Detecting zero morphisms via an open subscheme and its complement. In the setting described in Bernstein, Beilinson, and Deligne, associated to a scheme $X$, a closed subscheme $i: Z \to X$ and its open complement $j: U \to X$ we have six functors between the corresponding derived categories of étale sheaves $i^*, i_*, i^!, j_!, j^*, j_*$ (they are all derived but I will ommit the L's and R's).
My question is: if $X$ is a variety over a field $k$ with structural morphism $f: X \to k$ and $F$ is an object of $D^b(k)$, then is the canonical morphism $f^*F \to i_*i^*f^*F \oplus j_*j^*f^*F$ ever a monomorphism in the triangulated category $D^b(X)$?
 A: Let me gather my comments into an answer. Take $X$ to be a smooth projective curve, $Z$ a closed point, $U$ its complement, $F=k$. We are asking if the map $\mathcal{O}_X\to \mathcal{O}_Z\oplus \mathcal{O}_U$ is a monomorphism in the derived category.
Take $E=\omega_X^{-1}$. Let $E[-1]\to \mathcal{O}_X$ be the map corresponding to the nontrivial element of $Ext^1(\omega_X^{-1}, \mathcal{O}_X) = H^1(X, \omega_X)=k$. Then since $H^1(U, E|_U)$ and $H^1(Z, E|_Z)$ are both zero (since $Z$ and $U$ are affine, the composition $E[-1]\to \mathcal{O}_X\to \mathcal{O}_Z\oplus\mathcal{O}_U$ is zero.
It may also happen that the map $\mathcal{O}\to \mathcal{O}_Z\oplus\mathcal{O}_U$ is not a monomorphism even in the category of sheaves, e.g. $X = \mathrm{Spec} k[x,y]/(xy, y^2)$ be the line with an embedded point, $Z$ be the reduced point 0 and $U$ the complement. Then the nilpotent global section $x$ of $\mathcal{O}_X$ maps to zero in both $\mathcal{O}_Z$ and $\mathcal{O}_U$.
A: What about $X=\mathbb P^1_k$, $Z$ a point, $U=\mathbb A^1_k$ the open complement, and $F=Q_\ell$ (constant sheaf) ?
Then the map 
$$ Hom_{D(X)}(f^*F[-2],f^*F) \longrightarrow  Hom_{D(X)}(f^*F[-2],i_*i^*f^*F) \oplus Hom_{D(X)}(f^*F[-2],j_*j^*f^*F)$$ gets identified with the restriction map
$$ H^2(\mathbb P^1_k, Q_\ell) \longrightarrow  H^2(pt, Q_\ell) \oplus H^2(\mathbb A^1_k, Q_\ell)$$
which is clearly non injective.
