Let \(\ts{\mcf_i}\) be an inverse system of sheaves on \(X\). Show that the presheaf \(U \mapsto \varprojlim \mcf_i(U)\) is a sheaf. It is called the inverse limit of the system and is denoted \(\varprojlim \mcf_i\). Show that it has the universal property of an inverse limit in the category of sheaves.
Solution
Let \[ \mcp(U)=\varprojlim_i\mcf_i(U). \] Thus an element of \(\mcp(U)\) is a compatible tuple \[ (s_i)_i, \qquad s_i\in\mcf_i(U), \] such that every transition map of the inverse system sends the appropriate component to the next one. Restriction maps are defined componentwise.
1. The presheaf \(\mcp\) satisfies local uniqueness.
Let \[ U=\bigcup_{\alpha}U_\alpha \] be an open cover, and suppose \[ s=(s_i)_i, \qquad t=(t_i)_i \] in \(\mcp(U)\) have equal restrictions to every \(U_\alpha\).
For every index \(i\), this says \[ s_i|_{U_\alpha}=t_i|_{U_\alpha} \qquad \text{for every }\alpha. \] Since \(\mcf_i\) is a sheaf, \[ s_i=t_i. \] This holds for every \(i\), hence the tuples \(s\) and \(t\) are equal in \(\mcp(U)\).
2. The presheaf \(\mcp\) satisfies gluing.
Let \[ U=\bigcup_\alpha U_\alpha \] and suppose compatible local sections \[ s_\alpha=(s_{\alpha,i})_i \in\mcp(U_\alpha) \] are given.
Fix an index \(i\). Compatibility of the \(s_\alpha\) on overlaps implies that the component sections \[ s_{\alpha,i}\in\mcf_i(U_\alpha) \] agree on every \(U_\alpha\cap U_\beta\). Since \(\mcf_i\) is a sheaf, they glue uniquely to \[ s_i\in\mcf_i(U). \]
It remains to check that the tuple \((s_i)_i\) is compatible with the transition maps. Let \[ \theta_{ji}:\mcf_j\longrightarrow\mcf_i \] be a transition morphism. On every \(U_\alpha\), \[ \theta_{ji}(s_j)|_{U_\alpha} =\theta_{ji}(s_{\alpha,j}) =s_{\alpha,i} =s_i|_{U_\alpha}, \] because each local tuple \(s_\alpha\) already lies in the inverse limit. The sheaf uniqueness axiom for \(\mcf_i\) therefore gives \[ \theta_{ji}(s_j)=s_i. \] Hence \[ s=(s_i)_i\in\mcp(U). \] By construction, \(s\) restricts to every \(s_\alpha\).
3. Therefore\[ \boxed{ U\longmapsto\varprojlim_i\mcf_i(U) } \]is already a sheaf.
Step 1 proves uniqueness and 2 proves existence of glued sections for every open cover. These are precisely the sheaf axioms.
4. For every \(i\), component projection defines a canonical sheaf morphism\[ \pi_i:\mcp\longrightarrow\mcf_i, \]and the \(\pi_i\) are compatible with the inverse system.
On each open set \(U\), define \[ \pi_i(U)((s_j)_j)=s_i. \] This commutes with restrictions because restrictions in \(\mcp\) are componentwise. Compatibility with transition maps is exactly the defining compatibility condition on inverse-limit tuples.
5. Let \(\mcg\) be a sheaf with a compatible family of morphisms\[ \phi_i:\mcg\longrightarrow\mcf_i. \]There is a sheaf morphism\[ \Phi:\mcg\longrightarrow\mcp \]defined by\[ \boxed{ \Phi(U)(s)=(\phi_i(U)(s))_i. } \]
For \(s\in\mcg(U)\), compatibility of the morphisms \(\phi_i\) with the inverse system says exactly that the tuple \[ (\phi_i(U)(s))_i \] belongs to \[ \varprojlim_i\mcf_i(U)=\mcp(U). \]
Each \(\phi_i\) commutes with restriction, so the tuple-valued maps \(\Phi(U)\) do as well. Thus they define a morphism of sheaves \[ \Phi:\mcg\to\mcp. \] By construction, \[ \pi_i\circ\Phi=\phi_i \] for every \(i\).
6. The morphism \(\Phi\) in 5 is unique with this property.
Suppose \[ \Psi:\mcg\longrightarrow\mcp \] also satisfies \[ \pi_i\circ\Psi=\phi_i \] for every \(i\).
For an open \(U\) and a section \(s\in\mcg(U)\), the \(i\)th component of \(\Psi(U)(s)\) is \[ \pi_i(U)(\Psi(U)(s)) =\phi_i(U)(s). \] Thus every component of \(\Psi(U)(s)\) agrees with the corresponding component of \(\Phi(U)(s)\), so \[ \Psi(U)(s)=\Phi(U)(s). \] Hence \(\Psi=\Phi\).
7. Consequently, the sheaf \(\mcp\) with its projections \(\pi_i\) is the inverse limit in the category of sheaves:\[ \boxed{ \varprojlim_i\mcf_i(U) = \left(\varprojlim_i\mcf_i\right)(U) \quad\text{for every open }U. } \]
Steps 5 and 6 are exactly the existence and uniqueness clauses in the universal property of a categorical inverse limit.
8. Q.E.D.
Step 3 proves the sectionwise construction is a sheaf, and 7 proves it is the inverse limit in sheaves.