Cohomological range of a perverse sheaf

Question

I want to prove that, let $X$ be a smooth variety, then for a local system $L$ on $X$, $L[\dim X]$ has no quotient object in the abelian category of perverse sheaves supported on a proper closed subset $Z$, in order to prove that $L[n]$ is the intermediate extension of $L\rvert_{X\setminus Z}[\dim X]$.

Then my idea is that let $L[\dim X] \rightarrow P$ be an epimorphism that $P$ is supported on $Z$. Then I want to claim that $P \in \mathfrak{D}^{\ge -{\dim Z}}$ in the natural t-structure. But I cannot prove it nor find a counter example.

Now I have another way to prove that $L[\dim X]$ is an intermediate extension. But I am still curious about the cohomology range of a perverse sheaf.

Answer 1

The way I know how to do is to consider the truncation $P^{\leq -\dim Z-1}$ of P in degree $\leq -\dim Z -1$ in the usual $t$-structure. If $P \notin {\mathfrak D}^{\geq -\dim Z}$ then the natural map from $P^{\leq -\dim Z-1}$ to $P$ is nontrivial. But $P^{\leq -\dim Z-1}$ is concentrated in degree $\leq -\dim Z-1$ and supported in dimension $\leq D$ and thus lies in $\mathfrak D^{\le -1}$ for the perverse $t$-structure. But there are no nontrivial maps from $\mathfrak D^{\le -1}$ to $\mathfrak D^{\geq 0}$ by the definition of $t$-structure, so the map is trivial, and thus $P \in {\mathfrak D}^{\geq -\dim Z}$