwise in \(\mcg\). Since \(\mcf\) and \(\mcg\) are sheaves, there are unique sections \[ s\in\mcf(U),\qquad t\in\mcg(U) \] restricting to all the \(s_i\) and \(t_i\). Thus \((s,t)\) is the unique section of \(\mcf\oplus\mcg\) restricting to every \((s_i,t_i)\). This proves both existence and uniqueness in the sheaf gluing axiom.
2. With the projection morphisms\[ p_\mcf:\mcf\oplus\mcg\to\mcf, \qquad p_\mcg:\mcf\oplus\mcg\to\mcg, \]the sheaf \(\mcf\oplus\mcg\) is a categorical product of \(\mcf\) and \(\mcg\).
Let \(\mch\) be a sheaf with morphisms \[ f:\mch\to\mcf, \qquad g:\mch\to\mcg. \] For every open \(U\), define \[ h(U):\mch(U)\to\mcf(U)\oplus\mcg(U), \qquad u\mapsto(f(U)(u),g(U)(u)). \] These maps commute with restrictions because \(f\) and \(g\) do, so they define a sheaf morphism \[ h:\mch\to\mcf\oplus\mcg. \] By construction, \[ p_\mcf h=f, \qquad p_\mcg h=g. \] Any morphism with these two properties must agree with \(h\) on every section, since both components are prescribed. Hence \(h\) is unique, proving the product universal property.
3. With the inclusion morphisms\[ i_\mcf:\mcf\to\mcf\oplus\mcg, \qquad i_\mcg:\mcg\to\mcf\oplus\mcg, \]the sheaf \(\mcf\oplus\mcg\) is a categorical coproduct of \(\mcf\) and \(\mcg\).
Let \(\mch\) be a sheaf with morphisms \[ f:\mcf\to\mch, \qquad g:\mcg\to\mch. \] For each open \(U\), define \[ h(U):\mcf(U)\oplus\mcg(U)\to\mch(U), \qquad (s,t)\mapsto f(U)(s)+g(U)(t). \] Again these maps commute with restrictions, so they define a sheaf morphism \[ h:\mcf\oplus\mcg\to\mch. \] By construction, \[ hi_\mcf=f, \qquad hi_\mcg=g. \] Every section \((s,t)\) decomposes as \[ (s,t)=(s,0)+(0,t), \] so any morphism satisfying those two identities must equal \(h\). Thus \(h\) is unique, proving the coproduct universal property.
4. Hence \(\mcf\oplus\mcg\) is both product and coproduct in the category of sheaves of abelian groups.
Step 2 proves the product universal property, and step 3 proves the coproduct universal property for the same sheaf. This common object is therefore the biproduct, or direct sum, of \(\mcf\) and \(\mcg\).
5. Q.E.D.
Step 1 proves that the sectionwise direct sum is a sheaf, and steps <1>2–<1>4 prove both required universal properties.