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 $D^b(X)$?



Detecting zero morphisms via an open subscheme and its complement.
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 $\mathcal{O}\_U$.


