Let $(X, \mathcal O\_X)$ be a scheme and the following an infinite, exact sequence of injective sheaves of $\mathcal O\_X$-modules:
$$
\cdots \overset{f\_5}\longrightarrow I\_5\overset{f\_4}\longrightarrow I\_4 \overset{f\_3}\longrightarrow I\_3 \overset{f\_2}\longrightarrow I\_2\overset{f\_1}\longrightarrow I\_1 \overset{f\_0}\longrightarrow 0\tag{\*}
$$
Now, given an element $i\in \Gamma(X, I\_n)$ with $f\_{n-1}(i) = 0$ (or, rather, $f\_{n-1, X}$), is there any hope of finding a $j \in \Gamma(X, I\_{n+1})$ such that $f\_n(j) = i$? If assuming $X$ to be Noetherian is necessary, then that is fine.



Homology in the sections of an infinite exact sequence of injective sheaves of $\mathcal O_X$-modules?
Partial answer: if $X$ has underlying topological space of finite dimension $k$, then what you want is true, because $H^i(X, F)=0$ for any $O\_X$-module $F$ and $i>k$. Let $F=\ker(f\_{k+n})$, so that we have an injective resolution
$$ 0\to F\to I\_{n+k+1} \to I\_{n+k} \to \ldots \to I\_{n+1}\to I\_n \to I\_{n-1} \to\ldots $$
So if your $j$ didn't exist, $i$ would give a nonzero class in $H^{k+1}(X, F)=0$.


